diff options
Diffstat (limited to 'src/share/algebra/browse.daase')
| -rw-r--r-- | src/share/algebra/browse.daase | 596 |
1 files changed, 298 insertions, 298 deletions
diff --git a/src/share/algebra/browse.daase b/src/share/algebra/browse.daase index 969bd427..826b5d14 100644 --- a/src/share/algebra/browse.daase +++ b/src/share/algebra/browse.daase @@ -1,5 +1,5 @@ -(2280951 . 3485824333) +(2279591 . 3485856132) (-18 A S) ((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) NIL @@ -56,7 +56,7 @@ NIL ((|constructor| (NIL "This domain represents the syntax for an add-expression.")) (|body| (((|SpadAst|) $) "base(\\spad{d}) returns the actual body of the add-domain expression \\spad{`d'}.")) (|base| (((|SpadAst|) $) "\\spad{base(d)} returns the base domain(\\spad{s}) of the add-domain expression."))) NIL NIL -(-32 R -3029) +(-32 R -3027) ((|constructor| (NIL "This package provides algebraic functions over an integral domain.")) (|iroot| ((|#2| |#1| (|Integer|)) "\\spad{iroot(p, n)} should be a non-exported function.")) (|definingPolynomial| ((|#2| |#2|) "\\spad{definingPolynomial(f)} returns the defining polynomial of \\spad{f} as an element of \\spad{F}. Error: if \\spad{f} is not a kernel.")) (|minPoly| (((|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{minPoly(k)} returns the defining polynomial of \\spad{k}.")) (** ((|#2| |#2| (|Fraction| (|Integer|))) "\\spad{x ** q} is \\spad{x} raised to the rational power \\spad{q}.")) (|droot| (((|OutputForm|) (|List| |#2|)) "\\spad{droot(l)} should be a non-exported function.")) (|inrootof| ((|#2| (|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{inrootof(p, x)} should be a non-exported function.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}. Error: if \\spad{op} is not an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|rootOf| ((|#2| (|SparseUnivariatePolynomial| |#2|) (|Symbol|)) "\\spad{rootOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}."))) NIL ((|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) @@ -88,11 +88,11 @@ NIL ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in \\spadtype{AlgebraicNumber}.")) (|doublyTransitive?| (((|Boolean|) |#1|) "\\spad{doublyTransitive?(p)} is \\spad{true} if \\spad{p} is irreducible over over the field \\spad{K} generated by its coefficients,{} and if \\spad{p(X) / (X - a)} is irreducible over \\spad{K(a)} where \\spad{p(a) = 0}.")) (|split| (((|Factored| |#1|) |#1|) "\\spad{split(p)} returns a prime factorisation of \\spad{p} over its splitting field.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p} over the field generated by its coefficients.") (((|Factored| |#1|) |#1| (|List| (|AlgebraicNumber|))) "\\spad{factor(p, [a1,...,an])} returns a prime factorisation of \\spad{p} over the field generated by its coefficients and a1,{}...,{}an."))) NIL NIL -(-40 -3029 UP UPUP -3486) +(-40 -3027 UP UPUP -1444) ((|constructor| (NIL "Function field defined by \\spad{f}(\\spad{x},{} \\spad{y}) = 0.")) (|knownInfBasis| (((|Void|) (|NonNegativeInteger|)) "\\spad{knownInfBasis(n)} \\undocumented{}"))) ((-4453 |has| (-418 |#2|) (-373)) (-4458 |has| (-418 |#2|) (-373)) (-4452 |has| (-418 |#2|) (-373)) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-418 |#2|) (QUOTE (-146))) (|HasCategory| (-418 |#2|) (QUOTE (-148))) (|HasCategory| (-418 |#2|) (QUOTE (-359))) (-3765 (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-378))) (-3765 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3765 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3765 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-359))))) (-3765 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -650) (QUOTE (-575)))) (-3765 (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-378))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) -(-41 R -3029) +((|HasCategory| (-418 |#2|) (QUOTE (-146))) (|HasCategory| (-418 |#2|) (QUOTE (-148))) (|HasCategory| (-418 |#2|) (QUOTE (-359))) (-3763 (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-378))) (-3763 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3763 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3763 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-359))))) (-3763 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -650) (QUOTE (-575)))) (-3763 (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-378))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) +(-41 R -3027) ((|constructor| (NIL "AlgebraicManipulations provides functions to simplify and expand expressions involving algebraic operators.")) (|rootKerSimp| ((|#2| (|BasicOperator|) |#2| (|NonNegativeInteger|)) "\\spad{rootKerSimp(op,f,n)} should be local but conditional.")) (|rootSimp| ((|#2| |#2|) "\\spad{rootSimp(f)} transforms every radical of the form \\spad{(a * b**(q*n+r))**(1/n)} appearing in \\spad{f} into \\spad{b**q * (a * b**r)**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{b}.")) (|rootProduct| ((|#2| |#2|) "\\spad{rootProduct(f)} combines every product of the form \\spad{(a**(1/n))**m * (a**(1/s))**t} into a single power of a root of \\spad{a},{} and transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form.")) (|rootPower| ((|#2| |#2|) "\\spad{rootPower(f)} transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form if \\spad{m} and \\spad{n} have a common factor.")) (|ratPoly| (((|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{ratPoly(f)} returns a polynomial \\spad{p} such that \\spad{p} has no algebraic coefficients,{} and \\spad{p(f) = 0}.")) (|ratDenom| ((|#2| |#2| (|List| (|Kernel| |#2|))) "\\spad{ratDenom(f, [a1,...,an])} removes the \\spad{ai}\\spad{'s} which are algebraic from the denominators in \\spad{f}.") ((|#2| |#2| (|List| |#2|)) "\\spad{ratDenom(f, [a1,...,an])} removes the \\spad{ai}\\spad{'s} which are algebraic kernels from the denominators in \\spad{f}.") ((|#2| |#2| |#2|) "\\spad{ratDenom(f, a)} removes \\spad{a} from the denominators in \\spad{f} if \\spad{a} is an algebraic kernel.") ((|#2| |#2|) "\\spad{ratDenom(f)} rationalizes the denominators appearing in \\spad{f} by moving all the algebraic quantities into the numerators.")) (|rootSplit| ((|#2| |#2|) "\\spad{rootSplit(f)} transforms every radical of the form \\spad{(a/b)**(1/n)} appearing in \\spad{f} into \\spad{a**(1/n) / b**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{a} and \\spad{b}.")) (|coerce| (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(x)} \\undocumented")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(x)} \\undocumented")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(x)} \\undocumented"))) NIL ((-12 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -441) (|devaluate| |#1|))))) @@ -111,7 +111,7 @@ NIL (-45 |Key| |Entry|) ((|constructor| (NIL "\\spadtype{AssociationList} implements association lists. These may be viewed as lists of pairs where the first part is a key and the second is the stored value. For example,{} the key might be a string with a persons employee identification number and the value might be a record with personnel data."))) ((-4460 . T) (-4461 . T)) -((-3765 (-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|))))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|))))))) +((-3763 (-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|))))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|))))))) (-46 S R E) ((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#2|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#2| $ |#3|) "\\spad{coefficient(p,e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#2| |#3|) "\\spad{monomial(r,e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#3| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) NIL @@ -144,7 +144,7 @@ NIL ((|constructor| (NIL "\\spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate skew polynomials to be applied to appropriate modules.")) (|apply| ((|#2| |#3| (|Mapping| |#2| |#2|) |#2|) "\\spad{apply(p, f, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = f(m)}. \\spad{f} must be an \\spad{R}-pseudo linear map on \\spad{M}."))) NIL NIL -(-54 |Base| R -3029) +(-54 |Base| R -3027) ((|constructor| (NIL "This package apply rewrite rules to expressions,{} calling the pattern matcher.")) (|localUnquote| ((|#3| |#3| (|List| (|Symbol|))) "\\spad{localUnquote(f,ls)} is a local function.")) (|applyRules| ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3| (|PositiveInteger|)) "\\spad{applyRules([r1,...,rn], expr, n)} applies the rules \\spad{r1},{}...,{}\\spad{rn} to \\spad{f} a most \\spad{n} times.") ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3|) "\\spad{applyRules([r1,...,rn], expr)} applies the rules \\spad{r1},{}...,{}\\spad{rn} to \\spad{f} an unlimited number of times,{} \\spadignore{i.e.} until none of \\spad{r1},{}...,{}\\spad{rn} is applicable to the expression."))) NIL NIL @@ -167,11 +167,11 @@ NIL (-59 S) ((|constructor| (NIL "This is the domain of 1-based one dimensional arrays")) (|oneDimensionalArray| (($ (|NonNegativeInteger|) |#1|) "\\spad{oneDimensionalArray(n,s)} creates an array from \\spad{n} copies of element \\spad{s}") (($ (|List| |#1|)) "\\spad{oneDimensionalArray(l)} creates an array from a list of elements \\spad{l}"))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-60 R) ((|constructor| (NIL "\\indented{1}{A TwoDimensionalArray is a two dimensional array with} 1-based indexing for both rows and columns.")) (|shallowlyMutable| ((|attribute|) "One may destructively alter TwoDimensionalArray\\spad{'s}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-61 -1777) ((|constructor| (NIL "\\spadtype{ASP10} produces Fortran for Type 10 ASPs,{} needed for NAG routine \\axiomOpFrom{d02kef}{d02Package}. This ASP computes the values of a set of functions,{} for example:\\begin{verbatim} SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT) DOUBLE PRECISION ELAM,P,Q,X,DQDL INTEGER JINT P=1.0D0 Q=((-1.0D0*X**3)+ELAM*X*X-2.0D0)/(X*X) DQDL=1.0D0 RETURN END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE JINT) (QUOTE X) (QUOTE ELAM)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) NIL @@ -295,7 +295,7 @@ NIL (-91 S) ((|constructor| (NIL "A stack represented as a flexible array.")) (|arrayStack| (($ (|List| |#1|)) "\\spad{arrayStack([x,y,...,z])} creates an array stack with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-92 S) ((|constructor| (NIL "This is the category of Spad abstract syntax trees."))) NIL @@ -343,7 +343,7 @@ NIL (-103 S) ((|constructor| (NIL "\\spadtype{BalancedBinaryTree(S)} is the domain of balanced binary trees (bbtree). A balanced binary tree of \\spad{2**k} leaves,{} for some \\spad{k > 0},{} is symmetric,{} that is,{} the left and right subtree of each interior node have identical shape. In general,{} the left and right subtree of a given node can differ by at most leaf node.")) (|mapDown!| (($ $ |#1| (|Mapping| (|List| |#1|) |#1| |#1| |#1|)) "\\spad{mapDown!(t,p,f)} returns \\spad{t} after traversing \\spad{t} in \"preorder\" (node then left then right) fashion replacing the successive interior nodes as follows. Let \\spad{l} and \\spad{r} denote the left and right subtrees of \\spad{t}. The root value \\spad{x} of \\spad{t} is replaced by \\spad{p}. Then \\spad{f}(value \\spad{l},{} value \\spad{r},{} \\spad{p}),{} where \\spad{l} and \\spad{r} denote the left and right subtrees of \\spad{t},{} is evaluated producing two values \\spad{pl} and \\spad{pr}. Then \\spad{mapDown!(l,pl,f)} and \\spad{mapDown!(l,pr,f)} are evaluated.") (($ $ |#1| (|Mapping| |#1| |#1| |#1|)) "\\spad{mapDown!(t,p,f)} returns \\spad{t} after traversing \\spad{t} in \"preorder\" (node then left then right) fashion replacing the successive interior nodes as follows. The root value \\spad{x} is replaced by \\spad{q} \\spad{:=} \\spad{f}(\\spad{p},{}\\spad{x}). The mapDown!(\\spad{l},{}\\spad{q},{}\\spad{f}) and mapDown!(\\spad{r},{}\\spad{q},{}\\spad{f}) are evaluated for the left and right subtrees \\spad{l} and \\spad{r} of \\spad{t}.")) (|mapUp!| (($ $ $ (|Mapping| |#1| |#1| |#1| |#1| |#1|)) "\\spad{mapUp!(t,t1,f)} traverses \\spad{t} in an \"endorder\" (left then right then node) fashion returning \\spad{t} with the value at each successive interior node of \\spad{t} replaced by \\spad{f}(\\spad{l},{}\\spad{r},{}\\spad{l1},{}\\spad{r1}) where \\spad{l} and \\spad{r} are the values at the immediate left and right nodes. Values \\spad{l1} and \\spad{r1} are values at the corresponding nodes of a balanced binary tree \\spad{t1},{} of identical shape at \\spad{t}.") ((|#1| $ (|Mapping| |#1| |#1| |#1|)) "\\spad{mapUp!(t,f)} traverses balanced binary tree \\spad{t} in an \"endorder\" (left then right then node) fashion returning \\spad{t} with the value at each successive interior node of \\spad{t} replaced by \\spad{f}(\\spad{l},{}\\spad{r}) where \\spad{l} and \\spad{r} are the values at the immediate left and right nodes.")) (|setleaves!| (($ $ (|List| |#1|)) "\\spad{setleaves!(t, ls)} sets the leaves of \\spad{t} in left-to-right order to the elements of \\spad{ls}.")) (|balancedBinaryTree| (($ (|NonNegativeInteger|) |#1|) "\\spad{balancedBinaryTree(n, s)} creates a balanced binary tree with \\spad{n} nodes each with value \\spad{s}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-104 R UP M |Row| |Col|) ((|constructor| (NIL "\\spadtype{BezoutMatrix} contains functions for computing resultants and discriminants using Bezout matrices.")) (|bezoutDiscriminant| ((|#1| |#2|) "\\spad{bezoutDiscriminant(p)} computes the discriminant of a polynomial \\spad{p} by computing the determinant of a Bezout matrix.")) (|bezoutResultant| ((|#1| |#2| |#2|) "\\spad{bezoutResultant(p,q)} computes the resultant of the two polynomials \\spad{p} and \\spad{q} by computing the determinant of a Bezout matrix.")) (|bezoutMatrix| ((|#3| |#2| |#2|) "\\spad{bezoutMatrix(p,q)} returns the Bezout matrix for the two polynomials \\spad{p} and \\spad{q}.")) (|sylvesterMatrix| ((|#3| |#2| |#2|) "\\spad{sylvesterMatrix(p,q)} returns the Sylvester matrix for the two polynomials \\spad{p} and \\spad{q}."))) NIL @@ -363,7 +363,7 @@ NIL (-108) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating binary expansions.")) (|binary| (($ (|Fraction| (|Integer|))) "\\spad{binary(r)} converts a rational number to a binary expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(b)} returns the fractional part of a binary expansion."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3765 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) +((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3763 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) (-109) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Binding' is a name asosciated with a collection of properties.")) (|binding| (($ (|Identifier|) (|List| (|Property|))) "\\spad{binding(n,props)} constructs a binding with name \\spad{`n'} and property list `props'.")) (|properties| (((|List| (|Property|)) $) "\\spad{properties(b)} returns the properties associated with binding \\spad{b}.")) (|name| (((|Identifier|) $) "\\spad{name(b)} returns the name of binding \\spad{b}"))) NIL @@ -392,7 +392,7 @@ NIL ((|constructor| (NIL "A basic operator is an object that can be applied to a list of arguments from a set,{} the result being a kernel over that set.")) (|setProperties| (($ $ (|AssociationList| (|String|) (|None|))) "\\spad{setProperties(op, l)} sets the property list of \\spad{op} to \\spad{l}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|setProperty| (($ $ (|Identifier|) (|None|)) "\\spad{setProperty(op, p, v)} attaches property \\spad{p} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.") (($ $ (|String|) (|None|)) "\\spad{setProperty(op, s, v)} attaches property \\spad{s} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|property| (((|Maybe| (|None|)) $ (|Identifier|)) "\\spad{property(op, p)} returns the value of property \\spad{p} if it is attached to \\spad{op},{} otherwise \\spad{nothing}.") (((|Union| (|None|) "failed") $ (|String|)) "\\spad{property(op, s)} returns the value of property \\spad{s} if it is attached to \\spad{op},{} and \"failed\" otherwise.")) (|deleteProperty!| (($ $ (|Identifier|)) "\\spad{deleteProperty!(op, p)} unattaches property \\spad{p} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.") (($ $ (|String|)) "\\spad{deleteProperty!(op, s)} unattaches property \\spad{s} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|assert| (($ $ (|Identifier|)) "\\spad{assert(op, p)} attaches property \\spad{p} to \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|has?| (((|Boolean|) $ (|Identifier|)) "\\spad{has?(op,p)} tests if property \\spad{s} is attached to \\spad{op}.")) (|input| (((|Union| (|Mapping| (|InputForm|) (|List| (|InputForm|))) "failed") $) "\\spad{input(op)} returns the \"\\%input\" property of \\spad{op} if it has one attached,{} \"failed\" otherwise.") (($ $ (|Mapping| (|InputForm|) (|List| (|InputForm|)))) "\\spad{input(op, foo)} attaches foo as the \"\\%input\" property of \\spad{op}. If \\spad{op} has a \"\\%input\" property \\spad{f},{} then \\spad{op(a1,...,an)} gets converted to InputForm as \\spad{f(a1,...,an)}.")) (|display| (($ $ (|Mapping| (|OutputForm|) (|OutputForm|))) "\\spad{display(op, foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a)} gets converted to OutputForm as \\spad{f(a)}. Argument \\spad{op} must be unary.") (($ $ (|Mapping| (|OutputForm|) (|List| (|OutputForm|)))) "\\spad{display(op, foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a1,...,an)} gets converted to OutputForm as \\spad{f(a1,...,an)}.") (((|Union| (|Mapping| (|OutputForm|) (|List| (|OutputForm|))) "failed") $) "\\spad{display(op)} returns the \"\\%display\" property of \\spad{op} if it has one attached,{} and \"failed\" otherwise.")) (|comparison| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{comparison(op, foo?)} attaches foo? as the \"\\%less?\" property to \\spad{op}. If op1 and op2 have the same name,{} and one of them has a \"\\%less?\" property \\spad{f},{} then \\spad{f(op1, op2)} is called to decide whether \\spad{op1 < op2}.")) (|equality| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{equality(op, foo?)} attaches foo? as the \"\\%equal?\" property to \\spad{op}. If op1 and op2 have the same name,{} and one of them has an \"\\%equal?\" property \\spad{f},{} then \\spad{f(op1, op2)} is called to decide whether op1 and op2 should be considered equal.")) (|weight| (($ $ (|NonNegativeInteger|)) "\\spad{weight(op, n)} attaches the weight \\spad{n} to \\spad{op}.") (((|NonNegativeInteger|) $) "\\spad{weight(op)} returns the weight attached to \\spad{op}.")) (|nary?| (((|Boolean|) $) "\\spad{nary?(op)} tests if \\spad{op} has arbitrary arity.")) (|unary?| (((|Boolean|) $) "\\spad{unary?(op)} tests if \\spad{op} is unary.")) (|nullary?| (((|Boolean|) $) "\\spad{nullary?(op)} tests if \\spad{op} is nullary.")) (|operator| (($ (|Symbol|) (|Arity|)) "\\spad{operator(f, a)} makes \\spad{f} into an operator of arity \\spad{a}.") (($ (|Symbol|) (|NonNegativeInteger|)) "\\spad{operator(f, n)} makes \\spad{f} into an \\spad{n}-ary operator.") (($ (|Symbol|)) "\\spad{operator(f)} makes \\spad{f} into an operator with arbitrary arity.")) (|copy| (($ $) "\\spad{copy(op)} returns a copy of \\spad{op}.")) (|properties| (((|AssociationList| (|String|) (|None|)) $) "\\spad{properties(op)} returns the list of all the properties currently attached to \\spad{op}."))) NIL NIL -(-116 -3029 UP) +(-116 -3027 UP) ((|constructor| (NIL "\\spadtype{BoundIntegerRoots} provides functions to find lower bounds on the integer roots of a polynomial.")) (|integerBound| (((|Integer|) |#2|) "\\spad{integerBound(p)} returns a lower bound on the negative integer roots of \\spad{p},{} and 0 if \\spad{p} has no negative integer roots."))) NIL NIL @@ -403,7 +403,7 @@ NIL (-118 |p|) ((|constructor| (NIL "Stream-based implementation of \\spad{Qp:} numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-117 |#1|) (QUOTE (-924))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-148))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-117 |#1|) (QUOTE (-1039))) (|HasCategory| (-117 |#1|) (QUOTE (-831))) (-3765 (|HasCategory| (-117 |#1|) (QUOTE (-831))) (|HasCategory| (-117 |#1|) (QUOTE (-861)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-117 |#1|) (QUOTE (-1169))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| (-117 |#1|) (QUOTE (-237))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-117 |#1|) (QUOTE (-238))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -525) (QUOTE (-1194)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -318) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -295) (LIST (QUOTE -117) (|devaluate| |#1|)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (QUOTE (-316))) (|HasCategory| (-117 |#1|) (QUOTE (-556))) (|HasCategory| (-117 |#1|) (QUOTE (-861))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-924)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))))) +((|HasCategory| (-117 |#1|) (QUOTE (-924))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-148))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-117 |#1|) (QUOTE (-1039))) (|HasCategory| (-117 |#1|) (QUOTE (-831))) (-3763 (|HasCategory| (-117 |#1|) (QUOTE (-831))) (|HasCategory| (-117 |#1|) (QUOTE (-861)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-117 |#1|) (QUOTE (-1169))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| (-117 |#1|) (QUOTE (-237))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-117 |#1|) (QUOTE (-238))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -525) (QUOTE (-1194)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -318) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -295) (LIST (QUOTE -117) (|devaluate| |#1|)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (QUOTE (-316))) (|HasCategory| (-117 |#1|) (QUOTE (-556))) (|HasCategory| (-117 |#1|) (QUOTE (-861))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-924)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))))) (-119 A S) ((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,\"right\",b)} (also written \\axiom{\\spad{b} . right \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,\"left\",b)} (also written \\axiom{a . left \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) NIL @@ -419,7 +419,7 @@ NIL (-122 S) ((|constructor| (NIL "BinarySearchTree(\\spad{S}) is the domain of a binary trees where elements are ordered across the tree. A binary search tree is either empty or has a value which is an \\spad{S},{} and a right and left which are both BinaryTree(\\spad{S}) Elements are ordered across the tree.")) (|split| (((|Record| (|:| |less| $) (|:| |greater| $)) |#1| $) "\\spad{split(x,b)} splits binary tree \\spad{b} into two trees,{} one with elements greater than \\spad{x},{} the other with elements less than \\spad{x}.")) (|insertRoot!| (($ |#1| $) "\\spad{insertRoot!(x,b)} inserts element \\spad{x} as a root of binary search tree \\spad{b}.")) (|insert!| (($ |#1| $) "\\spad{insert!(x,b)} inserts element \\spad{x} as leaves into binary search tree \\spad{b}.")) (|binarySearchTree| (($ (|List| |#1|)) "\\spad{binarySearchTree(l)} \\undocumented"))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-123 S) ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}."))) NIL @@ -439,15 +439,15 @@ NIL (-127 S) ((|constructor| (NIL "\\spadtype{BinaryTournament(S)} is the domain of binary trees where elements are ordered down the tree. A binary search tree is either empty or is a node containing a \\spadfun{value} of type \\spad{S},{} and a \\spadfun{right} and a \\spadfun{left} which are both \\spadtype{BinaryTree(S)}")) (|insert!| (($ |#1| $) "\\spad{insert!(x,b)} inserts element \\spad{x} as leaves into binary tournament \\spad{b}.")) (|binaryTournament| (($ (|List| |#1|)) "\\spad{binaryTournament(ls)} creates a binary tournament with the elements of \\spad{ls} as values at the nodes."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-128 S) ((|constructor| (NIL "\\spadtype{BinaryTree(S)} is the domain of all binary trees. A binary tree over \\spad{S} is either empty or has a \\spadfun{value} which is an \\spad{S} and a \\spadfun{right} and \\spadfun{left} which are both binary trees.")) (|binaryTree| (($ $ |#1| $) "\\spad{binaryTree(l,v,r)} creates a binary tree with value \\spad{v} with left subtree \\spad{l} and right subtree \\spad{r}.") (($ |#1|) "\\spad{binaryTree(v)} is an non-empty binary tree with value \\spad{v},{} and left and right empty."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-129) ((|constructor| (NIL "ByteBuffer provides datatype for buffers of bytes. This domain differs from PrimitiveArray Byte in that it is not as rigid as PrimitiveArray Byte. That is,{} the typical use of ByteBuffer is to pre-allocate a vector of Byte of some capacity \\spad{`n'}. The array can then store up to \\spad{`n'} bytes. The actual interesting bytes count (the length of the buffer) is therefore different from the capacity. The length is no more than the capacity,{} but it can be set dynamically as needed. This functionality is used for example when reading bytes from input/output devices where we use buffers to transfer data in and out of the system. Note: a value of type ByteBuffer is 0-based indexed,{} as opposed \\indented{6}{Vector,{} but not unlike PrimitiveArray Byte.}")) (|finiteAggregate| ((|attribute|) "A ByteBuffer object is a finite aggregate")) (|setLength!| (((|NonNegativeInteger|) $ (|NonNegativeInteger|)) "\\spad{setLength!(buf,n)} sets the number of active bytes in the `buf'. Error if \\spad{`n'} is more than the capacity.")) (|capacity| (((|NonNegativeInteger|) $) "\\spad{capacity(buf)} returns the pre-allocated maximum size of `buf'.")) (|byteBuffer| (($ (|NonNegativeInteger|)) "\\spad{byteBuffer(n)} creates a buffer of capacity \\spad{n},{} and length 0."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| (-130) (QUOTE (-861))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130))))) (-12 (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130)))))) (-3765 (-12 (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130))))) (|HasCategory| (-130) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-130) (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| (-130) (QUOTE (-861))) (|HasCategory| (-130) (QUOTE (-1117)))) (|HasCategory| (-130) (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130)))))) +((-3763 (-12 (|HasCategory| (-130) (QUOTE (-861))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130))))) (-12 (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130)))))) (-3763 (-12 (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130))))) (|HasCategory| (-130) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-130) (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| (-130) (QUOTE (-861))) (|HasCategory| (-130) (QUOTE (-1117)))) (|HasCategory| (-130) (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-130) (QUOTE (-1117))) (|HasCategory| (-130) (LIST (QUOTE -318) (QUOTE (-130)))))) (-130) ((|constructor| (NIL "Byte is the datatype of 8-bit sized unsigned integer values.")) (|sample| (($) "\\spad{sample} gives a sample datum of type Byte.")) (|bitior| (($ $ $) "bitor(\\spad{x},{}\\spad{y}) returns the bitwise `inclusive or' of \\spad{`x'} and \\spad{`y'}.")) (|bitand| (($ $ $) "\\spad{bitand(x,y)} returns the bitwise `and' of \\spad{`x'} and \\spad{`y'}.")) (|byte| (($ (|NonNegativeInteger|)) "\\spad{byte(x)} injects the unsigned integer value \\spad{`v'} into the Byte algebra. \\spad{`v'} must be non-negative and less than 256."))) NIL @@ -472,11 +472,11 @@ NIL ((|constructor| (NIL "Members of the domain CardinalNumber are values indicating the cardinality of sets,{} both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. \\blankline If \\spad{x = \\#X} and \\spad{y = \\#Y} then \\indented{2}{\\spad{x+y\\space{2}= \\#(X+Y)}\\space{3}\\tab{30}disjoint union} \\indented{2}{\\spad{x-y\\space{2}= \\#(X-Y)}\\space{3}\\tab{30}relative complement} \\indented{2}{\\spad{x*y\\space{2}= \\#(X*Y)}\\space{3}\\tab{30}cartesian product} \\indented{2}{\\spad{x**y = \\#(X**Y)}\\space{2}\\tab{30}\\spad{X**Y = \\{g| g:Y->X\\}}} \\blankline The non-negative integers have a natural construction as cardinals \\indented{2}{\\spad{0 = \\#\\{\\}},{} \\spad{1 = \\{0\\}},{} \\spad{2 = \\{0, 1\\}},{} ...,{} \\spad{n = \\{i| 0 <= i < n\\}}.} \\blankline That \\spad{0} acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. \\blankline The generalized continuum hypothesis asserts \\center{\\spad{2**Aleph i = Aleph(i+1)}} and is independent of the axioms of set theory [Goedel 1940]. \\blankline Three commonly encountered cardinal numbers are \\indented{3}{\\spad{a = \\#Z}\\space{7}\\tab{30}countable infinity} \\indented{3}{\\spad{c = \\#R}\\space{7}\\tab{30}the continuum} \\indented{3}{\\spad{f = \\#\\{g| g:[0,1]->R\\}}} \\blankline In this domain,{} these values are obtained using \\indented{3}{\\spad{a := Aleph 0},{} \\spad{c := 2**a},{} \\spad{f := 2**c}.} \\blankline")) (|generalizedContinuumHypothesisAssumed| (((|Boolean|) (|Boolean|)) "\\spad{generalizedContinuumHypothesisAssumed(bool)} is used to dictate whether the hypothesis is to be assumed.")) (|generalizedContinuumHypothesisAssumed?| (((|Boolean|)) "\\spad{generalizedContinuumHypothesisAssumed?()} tests if the hypothesis is currently assumed.")) (|countable?| (((|Boolean|) $) "\\spad{countable?(\\spad{a})} determines whether \\spad{a} is a countable cardinal,{} \\spadignore{i.e.} an integer or \\spad{Aleph 0}.")) (|finite?| (((|Boolean|) $) "\\spad{finite?(\\spad{a})} determines whether \\spad{a} is a finite cardinal,{} \\spadignore{i.e.} an integer.")) (|Aleph| (($ (|NonNegativeInteger|)) "\\spad{Aleph(n)} provides the named (infinite) cardinal number.")) (** (($ $ $) "\\spad{x**y} returns \\spad{\\#(X**Y)} where \\spad{X**Y} is defined \\indented{1}{as \\spad{\\{g| g:Y->X\\}}.}")) (- (((|Union| $ "failed") $ $) "\\spad{x - y} returns an element \\spad{z} such that \\spad{z+y=x} or \"failed\" if no such element exists.")) (|commutative| ((|attribute| "*") "a domain \\spad{D} has \\spad{commutative(\"*\")} if it has an operation \\spad{\"*\": (D,D) -> D} which is commutative."))) (((-4462 "*") . T)) NIL -(-136 |minix| -2833 S T$) +(-136 |minix| -2831 S T$) ((|constructor| (NIL "This package provides functions to enable conversion of tensors given conversion of the components.")) (|map| (((|CartesianTensor| |#1| |#2| |#4|) (|Mapping| |#4| |#3|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{map(f,ts)} does a componentwise conversion of the tensor \\spad{ts} to a tensor with components of type \\spad{T}.")) (|reshape| (((|CartesianTensor| |#1| |#2| |#4|) (|List| |#4|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{reshape(lt,ts)} organizes the list of components \\spad{lt} into a tensor with the same shape as \\spad{ts}."))) NIL NIL -(-137 |minix| -2833 R) +(-137 |minix| -2831 R) ((|constructor| (NIL "CartesianTensor(minix,{}dim,{}\\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \\spad{R}. These tensors can have any number of indices. Each index takes values from \\spad{minix} to \\spad{minix + dim - 1}.")) (|sample| (($) "\\spad{sample()} returns an object of type \\%.")) (|unravel| (($ (|List| |#3|)) "\\spad{unravel(t)} produces a tensor from a list of components such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|ravel| (((|List| |#3|) $) "\\spad{ravel(t)} produces a list of components from a tensor such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|leviCivitaSymbol| (($) "\\spad{leviCivitaSymbol()} is the rank \\spad{dim} tensor defined by \\spad{leviCivitaSymbol()(i1,...idim) = +1/0/-1} if \\spad{i1,...,idim} is an even/is nota /is an odd permutation of \\spad{minix,...,minix+dim-1}.")) (|kroneckerDelta| (($) "\\spad{kroneckerDelta()} is the rank 2 tensor defined by \\indented{3}{\\spad{kroneckerDelta()(i,j)}} \\indented{6}{\\spad{= 1\\space{2}if i = j}} \\indented{6}{\\spad{= 0 if\\space{2}i \\~= j}}")) (|reindex| (($ $ (|List| (|Integer|))) "\\spad{reindex(t,[i1,...,idim])} permutes the indices of \\spad{t}. For example,{} if \\spad{r = reindex(t, [4,1,2,3])} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank for tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(l,i,j,k)}.}")) (|transpose| (($ $ (|Integer|) (|Integer|)) "\\spad{transpose(t,i,j)} exchanges the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices of \\spad{t}. For example,{} if \\spad{r = transpose(t,2,3)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(i,k,j,l)}.}") (($ $) "\\spad{transpose(t)} exchanges the first and last indices of \\spad{t}. For example,{} if \\spad{r = transpose(t)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(l,j,k,i)}.}")) (|contract| (($ $ (|Integer|) (|Integer|)) "\\spad{contract(t,i,j)} is the contraction of tensor \\spad{t} which sums along the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices. For example,{} if \\spad{r = contract(t,1,3)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 2 \\spad{(= 4 - 2)} tensor given by \\indented{4}{\\spad{r(i,j) = sum(h=1..dim,t(h,i,h,j))}.}") (($ $ (|Integer|) $ (|Integer|)) "\\spad{contract(t,i,s,j)} is the inner product of tenors \\spad{s} and \\spad{t} which sums along the \\spad{k1}\\spad{-}th index of \\spad{t} and the \\spad{k2}\\spad{-}th index of \\spad{s}. For example,{} if \\spad{r = contract(s,2,t,1)} for rank 3 tensors rank 3 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is the rank 4 \\spad{(= 3 + 3 - 2)} tensor given by \\indented{4}{\\spad{r(i,j,k,l) = sum(h=1..dim,s(i,h,j)*t(h,k,l))}.}")) (* (($ $ $) "\\spad{s*t} is the inner product of the tensors \\spad{s} and \\spad{t} which contracts the last index of \\spad{s} with the first index of \\spad{t},{} \\spadignore{i.e.} \\indented{4}{\\spad{t*s = contract(t,rank t, s, 1)}} \\indented{4}{\\spad{t*s = sum(k=1..N, t[i1,..,iN,k]*s[k,j1,..,jM])}} This is compatible with the use of \\spad{M*v} to denote the matrix-vector inner product.")) (|product| (($ $ $) "\\spad{product(s,t)} is the outer product of the tensors \\spad{s} and \\spad{t}. For example,{} if \\spad{r = product(s,t)} for rank 2 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is a rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = s(i,j)*t(k,l)}.}")) (|elt| ((|#3| $ (|List| (|Integer|))) "\\spad{elt(t,[i1,...,iN])} gives a component of a rank \\spad{N} tensor.") ((|#3| $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{elt(t,i,j,k,l)} gives a component of a rank 4 tensor.") ((|#3| $ (|Integer|) (|Integer|) (|Integer|)) "\\spad{elt(t,i,j,k)} gives a component of a rank 3 tensor.") ((|#3| $ (|Integer|) (|Integer|)) "\\spad{elt(t,i,j)} gives a component of a rank 2 tensor.") ((|#3| $) "\\spad{elt(t)} gives the component of a rank 0 tensor.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(t)} returns the tensorial rank of \\spad{t} (that is,{} the number of indices). This is the same as the graded module degree.")) (|coerce| (($ (|List| $)) "\\spad{coerce([t_1,...,t_dim])} allows tensors to be constructed using lists.") (($ (|List| |#3|)) "\\spad{coerce([r_1,...,r_dim])} allows tensors to be constructed using lists.") (($ (|SquareMatrix| |#2| |#3|)) "\\spad{coerce(m)} views a matrix as a rank 2 tensor.") (($ (|DirectProduct| |#2| |#3|)) "\\spad{coerce(v)} views a vector as a rank 1 tensor."))) NIL NIL @@ -499,7 +499,7 @@ NIL (-142) ((|constructor| (NIL "This domain allows classes of characters to be defined and manipulated efficiently.")) (|alphanumeric| (($) "\\spad{alphanumeric()} returns the class of all characters for which \\spadfunFrom{alphanumeric?}{Character} is \\spad{true}.")) (|alphabetic| (($) "\\spad{alphabetic()} returns the class of all characters for which \\spadfunFrom{alphabetic?}{Character} is \\spad{true}.")) (|lowerCase| (($) "\\spad{lowerCase()} returns the class of all characters for which \\spadfunFrom{lowerCase?}{Character} is \\spad{true}.")) (|upperCase| (($) "\\spad{upperCase()} returns the class of all characters for which \\spadfunFrom{upperCase?}{Character} is \\spad{true}.")) (|hexDigit| (($) "\\spad{hexDigit()} returns the class of all characters for which \\spadfunFrom{hexDigit?}{Character} is \\spad{true}.")) (|digit| (($) "\\spad{digit()} returns the class of all characters for which \\spadfunFrom{digit?}{Character} is \\spad{true}.")) (|charClass| (($ (|List| (|Character|))) "\\spad{charClass(l)} creates a character class which contains exactly the characters given in the list \\spad{l}.") (($ (|String|)) "\\spad{charClass(s)} creates a character class which contains exactly the characters given in the string \\spad{s}."))) ((-4460 . T) (-4450 . T) (-4461 . T)) -((-3765 (-12 (|HasCategory| (-145) (QUOTE (-378))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-145) (QUOTE (-378))) (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) +((-3763 (-12 (|HasCategory| (-145) (QUOTE (-378))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-145) (QUOTE (-378))) (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (-143 R Q A) ((|constructor| (NIL "CommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#3|) "\\spad{splitDenominator([q1,...,qn])} returns \\spad{[[p1,...,pn], d]} such that \\spad{qi = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|clearDenominator| ((|#3| |#3|) "\\spad{clearDenominator([q1,...,qn])} returns \\spad{[p1,...,pn]} such that \\spad{qi = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|commonDenominator| ((|#1| |#3|) "\\spad{commonDenominator([q1,...,qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}\\spad{qn}."))) NIL @@ -524,7 +524,7 @@ NIL ((|constructor| (NIL "Rings of Characteristic Zero."))) ((-4457 . T)) NIL -(-149 -3029 UP UPUP) +(-149 -3027 UP UPUP) ((|constructor| (NIL "Tools to send a point to infinity on an algebraic curve.")) (|chvar| (((|Record| (|:| |func| |#3|) (|:| |poly| |#3|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) |#3| |#3|) "\\spad{chvar(f(x,y), p(x,y))} returns \\spad{[g(z,t), q(z,t), c1(z), c2(z), n]} such that under the change of variable \\spad{x = c1(z)},{} \\spad{y = t * c2(z)},{} one gets \\spad{f(x,y) = g(z,t)}. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x, y) = 0}. The algebraic relation between \\spad{z} and \\spad{t} is \\spad{q(z, t) = 0}.")) (|eval| ((|#3| |#3| (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{eval(p(x,y), f(x), g(x))} returns \\spad{p(f(x), y * g(x))}.")) (|goodPoint| ((|#1| |#3| |#3|) "\\spad{goodPoint(p, q)} returns an integer a such that a is neither a pole of \\spad{p(x,y)} nor a branch point of \\spad{q(x,y) = 0}.")) (|rootPoly| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| (|Fraction| |#2|)) (|:| |radicand| |#2|)) (|Fraction| |#2|) (|NonNegativeInteger|)) "\\spad{rootPoly(g, n)} returns \\spad{[m, c, P]} such that \\spad{c * g ** (1/n) = P ** (1/m)} thus if \\spad{y**n = g},{} then \\spad{z**m = P} where \\spad{z = c * y}.")) (|radPoly| (((|Union| (|Record| (|:| |radicand| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) "failed") |#3|) "\\spad{radPoly(p(x, y))} returns \\spad{[c(x), n]} if \\spad{p} is of the form \\spad{y**n - c(x)},{} \"failed\" otherwise.")) (|mkIntegral| (((|Record| (|:| |coef| (|Fraction| |#2|)) (|:| |poly| |#3|)) |#3|) "\\spad{mkIntegral(p(x,y))} returns \\spad{[c(x), q(x,z)]} such that \\spad{z = c * y} is integral. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x, y) = 0}. The algebraic relation between \\spad{x} and \\spad{z} is \\spad{q(x, z) = 0}."))) NIL NIL @@ -564,7 +564,7 @@ NIL ((|constructor| (NIL "Color() specifies a domain of 27 colors provided in the \\Language{} system (the colors mix additively).")) (|color| (($ (|Integer|)) "\\spad{color(i)} returns a color of the indicated hue \\spad{i}.")) (|numberOfHues| (((|PositiveInteger|)) "\\spad{numberOfHues()} returns the number of total hues,{} set in totalHues.")) (|hue| (((|Integer|) $) "\\spad{hue(c)} returns the hue index of the indicated color \\spad{c}.")) (|blue| (($) "\\spad{blue()} returns the position of the blue hue from total hues.")) (|green| (($) "\\spad{green()} returns the position of the green hue from total hues.")) (|yellow| (($) "\\spad{yellow()} returns the position of the yellow hue from total hues.")) (|red| (($) "\\spad{red()} returns the position of the red hue from total hues.")) (+ (($ $ $) "\\spad{c1 + c2} additively mixes the two colors \\spad{c1} and \\spad{c2}.")) (* (($ (|DoubleFloat|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}.") (($ (|PositiveInteger|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}."))) NIL NIL -(-159 R -3029) +(-159 R -3027) ((|constructor| (NIL "Provides combinatorial functions over an integral domain.")) (|ipow| ((|#2| (|List| |#2|)) "\\spad{ipow(l)} should be local but conditional.")) (|iidprod| ((|#2| (|List| |#2|)) "\\spad{iidprod(l)} should be local but conditional.")) (|iidsum| ((|#2| (|List| |#2|)) "\\spad{iidsum(l)} should be local but conditional.")) (|iipow| ((|#2| (|List| |#2|)) "\\spad{iipow(l)} should be local but conditional.")) (|iiperm| ((|#2| (|List| |#2|)) "\\spad{iiperm(l)} should be local but conditional.")) (|iibinom| ((|#2| (|List| |#2|)) "\\spad{iibinom(l)} should be local but conditional.")) (|iifact| ((|#2| |#2|) "\\spad{iifact(x)} should be local but conditional.")) (|product| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{product(f(n), n = a..b)} returns \\spad{f}(a) * ... * \\spad{f}(\\spad{b}) as a formal product.") ((|#2| |#2| (|Symbol|)) "\\spad{product(f(n), n)} returns the formal product \\spad{P}(\\spad{n}) which verifies \\spad{P}(\\spad{n+1})\\spad{/P}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|summation| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{summation(f(n), n = a..b)} returns \\spad{f}(a) + ... + \\spad{f}(\\spad{b}) as a formal sum.") ((|#2| |#2| (|Symbol|)) "\\spad{summation(f(n), n)} returns the formal sum \\spad{S}(\\spad{n}) which verifies \\spad{S}(\\spad{n+1}) - \\spad{S}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|factorials| ((|#2| |#2| (|Symbol|)) "\\spad{factorials(f, x)} rewrites the permutations and binomials in \\spad{f} involving \\spad{x} in terms of factorials.") ((|#2| |#2|) "\\spad{factorials(f)} rewrites the permutations and binomials in \\spad{f} in terms of factorials.")) (|factorial| ((|#2| |#2|) "\\spad{factorial(n)} returns the factorial of \\spad{n},{} \\spadignore{i.e.} \\spad{n!}.")) (|permutation| ((|#2| |#2| |#2|) "\\spad{permutation(n, r)} returns the number of permutations of \\spad{n} objects taken \\spad{r} at a time,{} \\spadignore{i.e.} \\spad{n!/}(\\spad{n}-\\spad{r})!.")) (|binomial| ((|#2| |#2| |#2|) "\\spad{binomial(n, r)} returns the number of subsets of \\spad{r} objects taken among \\spad{n} objects,{} \\spadignore{i.e.} \\spad{n!/}(\\spad{r!} * (\\spad{n}-\\spad{r})!).")) (** ((|#2| |#2| |#2|) "\\spad{a ** b} is the formal exponential a**b.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a combinatorial operator.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a combinatorial operator."))) NIL NIL @@ -598,7 +598,7 @@ NIL ((|HasCategory| |#2| (QUOTE (-924))) (|HasCategory| |#2| (QUOTE (-556))) (|HasCategory| |#2| (QUOTE (-1019))) (|HasCategory| |#2| (QUOTE (-1220))) (|HasCategory| |#2| (QUOTE (-1077))) (|HasCategory| |#2| (QUOTE (-1039))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4456)) (|HasAttribute| |#2| (QUOTE -4459)) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-567)))) (-167 R) ((|constructor| (NIL "This category represents the extension of a ring by a square root of \\spad{-1}.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#1|) (|:| |phi| |#1|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(x, r)} returns the exact quotient of \\spad{x} by \\spad{r},{} or \"failed\" if \\spad{r} does not divide \\spad{x} exactly.")) (|norm| ((|#1| $) "\\spad{norm(x)} returns \\spad{x} * conjugate(\\spad{x})")) (|real| ((|#1| $) "\\spad{real(x)} returns real part of \\spad{x}.")) (|imag| ((|#1| $) "\\spad{imag(x)} returns imaginary part of \\spad{x}.")) (|conjugate| (($ $) "\\spad{conjugate(x + \\%i y)} returns \\spad{x} - \\%\\spad{i} \\spad{y}.")) (|imaginary| (($) "\\spad{imaginary()} = sqrt(\\spad{-1}) = \\%\\spad{i}.")) (|complex| (($ |#1| |#1|) "\\spad{complex(x,y)} constructs \\spad{x} + \\%i*y.") ((|attribute|) "indicates that \\% has sqrt(\\spad{-1})"))) -((-4453 -3765 (|has| |#1| (-567)) (-12 (|has| |#1| (-316)) (|has| |#1| (-924)))) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4456 |has| |#1| (-6 -4456)) (-4459 |has| |#1| (-6 -4459)) (-3502 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-4453 -3763 (|has| |#1| (-567)) (-12 (|has| |#1| (-316)) (|has| |#1| (-924)))) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4456 |has| |#1| (-6 -4456)) (-4459 |has| |#1| (-6 -4459)) (-3501 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-168 RR PR) ((|constructor| (NIL "\\indented{1}{Author:} Date Created: Date Last Updated: Basic Functions: Related Constructors: Complex,{} UnivariatePolynomial Also See: AMS Classifications: Keywords: complex,{} polynomial factorization,{} factor References:")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} factorizes the polynomial \\spad{p} with complex coefficients."))) @@ -614,8 +614,8 @@ NIL NIL (-171 R) ((|constructor| (NIL "\\spadtype {Complex(R)} creates the domain of elements of the form \\spad{a + b * i} where \\spad{a} and \\spad{b} come from the ring \\spad{R},{} and \\spad{i} is a new element such that \\spad{i**2 = -1}."))) -((-4453 -3765 (|has| |#1| (-567)) (-12 (|has| |#1| (-316)) (|has| |#1| (-924)))) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4456 |has| |#1| (-6 -4456)) (-4459 |has| |#1| (-6 -4459)) (-3502 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . 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T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . 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(|findConstructor| (((|Maybe| $) (|Identifier|)) "\\spad{findConstructor(s)} attempts to find a constructor named \\spad{s}. If successful,{} returns that constructor; otherwise,{} returns \\spad{nothing}."))) NIL NIL -(-190 R -3029) +(-190 R -3027) ((|constructor| (NIL "\\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.")) (|complexForm| (((|Complex| (|Expression| |#1|)) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| (((|Expression| |#1|) |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| (((|Expression| |#1|) |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) NIL NIL @@ -796,23 +796,23 @@ NIL ((|constructor| (NIL "\\indented{1}{This domain implements a simple view of a database whose fields are} indexed by symbols")) (- (($ $ $) "\\spad{db1-db2} returns the difference of databases \\spad{db1} and \\spad{db2} \\spadignore{i.e.} consisting of elements in \\spad{db1} but not in \\spad{db2}")) (+ (($ $ $) "\\spad{db1+db2} returns the merge of databases \\spad{db1} and \\spad{db2}")) (|fullDisplay| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{fullDisplay(db,start,end )} prints full details of entries in the range \\axiom{\\spad{start}..end} in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(db)} prints full details of each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(x)} displays \\spad{x} in detail")) (|display| (((|Void|) $) "\\spad{display(db)} prints a summary line for each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{display(x)} displays \\spad{x} in some form")) (|elt| (((|DataList| (|String|)) $ (|Symbol|)) "\\spad{elt(db,s)} returns the \\axiom{\\spad{s}} field of each element of \\axiom{\\spad{db}}.") (($ $ (|QueryEquation|)) "\\spad{elt(db,q)} returns all elements of \\axiom{\\spad{db}} which satisfy \\axiom{\\spad{q}}.") (((|String|) $ (|Symbol|)) "\\spad{elt(x,s)} returns an element of \\spad{x} indexed by \\spad{s}"))) NIL NIL -(-217 -3029 UP UPUP R) +(-217 -3027 UP UPUP R) ((|constructor| (NIL "This package provides functions for computing the residues of a function on an algebraic curve.")) (|doubleResultant| ((|#2| |#4| (|Mapping| |#2| |#2|)) "\\spad{doubleResultant(f, ')} returns \\spad{p}(\\spad{x}) whose roots are rational multiples of the residues of \\spad{f} at all its finite poles. Argument ' is the derivation to use."))) NIL NIL -(-218 -3029 FP) +(-218 -3027 FP) ((|constructor| (NIL "Package for the factorization of a univariate polynomial with coefficients in a finite field. The algorithm used is the \"distinct degree\" algorithm of Cantor-Zassenhaus,{} modified to use trace instead of the norm and a table for computing Frobenius as suggested by Naudin and Quitte .")) (|irreducible?| (((|Boolean|) |#2|) "\\spad{irreducible?(p)} tests whether the polynomial \\spad{p} is irreducible.")) (|tracePowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{tracePowMod(u,k,v)} produces the sum of \\spad{u**(q**i)} for \\spad{i} running and \\spad{q=} size \\spad{F}")) (|trace2PowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{trace2PowMod(u,k,v)} produces the sum of \\spad{u**(2**i)} for \\spad{i} running from 1 to \\spad{k} all computed modulo the polynomial \\spad{v}.")) (|exptMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{exptMod(u,k,v)} raises the polynomial \\spad{u} to the \\spad{k}th power modulo the polynomial \\spad{v}.")) (|separateFactors| (((|List| |#2|) (|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|)))) "\\spad{separateFactors(lfact)} takes the list produced by \\spadfunFrom{separateDegrees}{DistinctDegreeFactorization} and produces the complete list of factors.")) (|separateDegrees| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|))) |#2|) "\\spad{separateDegrees(p)} splits the square free polynomial \\spad{p} into factors each of which is a product of irreducibles of the same degree.")) (|distdfact| (((|Record| (|:| |cont| |#1|) (|:| |factors| (|List| (|Record| (|:| |irr| |#2|) (|:| |pow| (|Integer|)))))) |#2| (|Boolean|)) "\\spad{distdfact(p,sqfrflag)} produces the complete factorization of the polynomial \\spad{p} returning an internal data structure. If argument \\spad{sqfrflag} is \\spad{true},{} the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#2|) |#2|) "\\spad{factorSquareFree(p)} produces the complete factorization of the square free polynomial \\spad{p}.")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} produces the complete factorization of the polynomial \\spad{p}."))) NIL NIL (-219) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions.")) (|decimal| (($ (|Fraction| (|Integer|))) "\\spad{decimal(r)} converts a rational number to a decimal expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(d)} returns the fractional part of a decimal expansion."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3765 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) +((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3763 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) (-220) ((|constructor| (NIL "This domain represents the syntax of a definition.")) (|body| (((|SpadAst|) $) "\\spad{body(d)} returns the right hand side of the definition \\spad{`d'}.")) (|signature| (((|Signature|) $) "\\spad{signature(d)} returns the signature of the operation being defined. Note that this list may be partial in that it contains only the types actually specified in the definition.")) (|head| (((|HeadAst|) $) "\\spad{head(d)} returns the head of the definition \\spad{`d'}. This is a list of identifiers starting with the name of the operation followed by the name of the parameters,{} if any."))) NIL NIL -(-221 R -3029) +(-221 R -3027) ((|constructor| (NIL "\\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.")) (|innerint| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{innerint(f, x, a, b, ignore?)} should be local but conditional")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|)) (|String|)) "\\spad{integrate(f, x = a..b, \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|))) "\\spad{integrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) NIL NIL @@ -827,18 +827,18 @@ NIL (-224 S) ((|constructor| (NIL "Linked list implementation of a Dequeue")) (|dequeue| (($ (|List| |#1|)) "\\spad{dequeue([x,y,...,z])} creates a dequeue with first (top or front) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-225 |CoefRing| |listIndVar|) ((|constructor| (NIL "The deRham complex of Euclidean space,{} that is,{} the class of differential forms of arbitary degree over a coefficient ring. See Flanders,{} Harley,{} Differential Forms,{} With Applications to the Physical Sciences,{} New York,{} Academic Press,{} 1963.")) (|exteriorDifferential| (($ $) "\\spad{exteriorDifferential(df)} returns the exterior derivative (gradient,{} curl,{} divergence,{} ...) of the differential form \\spad{df}.")) (|totalDifferential| (($ (|Expression| |#1|)) "\\spad{totalDifferential(x)} returns the total differential (gradient) form for element \\spad{x}.")) (|map| (($ (|Mapping| (|Expression| |#1|) (|Expression| |#1|)) $) "\\spad{map(f,df)} replaces each coefficient \\spad{x} of differential form \\spad{df} by \\spad{f(x)}.")) (|degree| (((|Integer|) $) "\\spad{degree(df)} returns the homogeneous degree of differential form \\spad{df}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(df)} tests if differential form \\spad{df} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{df}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(df)} tests if all of the terms of differential form \\spad{df} have the same degree.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th basis term for a differential form.")) (|coefficient| (((|Expression| |#1|) $ $) "\\spad{coefficient(df,u)},{} where \\spad{df} is a differential form,{} returns the coefficient of \\spad{df} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise.")) (|reductum| (($ $) "\\spad{reductum(df)},{} where \\spad{df} is a differential form,{} returns \\spad{df} minus the leading term of \\spad{df} if \\spad{df} has two or more terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(df)} returns the leading basis term of differential form \\spad{df}.")) (|leadingCoefficient| (((|Expression| |#1|) $) "\\spad{leadingCoefficient(df)} returns the leading coefficient of differential form \\spad{df}."))) ((-4457 . T)) NIL -(-226 R -3029) +(-226 R -3027) ((|constructor| (NIL "\\spadtype{DefiniteIntegrationTools} provides common tools used by the definite integration of both rational and elementary functions.")) (|checkForZero| (((|Union| (|Boolean|) "failed") (|SparseUnivariatePolynomial| |#2|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p, a, b, incl?)} is \\spad{true} if \\spad{p} has a zero between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.") (((|Union| (|Boolean|) "failed") (|Polynomial| |#1|) (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p, x, a, b, incl?)} is \\spad{true} if \\spad{p} has a zero for \\spad{x} between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.")) (|computeInt| (((|Union| (|OrderedCompletion| |#2|) "failed") (|Kernel| |#2|) |#2| (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{computeInt(x, g, a, b, eval?)} returns the integral of \\spad{f} for \\spad{x} between a and \\spad{b},{} assuming that \\spad{g} is an indefinite integral of \\spad{f} and \\spad{f} has no pole between a and \\spad{b}. If \\spad{eval?} is \\spad{true},{} then \\spad{g} can be evaluated safely at \\spad{a} and \\spad{b},{} provided that they are finite values. Otherwise,{} limits must be computed.")) (|ignore?| (((|Boolean|) (|String|)) "\\spad{ignore?(s)} is \\spad{true} if \\spad{s} is the string that tells the integrator to assume that the function has no pole in the integration interval."))) NIL NIL (-227) ((|constructor| (NIL "\\indented{1}{\\spadtype{DoubleFloat} is intended to make accessible} hardware floating point arithmetic in \\Language{},{} either native double precision,{} or IEEE. On most machines,{} there will be hardware support for the arithmetic operations: \\spadfunFrom{+}{DoubleFloat},{} \\spadfunFrom{*}{DoubleFloat},{} \\spadfunFrom{/}{DoubleFloat} and possibly also the \\spadfunFrom{sqrt}{DoubleFloat} operation. The operations \\spadfunFrom{exp}{DoubleFloat},{} \\spadfunFrom{log}{DoubleFloat},{} \\spadfunFrom{sin}{DoubleFloat},{} \\spadfunFrom{cos}{DoubleFloat},{} \\spadfunFrom{atan}{DoubleFloat} are normally coded in software based on minimax polynomial/rational approximations. Note that under Lisp/VM,{} \\spadfunFrom{atan}{DoubleFloat} is not available at this time. Some general comments about the accuracy of the operations: the operations \\spadfunFrom{+}{DoubleFloat},{} \\spadfunFrom{*}{DoubleFloat},{} \\spadfunFrom{/}{DoubleFloat} and \\spadfunFrom{sqrt}{DoubleFloat} are expected to be fully accurate. The operations \\spadfunFrom{exp}{DoubleFloat},{} \\spadfunFrom{log}{DoubleFloat},{} \\spadfunFrom{sin}{DoubleFloat},{} \\spadfunFrom{cos}{DoubleFloat} and \\spadfunFrom{atan}{DoubleFloat} are not expected to be fully accurate. In particular,{} \\spadfunFrom{sin}{DoubleFloat} and \\spadfunFrom{cos}{DoubleFloat} will lose all precision for large arguments. \\blankline The \\spadtype{Float} domain provides an alternative to the \\spad{DoubleFloat} domain. It provides an arbitrary precision model of floating point arithmetic. This means that accuracy problems like those above are eliminated by increasing the working precision where necessary. \\spadtype{Float} provides some special functions such as \\spadfunFrom{erf}{DoubleFloat},{} the error function in addition to the elementary functions. The disadvantage of \\spadtype{Float} is that it is much more expensive than small floats when the latter can be used.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n, b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)} (that is,{} \\spad{|(r-f)/f| < b**(-n)}).") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|Beta| (($ $ $) "\\spad{Beta(x,y)} is \\spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.")) (|Gamma| (($ $) "\\spad{Gamma(x)} is the Euler Gamma function.")) (|atan| (($ $ $) "\\spad{atan(x,y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm with base 10 for \\spad{x}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm with base 2 for \\spad{x}.")) (|exp1| (($) "\\spad{exp1()} returns the natural log base \\spad{2.718281828...}.")) (** (($ $ $) "\\spad{x ** y} returns the \\spad{y}th power of \\spad{x} (equal to \\spad{exp(y log x)}).")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) -((-3494 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-3493 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-228) ((|constructor| (NIL "This package provides special functions for double precision real and complex floating point.")) (|hypergeometric0F1| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{hypergeometric0F1(c,z)} is the hypergeometric function \\spad{0F1(; c; z)}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{hypergeometric0F1(c,z)} is the hypergeometric function \\spad{0F1(; c; z)}.")) (|airyBi| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Bi''(x) - x * Bi(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Bi''(x) - x * Bi(x) = 0}.}")) (|airyAi| (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Ai''(x) - x * Ai(x) = 0}.}") (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Ai''(x) - x * Ai(x) = 0}.}")) (|besselK| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselK(v,x)} is the modified Bessel function of the first kind,{} \\spad{K(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{K(v,x) = \\%pi/2*(I(-v,x) - I(v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselK(v,x)} is the modified Bessel function of the first kind,{} \\spad{K(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{K(v,x) = \\%pi/2*(I(-v,x) - I(v,x))/sin(v*\\%pi)}.} so is not valid for integer values of \\spad{v}.")) (|besselI| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselI(v,x)} is the modified Bessel function of the first kind,{} \\spad{I(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselI(v,x)} is the modified Bessel function of the first kind,{} \\spad{I(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}")) (|besselY| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselY(v,x)} is the Bessel function of the second kind,{} \\spad{Y(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{Y(v,x) = (J(v,x) cos(v*\\%pi) - J(-v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselY(v,x)} is the Bessel function of the second kind,{} \\spad{Y(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{Y(v,x) = (J(v,x) cos(v*\\%pi) - J(-v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.")) (|besselJ| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselJ(v,x)} is the Bessel function of the first kind,{} \\spad{J(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselJ(v,x)} is the Bessel function of the first kind,{} \\spad{J(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}")) (|polygamma| (((|Complex| (|DoubleFloat|)) (|NonNegativeInteger|) (|Complex| (|DoubleFloat|))) "\\spad{polygamma(n, x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.") (((|DoubleFloat|) (|NonNegativeInteger|) (|DoubleFloat|)) "\\spad{polygamma(n, x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.")) (|digamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}")) (|logGamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.")) (|Beta| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Beta(x, y)} is the Euler beta function,{} \\spad{B(x,y)},{} defined by \\indented{2}{\\spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{Beta(x, y)} is the Euler beta function,{} \\spad{B(x,y)},{} defined by \\indented{2}{\\spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}")) (|Gamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\\%infinity)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\\%infinity)}.}"))) @@ -847,7 +847,7 @@ NIL (-229 R) ((|constructor| (NIL "\\indented{1}{A Denavit-Hartenberg Matrix is a 4x4 Matrix of the form:} \\indented{1}{\\spad{nx ox ax px}} \\indented{1}{\\spad{ny oy ay py}} \\indented{1}{\\spad{nz oz az pz}} \\indented{2}{\\spad{0\\space{2}0\\space{2}0\\space{2}1}} (\\spad{n},{} \\spad{o},{} and a are the direction cosines)")) (|translate| (($ |#1| |#1| |#1|) "\\spad{translate(X,Y,Z)} returns a dhmatrix for translation by \\spad{X},{} \\spad{Y},{} and \\spad{Z}")) (|scale| (($ |#1| |#1| |#1|) "\\spad{scale(sx,sy,sz)} returns a dhmatrix for scaling in the \\spad{X},{} \\spad{Y} and \\spad{Z} directions")) (|rotatez| (($ |#1|) "\\spad{rotatez(r)} returns a dhmatrix for rotation about axis \\spad{Z} for \\spad{r} degrees")) (|rotatey| (($ |#1|) "\\spad{rotatey(r)} returns a dhmatrix for rotation about axis \\spad{Y} for \\spad{r} degrees")) (|rotatex| (($ |#1|) "\\spad{rotatex(r)} returns a dhmatrix for rotation about axis \\spad{X} for \\spad{r} degrees")) (|identity| (($) "\\spad{identity()} create the identity dhmatrix")) (* (((|Point| |#1|) $ (|Point| |#1|)) "\\spad{t*p} applies the dhmatrix \\spad{t} to point \\spad{p}"))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-567))) (|HasAttribute| |#1| (QUOTE (-4462 "*"))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-567))) (|HasAttribute| |#1| (QUOTE (-4462 "*"))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-230 A S) ((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) NIL @@ -896,22 +896,22 @@ NIL ((|constructor| (NIL "any solution of a homogeneous linear Diophantine equation can be represented as a sum of minimal solutions,{} which form a \"basis\" (a minimal solution cannot be represented as a nontrivial sum of solutions) in the case of an inhomogeneous linear Diophantine equation,{} each solution is the sum of a inhomogeneous solution and any number of homogeneous solutions therefore,{} it suffices to compute two sets: \\indented{3}{1. all minimal inhomogeneous solutions} \\indented{3}{2. all minimal homogeneous solutions} the algorithm implemented is a completion procedure,{} which enumerates all solutions in a recursive depth-first-search it can be seen as finding monotone paths in a graph for more details see Reference")) (|dioSolve| (((|Record| (|:| |varOrder| (|List| (|Symbol|))) (|:| |inhom| (|Union| (|List| (|Vector| (|NonNegativeInteger|))) "failed")) (|:| |hom| (|List| (|Vector| (|NonNegativeInteger|))))) (|Equation| (|Polynomial| (|Integer|)))) "\\spad{dioSolve(u)} computes a basis of all minimal solutions for linear homogeneous Diophantine equation \\spad{u},{} then all minimal solutions of inhomogeneous equation"))) NIL NIL -(-242 S -2833 R) +(-242 S -2831 R) ((|constructor| (NIL "\\indented{2}{This category represents a finite cartesian product of a given type.} Many categorical properties are preserved under this construction.")) (|dot| ((|#3| $ $) "\\spad{dot(x,y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#3|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) NIL ((|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (QUOTE (-804))) (|HasCategory| |#3| (QUOTE (-861))) (|HasAttribute| |#3| (QUOTE -4457)) (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-378))) (|HasCategory| |#3| (QUOTE (-737))) (|HasCategory| |#3| (QUOTE (-21))) (|HasCategory| |#3| (QUOTE (-23))) (|HasCategory| |#3| (QUOTE (-132))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-1066))) (|HasCategory| |#3| (QUOTE (-1117)))) -(-243 -2833 R) +(-243 -2831 R) ((|constructor| (NIL "\\indented{2}{This category represents a finite cartesian product of a given type.} Many categorical properties are preserved under this construction.")) (|dot| ((|#2| $ $) "\\spad{dot(x,y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#2|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) ((-4454 |has| |#2| (-1066)) (-4455 |has| |#2| (-1066)) (-4457 |has| |#2| (-6 -4457)) (-4460 . T)) NIL -(-244 -2833 A B) +(-244 -2831 A B) ((|constructor| (NIL "\\indented{2}{This package provides operations which all take as arguments} direct products of elements of some type \\spad{A} and functions from \\spad{A} to another type \\spad{B}. The operations all iterate over their vector argument and either return a value of type \\spad{B} or a direct product over \\spad{B}.")) (|map| (((|DirectProduct| |#1| |#3|) (|Mapping| |#3| |#2|) (|DirectProduct| |#1| |#2|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values.")) (|reduce| ((|#3| (|Mapping| |#3| |#2| |#3|) (|DirectProduct| |#1| |#2|) |#3|) "\\spad{reduce(func,vec,ident)} combines the elements in \\spad{vec} using the binary function \\spad{func}. Argument \\spad{ident} is returned if the vector is empty.")) (|scan| (((|DirectProduct| |#1| |#3|) (|Mapping| |#3| |#2| |#3|) (|DirectProduct| |#1| |#2|) |#3|) "\\spad{scan(func,vec,ident)} creates a new vector whose elements are the result of applying reduce to the binary function \\spad{func},{} increasing initial subsequences of the vector \\spad{vec},{} and the element \\spad{ident}."))) NIL NIL -(-245 -2833 R) +(-245 -2831 R) ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying component type. 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(|sayLength| (((|Integer|) (|List| (|String|))) "\\spad{sayLength(l)} returns the length of a list of strings \\spad{l} as an integer.") (((|Integer|) (|String|)) "\\spad{sayLength(s)} returns the length of a string \\spad{s} as an integer.")) (|say| (((|Void|) (|List| (|String|))) "\\spad{say(l)} sends a list of strings \\spad{l} to output.") (((|Void|) (|String|)) "\\spad{say(s)} sends a string \\spad{s} to output.")) (|center| (((|List| (|String|)) (|List| (|String|)) (|Integer|) (|String|)) "\\spad{center(l,i,s)} takes a list of strings \\spad{l},{} and centers them within a list of strings which is \\spad{i} characters long,{} in which the remaining spaces are filled with strings composed of as many repetitions as possible of the last string parameter \\spad{s}.") (((|String|) (|String|) (|Integer|) (|String|)) "\\spad{center(s,i,s)} takes the first string \\spad{s},{} and centers it within a string of length \\spad{i},{} in which the other elements of the string are composed of as many replications as possible of the second indicated string,{} \\spad{s} which must have a length greater than that of an empty string.")) (|copies| (((|String|) (|Integer|) (|String|)) "\\spad{copies(i,s)} will take a string \\spad{s} and create a new string composed of \\spad{i} copies of \\spad{s}.")) (|newLine| (((|String|)) "\\spad{newLine()} sends a new line command to output.")) (|bright| (((|List| (|String|)) (|List| (|String|))) "\\spad{bright(l)} sets the font property of a list of strings,{} \\spad{l},{} to bold-face type.") (((|List| (|String|)) (|String|)) "\\spad{bright(s)} sets the font property of the string \\spad{s} to bold-face type."))) NIL @@ -931,7 +931,7 @@ NIL (-250 S) ((|constructor| (NIL "This domain provides some nice functions on lists")) (|elt| (((|NonNegativeInteger|) $ "count") "\\axiom{\\spad{l}.\"count\"} returns the number of elements in \\axiom{\\spad{l}}.") (($ $ "sort") "\\axiom{\\spad{l}.sort} returns \\axiom{\\spad{l}} with elements sorted. Note: \\axiom{\\spad{l}.sort = sort(\\spad{l})}") (($ $ "unique") "\\axiom{\\spad{l}.unique} returns \\axiom{\\spad{l}} with duplicates removed. Note: \\axiom{\\spad{l}.unique = removeDuplicates(\\spad{l})}.")) (|datalist| (($ (|List| |#1|)) "\\spad{datalist(l)} creates a datalist from \\spad{l}"))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-251 M) ((|constructor| (NIL "DiscreteLogarithmPackage implements help functions for discrete logarithms in monoids using small cyclic groups.")) (|shanksDiscLogAlgorithm| (((|Union| (|NonNegativeInteger|) "failed") |#1| |#1| (|NonNegativeInteger|)) "\\spad{shanksDiscLogAlgorithm(b,a,p)} computes \\spad{s} with \\spad{b**s = a} for assuming that \\spad{a} and \\spad{b} are elements in a 'small' cyclic group of order \\spad{p} by Shank\\spad{'s} algorithm. Note: this is a subroutine of the function \\spadfun{discreteLog}.")) (** ((|#1| |#1| (|Integer|)) "\\spad{x ** n} returns \\spad{x} raised to the integer power \\spad{n}"))) NIL @@ -939,7 +939,7 @@ NIL (-252 |vl| R) ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is lexicographic specified by the variable list parameter with the most significant variable first in the list.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) (((-4462 "*") |has| |#2| (-174)) (-4453 |has| |#2| (-567)) (-4458 |has| |#2| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#2| (QUOTE (-924))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) +((|HasCategory| |#2| (QUOTE (-924))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) (-253) ((|showSummary| (((|Void|) $) "\\spad{showSummary(d)} prints out implementation detail information of domain \\spad{`d'}.")) (|reflect| (($ (|ConstructorCall| (|DomainConstructor|))) "\\spad{reflect cc} returns the domain object designated by the ConstructorCall syntax `cc'. The constructor implied by `cc' must be known to the system since it is instantiated.")) (|reify| (((|ConstructorCall| (|DomainConstructor|)) $) "\\spad{reify(d)} returns the abstract syntax for the domain \\spad{`x'}.")) (|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Create: October 18,{} 2007. Date Last Updated: December 20,{} 2008. Basic Operations: coerce,{} reify Related Constructors: Type,{} Syntax,{} OutputForm Also See: Type,{} ConstructorCall") (((|DomainConstructor|) $) "\\spad{constructor(d)} returns the domain constructor that is instantiated to the domain object \\spad{`d'}."))) 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functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \\blankline")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#4| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note: an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#3|) "\\spad{weight(p, s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#3|) "\\spad{weights(p, s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p, s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{order(p,s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#3|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} \\spad{:=} makeVariable(\\spad{p}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#3|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) NIL @@ -1019,7 +1019,7 @@ NIL (-272 R S V) ((|constructor| (NIL "\\spadtype{DifferentialSparseMultivariatePolynomial} implements an ordinary differential polynomial ring by combining a domain belonging to the category \\spadtype{DifferentialVariableCategory} with the domain \\spadtype{SparseMultivariatePolynomial}. \\blankline"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#3| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#3| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#3| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#3| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#3| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) +((|HasCategory| |#1| (QUOTE (-924))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#3| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#3| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#3| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#3| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#3| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-273 A S) ((|constructor| (NIL "\\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \\spad{x},{}...,{}\\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \\spad{x},{}\\spad{x'},{}\\spad{x''},{}...,{} \\spad{y},{}\\spad{y'},{}\\spad{y''},{}... (Note: in the interpreter,{} the \\spad{n}-th derivative of \\spad{y} is displayed as \\spad{y} with a subscript \\spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \\blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \\spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \\spadfun{<}. For example,{} one may define one derivative to be less than another by lexicographically comparing first the \\spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \\blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{} this constructor provides a function \\spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \\spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.")) (|coerce| (($ |#2|) "\\spad{coerce(s)} returns \\spad{s},{} viewed as the zero-th order derivative of \\spad{s}.")) (|weight| (((|NonNegativeInteger|) $) "\\spad{weight(v)} returns the weight of the derivative \\spad{v}.")) (|variable| ((|#2| $) "\\spad{variable(v)} returns \\spad{s} if \\spad{v} is any derivative of the differential indeterminate \\spad{s}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(v)} returns \\spad{n} if \\spad{v} is the \\spad{n}-th derivative of any differential indeterminate.")) (|makeVariable| (($ |#2| (|NonNegativeInteger|)) "\\spad{makeVariable(s, n)} returns the \\spad{n}-th derivative of a differential indeterminate \\spad{s} as an algebraic indeterminate."))) NIL @@ -1064,11 +1064,11 @@ NIL ((|constructor| (NIL "A domain used in the construction of the exterior algebra on a set \\spad{X} over a ring \\spad{R}. This domain represents the set of all ordered subsets of the set \\spad{X},{} assumed to be in correspondance with {1,{}2,{}3,{} ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements which can only be 0 or 1. \\blankline The multiplicative identity element of the exterior algebra corresponds to the empty subset of \\spad{X}. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.")) (|Nul| (($ (|NonNegativeInteger|)) "\\spad{Nul()} gives the basis element 1 for the algebra generated by \\spad{n} generators.")) (|exponents| (((|List| (|Integer|)) $) "\\spad{exponents(x)} converts a domain element into a list of zeros and ones corresponding to the exponents in the basis element that \\spad{x} represents.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(x)} gives the numbers of 1\\spad{'s} in \\spad{x},{} \\spadignore{i.e.} the number of non-zero exponents in the basis element that \\spad{x} represents.")) (|coerce| (($ (|List| (|Integer|))) "\\spad{coerce(l)} converts a list of 0\\spad{'s} and 1\\spad{'s} into a basis element,{} where 1 (respectively 0) designates that the variable of the corresponding index of \\spad{l} is (respectively,{} is not) present. Error: if an element of \\spad{l} is not 0 or 1."))) NIL NIL -(-284 R -3029) +(-284 R -3027) ((|constructor| (NIL "Provides elementary functions over an integral domain.")) (|localReal?| (((|Boolean|) |#2|) "\\spad{localReal?(x)} should be local but conditional")) (|specialTrigs| (((|Union| |#2| "failed") |#2| (|List| (|Record| (|:| |func| |#2|) (|:| |pole| (|Boolean|))))) "\\spad{specialTrigs(x,l)} should be local but conditional")) (|iiacsch| ((|#2| |#2|) "\\spad{iiacsch(x)} should be local but conditional")) (|iiasech| ((|#2| |#2|) "\\spad{iiasech(x)} should be local but conditional")) (|iiacoth| ((|#2| |#2|) "\\spad{iiacoth(x)} should be local but conditional")) (|iiatanh| ((|#2| |#2|) "\\spad{iiatanh(x)} should be local but conditional")) (|iiacosh| ((|#2| |#2|) "\\spad{iiacosh(x)} should be local but conditional")) (|iiasinh| ((|#2| |#2|) "\\spad{iiasinh(x)} should be local but conditional")) (|iicsch| ((|#2| |#2|) "\\spad{iicsch(x)} should be local but conditional")) (|iisech| ((|#2| |#2|) "\\spad{iisech(x)} should be local but conditional")) (|iicoth| ((|#2| |#2|) "\\spad{iicoth(x)} should be local but conditional")) (|iitanh| ((|#2| |#2|) "\\spad{iitanh(x)} should be local but conditional")) (|iicosh| ((|#2| |#2|) "\\spad{iicosh(x)} should be local but conditional")) (|iisinh| ((|#2| |#2|) "\\spad{iisinh(x)} should be local but conditional")) (|iiacsc| ((|#2| |#2|) "\\spad{iiacsc(x)} should be local but conditional")) (|iiasec| ((|#2| |#2|) "\\spad{iiasec(x)} should be local but conditional")) (|iiacot| ((|#2| |#2|) "\\spad{iiacot(x)} should be local but conditional")) (|iiatan| ((|#2| |#2|) "\\spad{iiatan(x)} should be local but conditional")) (|iiacos| ((|#2| |#2|) "\\spad{iiacos(x)} should be local but conditional")) (|iiasin| ((|#2| |#2|) "\\spad{iiasin(x)} should be local but conditional")) (|iicsc| ((|#2| |#2|) "\\spad{iicsc(x)} should be local but conditional")) (|iisec| ((|#2| |#2|) "\\spad{iisec(x)} should be local but conditional")) (|iicot| ((|#2| |#2|) "\\spad{iicot(x)} should be local but conditional")) (|iitan| ((|#2| |#2|) "\\spad{iitan(x)} should be local but conditional")) (|iicos| ((|#2| |#2|) "\\spad{iicos(x)} should be local but conditional")) (|iisin| ((|#2| |#2|) "\\spad{iisin(x)} should be local but conditional")) (|iilog| ((|#2| |#2|) "\\spad{iilog(x)} should be local but conditional")) (|iiexp| ((|#2| |#2|) "\\spad{iiexp(x)} should be local but conditional")) (|iisqrt3| ((|#2|) "\\spad{iisqrt3()} should be local but conditional")) (|iisqrt2| ((|#2|) "\\spad{iisqrt2()} should be local but conditional")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(p)} returns an elementary operator with the same symbol as \\spad{p}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(p)} returns \\spad{true} if operator \\spad{p} is elementary")) (|pi| ((|#2|) "\\spad{pi()} returns the \\spad{pi} operator")) (|acsch| ((|#2| |#2|) "\\spad{acsch(x)} applies the inverse hyperbolic cosecant operator to \\spad{x}")) (|asech| ((|#2| |#2|) "\\spad{asech(x)} applies the inverse hyperbolic secant operator to \\spad{x}")) (|acoth| ((|#2| |#2|) "\\spad{acoth(x)} applies the inverse hyperbolic cotangent operator to \\spad{x}")) (|atanh| ((|#2| |#2|) "\\spad{atanh(x)} applies the inverse hyperbolic tangent operator to \\spad{x}")) (|acosh| ((|#2| |#2|) "\\spad{acosh(x)} applies the inverse hyperbolic cosine operator to \\spad{x}")) (|asinh| ((|#2| |#2|) "\\spad{asinh(x)} applies the inverse hyperbolic sine operator to \\spad{x}")) (|csch| ((|#2| |#2|) "\\spad{csch(x)} applies the hyperbolic cosecant operator to \\spad{x}")) (|sech| ((|#2| |#2|) "\\spad{sech(x)} applies the hyperbolic secant operator to \\spad{x}")) (|coth| ((|#2| |#2|) "\\spad{coth(x)} applies the hyperbolic cotangent operator to \\spad{x}")) (|tanh| ((|#2| |#2|) "\\spad{tanh(x)} applies the hyperbolic tangent operator to \\spad{x}")) (|cosh| ((|#2| |#2|) "\\spad{cosh(x)} applies the hyperbolic cosine operator to \\spad{x}")) (|sinh| ((|#2| |#2|) "\\spad{sinh(x)} applies the hyperbolic sine operator to \\spad{x}")) (|acsc| ((|#2| |#2|) "\\spad{acsc(x)} applies the inverse cosecant operator to \\spad{x}")) (|asec| ((|#2| |#2|) "\\spad{asec(x)} applies the inverse secant operator to \\spad{x}")) (|acot| ((|#2| |#2|) "\\spad{acot(x)} applies the inverse cotangent operator to \\spad{x}")) (|atan| ((|#2| |#2|) "\\spad{atan(x)} applies the inverse tangent operator to \\spad{x}")) (|acos| ((|#2| |#2|) "\\spad{acos(x)} applies the inverse cosine operator to \\spad{x}")) (|asin| ((|#2| |#2|) "\\spad{asin(x)} applies the inverse sine operator to \\spad{x}")) (|csc| ((|#2| |#2|) "\\spad{csc(x)} applies the cosecant operator to \\spad{x}")) (|sec| ((|#2| |#2|) "\\spad{sec(x)} applies the secant operator to \\spad{x}")) (|cot| ((|#2| |#2|) "\\spad{cot(x)} applies the cotangent operator to \\spad{x}")) (|tan| ((|#2| |#2|) "\\spad{tan(x)} applies the tangent operator to \\spad{x}")) (|cos| ((|#2| |#2|) "\\spad{cos(x)} applies the cosine operator to \\spad{x}")) (|sin| ((|#2| |#2|) "\\spad{sin(x)} applies the sine operator to \\spad{x}")) (|log| ((|#2| |#2|) "\\spad{log(x)} applies the logarithm operator to \\spad{x}")) (|exp| ((|#2| |#2|) "\\spad{exp(x)} applies the exponential operator to \\spad{x}"))) NIL NIL -(-285 R -3029) +(-285 R -3027) ((|constructor| (NIL "ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions,{} using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.")) (|tanQ| ((|#2| (|Fraction| (|Integer|)) |#2|) "\\spad{tanQ(q,a)} is a local function with a conditional implementation.")) (|rootNormalize| ((|#2| |#2| (|Kernel| |#2|)) "\\spad{rootNormalize(f, k)} returns \\spad{f} rewriting either \\spad{k} which must be an \\spad{n}th-root in terms of radicals already in \\spad{f},{} or some radicals in \\spad{f} in terms of \\spad{k}.")) (|validExponential| (((|Union| |#2| "failed") (|List| (|Kernel| |#2|)) |#2| (|Symbol|)) "\\spad{validExponential([k1,...,kn],f,x)} returns \\spad{g} if \\spad{exp(f)=g} and \\spad{g} involves only \\spad{k1...kn},{} and \"failed\" otherwise.")) (|realElementary| ((|#2| |#2| (|Symbol|)) "\\spad{realElementary(f,x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log, exp, tan, atan}.") ((|#2| |#2|) "\\spad{realElementary(f)} rewrites \\spad{f} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log, exp, tan, atan}.")) (|rischNormalize| (((|Record| (|:| |func| |#2|) (|:| |kers| (|List| (|Kernel| |#2|))) (|:| |vals| (|List| |#2|))) |#2| (|Symbol|)) "\\spad{rischNormalize(f, x)} returns \\spad{[g, [k1,...,kn], [h1,...,hn]]} such that \\spad{g = normalize(f, x)} and each \\spad{ki} was rewritten as \\spad{hi} during the normalization.")) (|normalize| ((|#2| |#2| (|Symbol|)) "\\spad{normalize(f, x)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{normalize(f)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels."))) NIL NIL @@ -1120,7 +1120,7 @@ NIL ((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\\em Dom} to an image domain {\\em Im}.")) (|qsetelt!| ((|#2| $ |#1| |#2|) "\\spad{qsetelt!(u,x,y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#2| $ |#1| |#2|) "\\spad{setelt(u,x,y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#2| $ |#1|) "\\spad{qelt(u, x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#2| $ |#1| |#2|) "\\spad{elt(u, x, y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) NIL NIL -(-298 S R |Mod| -4256 -3794 |exactQuo|) +(-298 S R |Mod| -1745 -4167 |exactQuo|) ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{ModularRing},{} \\spadtype{ModularField}")) (|inv| (($ $) "\\spad{inv(x)} \\undocumented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} \\undocumented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#2| |#3|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#2| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#3| $) "\\spad{modulus(x)} \\undocumented"))) ((-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL @@ -1142,21 +1142,21 @@ NIL NIL (-303 S) ((|constructor| (NIL "Equations as mathematical objects. All properties of the basis domain,{} \\spadignore{e.g.} being an abelian group are carried over the equation domain,{} by performing the structural operations on the left and on the right hand side.")) (|subst| (($ $ $) "\\spad{subst(eq1,eq2)} substitutes \\spad{eq2} into both sides of \\spad{eq1} the \\spad{lhs} of \\spad{eq2} should be a kernel")) (|inv| (($ $) "\\spad{inv(x)} returns the multiplicative inverse of \\spad{x}.")) (/ (($ $ $) "\\spad{e1/e2} produces a new equation by dividing the left and right hand sides of equations e1 and e2.")) (|factorAndSplit| (((|List| $) $) "\\spad{factorAndSplit(eq)} make the right hand side 0 and factors the new left hand side. Each factor is equated to 0 and put into the resulting list without repetitions.")) (|rightOne| (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side.") (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side,{} if possible.")) (|leftOne| (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side.") (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side,{} if possible.")) (* (($ $ |#1|) "\\spad{eqn*x} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.") (($ |#1| $) "\\spad{x*eqn} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.")) (- (($ $ |#1|) "\\spad{eqn-x} produces a new equation by subtracting \\spad{x} from both sides of equation eqn.") (($ |#1| $) "\\spad{x-eqn} produces a new equation by subtracting both sides of equation eqn from \\spad{x}.")) (|rightZero| (($ $) "\\spad{rightZero(eq)} subtracts the right hand side.")) (|leftZero| (($ $) "\\spad{leftZero(eq)} subtracts the left hand side.")) (+ (($ $ |#1|) "\\spad{eqn+x} produces a new equation by adding \\spad{x} to both sides of equation eqn.") (($ |#1| $) "\\spad{x+eqn} produces a new equation by adding \\spad{x} to both sides of equation eqn.")) (|eval| (($ $ (|List| $)) "\\spad{eval(eqn, [x1=v1, ... xn=vn])} replaces \\spad{xi} by \\spad{vi} in equation \\spad{eqn}.") (($ $ $) "\\spad{eval(eqn, x=f)} replaces \\spad{x} by \\spad{f} in equation \\spad{eqn}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,eqn)} constructs a new equation by applying \\spad{f} to both sides of \\spad{eqn}.")) (|rhs| ((|#1| $) "\\spad{rhs(eqn)} returns the right hand side of equation \\spad{eqn}.")) (|lhs| ((|#1| $) "\\spad{lhs(eqn)} returns the left hand side of equation \\spad{eqn}.")) (|swap| (($ $) "\\spad{swap(eq)} interchanges left and right hand side of equation \\spad{eq}.")) (|equation| (($ |#1| |#1|) "\\spad{equation(a,b)} creates an equation.")) 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Thus keys are considered equal only if they are the same instance of a structure."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) (-305) ((|constructor| (NIL "ErrorFunctions implements error functions callable from the system interpreter. Typically,{} these functions would be called in user functions. The simple forms of the functions take one argument which is either a string (an error message) or a list of strings which all together make up a message. The list can contain formatting codes (see below). The more sophisticated versions takes two arguments where the first argument is the name of the function from which the error was invoked and the second argument is either a string or a list of strings,{} as above. When you use the one argument version in an interpreter function,{} the system will automatically insert the name of the function as the new first argument. Thus in the user interpreter function \\indented{2}{\\spad{f x == if x < 0 then error \"negative argument\" else x}} the call to error will actually be of the form \\indented{2}{\\spad{error(\"f\",\"negative argument\")}} because the interpreter will have created a new first argument. \\blankline Formatting codes: error messages may contain the following formatting codes (they should either start or end a string or else have blanks around them): \\indented{3}{\\spad{\\%l}\\space{6}start a new line} \\indented{3}{\\spad{\\%b}\\space{6}start printing in a bold font (where available)} \\indented{3}{\\spad{\\%d}\\space{6}stop\\space{2}printing in a bold font (where available)} \\indented{3}{\\spad{ \\%ceon}\\space{2}start centering message lines} \\indented{3}{\\spad{\\%ceoff}\\space{2}stop\\space{2}centering message lines} \\indented{3}{\\spad{\\%rjon}\\space{3}start displaying lines \"ragged left\"} \\indented{3}{\\spad{\\%rjoff}\\space{2}stop\\space{2}displaying lines \"ragged left\"} \\indented{3}{\\spad{\\%i}\\space{6}indent\\space{3}following lines 3 additional spaces} \\indented{3}{\\spad{\\%u}\\space{6}unindent following lines 3 additional spaces} \\indented{3}{\\spad{\\%xN}\\space{5}insert \\spad{N} blanks (eg,{} \\spad{\\%x10} inserts 10 blanks)} \\blankline")) (|error| (((|Exit|) (|String|) (|List| (|String|))) "\\spad{error(nam,lmsg)} displays error messages \\spad{lmsg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|String|) (|String|)) "\\spad{error(nam,msg)} displays error message \\spad{msg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|List| (|String|))) "\\spad{error(lmsg)} displays error message \\spad{lmsg} and terminates.") (((|Exit|) (|String|)) "\\spad{error(msg)} displays error message \\spad{msg} and terminates."))) NIL NIL -(-306 -3029 S) +(-306 -3027 S) ((|constructor| (NIL "This package allows a map from any expression space into any object to be lifted to a kernel over the expression set,{} using a given property of the operator of the kernel.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|String|) (|Kernel| |#1|)) "\\spad{map(f, p, k)} uses the property \\spad{p} of the operator of \\spad{k},{} in order to lift \\spad{f} and apply it to \\spad{k}."))) NIL NIL -(-307 E -3029) +(-307 E -3027) ((|constructor| (NIL "This package allows a mapping \\spad{E} \\spad{->} \\spad{F} to be lifted to a kernel over \\spad{E}; This lifting can fail if the operator of the kernel cannot be applied in \\spad{F}; Do not use this package with \\spad{E} = \\spad{F},{} since this may drop some properties of the operators.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|Kernel| |#1|)) "\\spad{map(f, k)} returns \\spad{g = op(f(a1),...,f(an))} where \\spad{k = op(a1,...,an)}."))) NIL NIL @@ -1204,7 +1204,7 @@ NIL ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions.")) (|eval| (($ $ (|List| (|Equation| |#1|))) "\\spad{eval(f, [x1 = v1,...,xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#1|)) "\\spad{eval(f,x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) NIL NIL -(-319 -3029) +(-319 -3027) ((|constructor| (NIL "This package is to be used in conjuction with \\indented{12}{the CycleIndicators package. It provides an evaluation} \\indented{12}{function for SymmetricPolynomials.}")) (|eval| ((|#1| (|Mapping| |#1| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{eval(f,s)} evaluates the cycle index \\spad{s} by applying \\indented{1}{the function \\spad{f} to each integer in a monomial partition,{}} \\indented{1}{forms their product and sums the results over all monomials.}"))) NIL NIL @@ -1219,7 +1219,7 @@ NIL (-322 R FE |var| |cen|) ((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent essential singularities of functions. Objects in this domain are quotients of sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) "\\spad{coerce(f)} converts a \\spadtype{UnivariatePuiseuxSeries} to an \\spadtype{ExponentialExpansion}.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> a+,f(var))}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . 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(LIST (QUOTE -1271) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (LIST (QUOTE -318) (LIST (QUOTE -1271) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (LIST (QUOTE -295) (LIST (QUOTE -1271) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)) (LIST (QUOTE -1271) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (QUOTE (-316))) (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (QUOTE (-556))) (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (QUOTE (-861))) (-12 (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (QUOTE (-924))) (|HasCategory| $ (QUOTE (-146)))) (-3763 (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (QUOTE (-146))) (-12 (|HasCategory| (-1271 |#1| |#2| |#3| |#4|) (QUOTE (-924))) (|HasCategory| $ (QUOTE (-146)))))) (-323 R S) ((|constructor| (NIL "Lifting of maps to Expressions. Date Created: 16 Jan 1989 Date Last Updated: 22 Jan 1990")) (|map| (((|Expression| |#2|) (|Mapping| |#2| |#1|) (|Expression| |#1|)) "\\spad{map(f, e)} applies \\spad{f} to all the constants appearing in \\spad{e}."))) NIL @@ -1230,9 +1230,9 @@ NIL NIL (-325 R) ((|constructor| (NIL "Expressions involving symbolic functions.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} \\undocumented{}")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} \\undocumented{}")) (|simplifyPower| (($ $ (|Integer|)) "simplifyPower?(\\spad{f},{}\\spad{n}) \\undocumented{}")) (|number?| (((|Boolean|) $) "\\spad{number?(f)} tests if \\spad{f} is rational")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic quantities present in \\spad{f} by applying their defining relations."))) -((-4457 -3765 (-12 (|has| |#1| (-567)) (-3765 (|has| |#1| (-1066)) (|has| |#1| (-484)))) (|has| |#1| (-1066)) (|has| |#1| (-484))) (-4455 |has| |#1| (-174)) (-4454 |has| |#1| (-174)) ((-4462 "*") |has| |#1| (-567)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-567)) (-4452 |has| |#1| (-567))) -((-3765 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))))) (|HasCategory| |#1| (QUOTE (-567))) (-3765 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (QUOTE (-21))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (QUOTE (-1066))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (LIST (QUOTE -650) 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|#1| (-567)) (-4458 |has| |#1| (-567)) (-4452 |has| |#1| (-567))) +((-3763 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))))) (|HasCategory| |#1| (QUOTE (-567))) (-3763 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (QUOTE (-21))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (QUOTE (-1066))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))))) (-3763 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#1| (QUOTE (-1129)))) (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (-3763 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-1066)))) (-3763 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-1066)))) (-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-1066)))) (-12 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567)))) (-3763 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (-3763 (|HasCategory| |#1| (QUOTE (-21))) (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))))) (-3763 (|HasCategory| |#1| (QUOTE (-25))) (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-1129)))) (-3763 (|HasCategory| |#1| (QUOTE (-25))) (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))))) (-3763 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#1| (QUOTE (-1066)))) (-3763 (-12 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1129))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| $ (QUOTE (-1066))) (|HasCategory| $ (LIST (QUOTE -1055) (QUOTE (-575))))) +(-326 R -3027) ((|constructor| (NIL "Taylor series solutions of explicit ODE\\spad{'s}.")) (|seriesSolve| (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq, y, x = a, [b0,...,bn])} is equivalent to \\spad{seriesSolve(eq = 0, y, x = a, [b0,...,b(n-1)])}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq, y, x = a, y a = b)} is equivalent to \\spad{seriesSolve(eq=0, y, x=a, y a = b)}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq, y, x = a, b)} is equivalent to \\spad{seriesSolve(eq = 0, y, x = a, y a = b)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq,y, x=a, b)} is equivalent to \\spad{seriesSolve(eq, y, x=a, y a = b)}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,[y1 a = b1,..., yn a = bn])} is equivalent to \\spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x = a, [y1 a = b1,..., yn a = bn])}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])} is equivalent to \\spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x=a, [b1,...,bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])} is equivalent to \\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a, [y1 a = b1,..., yn a = bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,...,eqn],[y1,...,yn],x = a,[y1 a = b1,...,yn a = bn])} returns a taylor series solution of \\spad{[eq1,...,eqn]} around \\spad{x = a} with initial conditions \\spad{yi(a) = bi}. Note: eqi must be of the form \\spad{fi(x, y1 x, y2 x,..., yn x) y1'(x) + gi(x, y1 x, y2 x,..., yn x) = h(x, y1 x, y2 x,..., yn x)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq,y,x=a,[b0,...,b(n-1)])} returns a Taylor series solution of \\spad{eq} around \\spad{x = a} with initial conditions \\spad{y(a) = b0},{} \\spad{y'(a) = b1},{} \\spad{y''(a) = b2},{} ...,{}\\spad{y(n-1)(a) = b(n-1)} \\spad{eq} must be of the form \\spad{f(x, y x, y'(x),..., y(n-1)(x)) y(n)(x) + g(x,y x,y'(x),...,y(n-1)(x)) = h(x,y x, y'(x),..., y(n-1)(x))}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq,y,x=a, y a = b)} returns a Taylor series solution of \\spad{eq} around \\spad{x} = a with initial condition \\spad{y(a) = b}. Note: \\spad{eq} must be of the form \\spad{f(x, y x) y'(x) + g(x, y x) = h(x, y x)}."))) NIL NIL @@ -1243,7 +1243,7 @@ NIL (-328 FE |var| |cen|) ((|constructor| (NIL "ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form \\spad{exp(f(x))},{} where \\spad{f(x)} is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity,{} with functions which tend more rapidly to zero or infinity considered to be larger. Thus,{} if \\spad{order(f(x)) < order(g(x))},{} \\spadignore{i.e.} the first non-zero term of \\spad{f(x)} has lower degree than the first non-zero term of \\spad{g(x)},{} then \\spad{exp(f(x)) > exp(g(x))}. If \\spad{order(f(x)) = order(g(x))},{} then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.")) (|exponentialOrder| (((|Fraction| (|Integer|)) $) "\\spad{exponentialOrder(exp(c * x **(-n) + ...))} returns \\spad{-n}. exponentialOrder(0) returns \\spad{0}.")) (|exponent| (((|UnivariatePuiseuxSeries| |#1| |#2| |#3|) $) "\\spad{exponent(exp(f(x)))} returns \\spad{f(x)}")) (|exponential| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{exponential(f(x))} returns \\spad{exp(f(x))}. Note: the function does NOT check that \\spad{f(x)} has no non-negative terms."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|)))) (|HasCategory| (-418 (-575)) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-3765 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasSignature| |#1| (LIST (QUOTE -2883) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3765 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4413) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|)))) (|HasCategory| (-418 (-575)) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasSignature| |#1| (LIST (QUOTE -2882) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3763 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4388) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) (-329 M) ((|constructor| (NIL "computes various functions on factored arguments.")) (|log| (((|List| (|Record| (|:| |coef| (|NonNegativeInteger|)) (|:| |logand| |#1|))) (|Factored| |#1|)) "\\spad{log(f)} returns \\spad{[(a1,b1),...,(am,bm)]} such that the logarithm of \\spad{f} is equal to \\spad{a1*log(b1) + ... + am*log(bm)}.")) (|nthRoot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#1|) (|:| |radicand| (|List| |#1|))) (|Factored| |#1|) (|NonNegativeInteger|)) "\\spad{nthRoot(f, n)} returns \\spad{(p, r, [r1,...,rm])} such that the \\spad{n}th-root of \\spad{f} is equal to \\spad{r * \\spad{p}th-root(r1 * ... * rm)},{} where \\spad{r1},{}...,{}\\spad{rm} are distinct factors of \\spad{f},{} each of which has an exponent smaller than \\spad{p} in \\spad{f}."))) NIL @@ -1275,12 +1275,12 @@ NIL (-336 S) ((|constructor| (NIL "\\indented{1}{A FlexibleArray is the notion of an array intended to allow for growth} at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) -(-337 S -3029) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +(-337 S -3027) ((|constructor| (NIL "FiniteAlgebraicExtensionField {\\em F} is the category of fields which are finite algebraic extensions of the field {\\em F}. If {\\em F} is finite then any finite algebraic extension of {\\em F} is finite,{} too. Let {\\em K} be a finite algebraic extension of the finite field {\\em F}. The exponentiation of elements of {\\em K} defines a \\spad{Z}-module structure on the multiplicative group of {\\em K}. The additive group of {\\em K} becomes a module over the ring of polynomials over {\\em F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em K},{} {\\em c,d} from {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)} where {\\em q=size()\\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#2|) "failed") $ $) "\\spad{linearAssociatedLog(b,a)} returns a polynomial {\\em g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals {\\em a}. If there is no such polynomial {\\em g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial {\\em g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals {\\em a}.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\\em g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#2|)) "\\spad{linearAssociatedExp(a,f)} is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em \\$},{} {\\em c,d} form {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)},{} where {\\em q=size()\\$F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: \\spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}.") ((|#2| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: norm(a,{}\\spad{d}) = reduce(*,{}[a**(\\spad{q**}(d*i)) for \\spad{i} in 0..\\spad{n/d}])") ((|#2| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#2|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\spad{\\$} as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\spad{\\$} as \\spad{F}-vectorspace."))) NIL ((|HasCategory| |#2| (QUOTE (-378)))) -(-338 -3029) +(-338 -3027) ((|constructor| (NIL "FiniteAlgebraicExtensionField {\\em F} is the category of fields which are finite algebraic extensions of the field {\\em F}. If {\\em F} is finite then any finite algebraic extension of {\\em F} is finite,{} too. Let {\\em K} be a finite algebraic extension of the finite field {\\em F}. The exponentiation of elements of {\\em K} defines a \\spad{Z}-module structure on the multiplicative group of {\\em K}. The additive group of {\\em K} becomes a module over the ring of polynomials over {\\em F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em K},{} {\\em c,d} from {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)} where {\\em q=size()\\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") $ $) "\\spad{linearAssociatedLog(b,a)} returns a polynomial {\\em g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals {\\em a}. If there is no such polynomial {\\em g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial {\\em g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals {\\em a}.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\\em g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#1|)) "\\spad{linearAssociatedExp(a,f)} is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em \\$},{} {\\em c,d} form {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)},{} where {\\em q=size()\\$F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: \\spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}.") ((|#1| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: norm(a,{}\\spad{d}) = reduce(*,{}[a**(\\spad{q**}(d*i)) for \\spad{i} in 0..\\spad{n/d}])") ((|#1| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#1|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\spad{\\$} as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\spad{\\$} as \\spad{F}-vectorspace."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL @@ -1304,15 +1304,15 @@ NIL ((|constructor| (NIL "\\indented{1}{Lift a map to finite divisors.} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 19 May 1993")) (|map| (((|FiniteDivisor| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{map(f,d)} \\undocumented{}"))) NIL NIL -(-344 S -3029 UP UPUP R) +(-344 S -3027 UP UPUP R) ((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|generator| (((|Union| |#5| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) (|:| |principalPart| |#5|)) $) "\\spad{decompose(d)} returns \\spad{[id, f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#5| |#3| |#3| |#3| |#2|) "\\spad{divisor(h, d, d', g, r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,discriminant)} contains the ramified zeros of \\spad{d}") (($ |#2| |#2| (|Integer|)) "\\spad{divisor(a, b, n)} makes the divisor \\spad{nP} where \\spad{P:} \\spad{(x = a, y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#2| |#2|) "\\spad{divisor(a, b)} makes the divisor \\spad{P:} \\spad{(x = a, y = b)}. Error: if \\spad{P} is singular.") (($ |#5|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) NIL NIL -(-345 -3029 UP UPUP R) +(-345 -3027 UP UPUP R) ((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|generator| (((|Union| |#4| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) (|:| |principalPart| |#4|)) $) "\\spad{decompose(d)} returns \\spad{[id, f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#4| |#2| |#2| |#2| |#1|) "\\spad{divisor(h, d, d', g, r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,discriminant)} contains the ramified zeros of \\spad{d}") (($ |#1| |#1| (|Integer|)) "\\spad{divisor(a, b, n)} makes the divisor \\spad{nP} where \\spad{P:} \\spad{(x = a, y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#1| |#1|) "\\spad{divisor(a, b)} makes the divisor \\spad{P:} \\spad{(x = a, y = b)}. Error: if \\spad{P} is singular.") (($ |#4|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) NIL NIL -(-346 -3029 UP UPUP R) +(-346 -3027 UP UPUP R) ((|constructor| (NIL "This domains implements finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|lSpaceBasis| (((|Vector| |#4|) $) "\\spad{lSpaceBasis(d)} returns a basis for \\spad{L(d) = {f | (f) >= -d}} as a module over \\spad{K[x]}.")) (|finiteBasis| (((|Vector| |#4|) $) "\\spad{finiteBasis(d)} returns a basis for \\spad{d} as a module over {\\em K[x]}."))) NIL NIL @@ -1332,26 +1332,26 @@ NIL ((|constructor| (NIL "Lifts a map from rings to function fields over them.")) (|map| ((|#8| (|Mapping| |#5| |#1|) |#4|) "\\spad{map(f, p)} lifts \\spad{f} to \\spad{F1} and applies it to \\spad{p}."))) NIL NIL -(-351 S -3029 UP UPUP) +(-351 S -3027 UP UPUP) ((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#2|))) "\\spad{rationalPoints()} returns the list of all the affine rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#2|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in u1,{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#3|) (|:| |derivden| |#3|) (|:| |gd| |#3|)) $ (|Mapping| |#3| |#3|)) "\\spad{algSplitSimple(f, D)} returns \\spad{[h,d,d',g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d, discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#3| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#3| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#2| $ |#2| |#2|) "\\spad{elt(f,a,b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a, y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#3| |#3|)) "\\spad{differentiate(x, d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#3|)) (|:| |den| |#3|)) (|Mapping| |#3| |#3|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(wi)} with respect to \\spad{(w1,...,wn)} where \\spad{(w1,...,wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#3|) |#3|) "\\spad{integralRepresents([A1,...,An], D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,...,wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,...,wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#3|) |#3|) "\\spad{represents([A0,...,A(n-1)],D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\spad{inverseIntegralMatrixAtInfinity()} returns \\spad{M} such that \\spad{M (v1,...,vn) = (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\spad{integralMatrixAtInfinity()} returns \\spad{M} such that \\spad{(v1,...,vn) = M (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\spad{inverseIntegralMatrix()} returns \\spad{M} such that \\spad{M (w1,...,wn) = (1, y, ..., y**(n-1))} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|integralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\spad{integralMatrix()} returns \\spad{M} such that \\spad{(w1,...,wn) = M (1, y, ..., y**(n-1))},{} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,...,bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,...,bn)} returns the complementary basis \\spad{(b1',...,bn')} of \\spad{(b1,...,bn)}.")) (|integral?| (((|Boolean|) $ |#3|) "\\spad{integral?(f, p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}.") (((|Boolean|) $ |#2|) "\\spad{integral?(f, a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.")) (|integralBasis| (((|Vector| $)) "\\spad{integralBasis()} returns the integral basis for the curve.")) (|ramified?| (((|Boolean|) |#3|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#2|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#3|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#2|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#3|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#2|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.")) (|rationalPoint?| (((|Boolean|) |#2| |#2|) "\\spad{rationalPoint?(a, b)} tests if \\spad{(x=a,y=b)} is on the curve.")) (|absolutelyIrreducible?| (((|Boolean|)) "\\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?")) (|genus| (((|NonNegativeInteger|)) "\\spad{genus()} returns the genus of one absolutely irreducible component")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\spad{numberOfComponents()} returns the number of absolutely irreducible components."))) NIL ((|HasCategory| |#2| (QUOTE (-378))) (|HasCategory| |#2| (QUOTE (-373)))) -(-352 -3029 UP UPUP) +(-352 -3027 UP UPUP) ((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#1|))) "\\spad{rationalPoints()} returns the list of all the affine rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in u1,{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#2|) (|:| |derivden| |#2|) (|:| |gd| |#2|)) $ (|Mapping| |#2| |#2|)) "\\spad{algSplitSimple(f, D)} returns \\spad{[h,d,d',g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d, discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#2| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#2| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#1| $ |#1| |#1|) "\\spad{elt(f,a,b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a, y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x, d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#2|)) (|:| |den| |#2|)) (|Mapping| |#2| |#2|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(wi)} with respect to \\spad{(w1,...,wn)} where \\spad{(w1,...,wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#2|) |#2|) "\\spad{integralRepresents([A1,...,An], D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,...,wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,...,wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#2|) |#2|) "\\spad{represents([A0,...,A(n-1)],D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\spad{inverseIntegralMatrixAtInfinity()} returns \\spad{M} such that \\spad{M (v1,...,vn) = (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\spad{integralMatrixAtInfinity()} returns \\spad{M} such that \\spad{(v1,...,vn) = M (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\spad{inverseIntegralMatrix()} returns \\spad{M} such that \\spad{M (w1,...,wn) = (1, y, ..., y**(n-1))} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|integralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\spad{integralMatrix()} returns \\spad{M} such that \\spad{(w1,...,wn) = M (1, y, ..., y**(n-1))},{} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,...,bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,...,bn)} returns the complementary basis \\spad{(b1',...,bn')} of \\spad{(b1,...,bn)}.")) (|integral?| (((|Boolean|) $ |#2|) "\\spad{integral?(f, p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}.") (((|Boolean|) $ |#1|) "\\spad{integral?(f, a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.")) (|integralBasis| (((|Vector| $)) "\\spad{integralBasis()} returns the integral basis for the curve.")) (|ramified?| (((|Boolean|) |#2|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#1|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#2|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#1|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#2|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#1|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.")) (|rationalPoint?| (((|Boolean|) |#1| |#1|) "\\spad{rationalPoint?(a, b)} tests if \\spad{(x=a,y=b)} is on the curve.")) (|absolutelyIrreducible?| (((|Boolean|)) "\\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?")) (|genus| (((|NonNegativeInteger|)) "\\spad{genus()} returns the genus of one absolutely irreducible component")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\spad{numberOfComponents()} returns the number of absolutely irreducible components."))) ((-4453 |has| (-418 |#2|) (-373)) (-4458 |has| (-418 |#2|) (-373)) (-4452 |has| (-418 |#2|) (-373)) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-353 |p| |extdeg|) ((|constructor| (NIL "FiniteFieldCyclicGroup(\\spad{p},{}\\spad{n}) implements a finite field extension of degee \\spad{n} over the prime field with \\spad{p} elements. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\\em createPrimitivePoly} from \\spadtype{FiniteFieldPolynomialPackage}. The Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| (-925 |#1|) (QUOTE (-146))) (|HasCategory| (-925 |#1|) (QUOTE (-378)))) (|HasCategory| (-925 |#1|) (QUOTE (-148))) (|HasCategory| (-925 |#1|) (QUOTE (-378))) (|HasCategory| (-925 |#1|) (QUOTE (-146)))) +((-3763 (|HasCategory| (-925 |#1|) (QUOTE (-146))) (|HasCategory| (-925 |#1|) (QUOTE (-378)))) (|HasCategory| (-925 |#1|) (QUOTE (-148))) (|HasCategory| (-925 |#1|) (QUOTE (-378))) (|HasCategory| (-925 |#1|) (QUOTE (-146)))) (-354 GF |defpol|) ((|constructor| (NIL "FiniteFieldCyclicGroupExtensionByPolynomial(\\spad{GF},{}defpol) implements a finite extension field of the ground field {\\em GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial {\\em defpol},{} which MUST be primitive (user responsibility). Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field it is used to perform additions in the field quickly."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) +((-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) (-355 GF |extdeg|) ((|constructor| (NIL "FiniteFieldCyclicGroupExtension(\\spad{GF},{}\\spad{n}) implements a extension of degree \\spad{n} over the ground field {\\em GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\\em createPrimitivePoly} from \\spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) +((-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) (-356 GF) ((|constructor| (NIL "FiniteFieldFunctions(\\spad{GF}) is a package with functions concerning finite extension fields of the finite ground field {\\em GF},{} \\spadignore{e.g.} Zech logarithms.")) (|createLowComplexityNormalBasis| (((|Union| (|SparseUnivariatePolynomial| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) (|PositiveInteger|)) "\\spad{createLowComplexityNormalBasis(n)} tries to find a a low complexity normal basis of degree {\\em n} over {\\em GF} and returns its multiplication matrix If no low complexity basis is found it calls \\axiomFunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}(\\spad{n}) to produce a normal polynomial of degree {\\em n} over {\\em GF}")) (|createLowComplexityTable| (((|Union| (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) "failed") (|PositiveInteger|)) "\\spad{createLowComplexityTable(n)} tries to find a low complexity normal basis of degree {\\em n} over {\\em GF} and returns its multiplication matrix Fails,{} if it does not find a low complexity basis")) (|sizeMultiplication| (((|NonNegativeInteger|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{sizeMultiplication(m)} returns the number of entries of the multiplication table {\\em m}.")) (|createMultiplicationMatrix| (((|Matrix| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{createMultiplicationMatrix(m)} forms the multiplication table {\\em m} into a matrix over the ground field.")) (|createMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createMultiplicationTable(f)} generates a multiplication table for the normal basis of the field extension determined by {\\em f}. This is needed to perform multiplications between elements represented as coordinate vectors to this basis. See \\spadtype{FFNBP},{} \\spadtype{FFNBX}.")) (|createZechTable| (((|PrimitiveArray| (|SingleInteger|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createZechTable(f)} generates a Zech logarithm table for the cyclic group representation of a extension of the ground field by the primitive polynomial {\\em f(x)},{} \\spadignore{i.e.} \\spad{Z(i)},{} defined by {\\em x**Z(i) = 1+x**i} is stored at index \\spad{i}. This is needed in particular to perform addition of field elements in finite fields represented in this way. See \\spadtype{FFCGP},{} \\spadtype{FFCGX}."))) NIL @@ -1368,31 +1368,31 @@ NIL ((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note: see \\spad{ch}.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of {\\em a}. Note: such a root is alway defined in finite fields."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL -(-360 R UP -3029) +(-360 R UP -3027) ((|constructor| (NIL "In this package \\spad{R} is a Euclidean domain and \\spad{F} is a framed algebra over \\spad{R}. The package provides functions to compute the integral closure of \\spad{R} in the quotient field of \\spad{F}. It is assumed that \\spad{char(R/P) = char(R)} for any prime \\spad{P} of \\spad{R}. A typical instance of this is when \\spad{R = K[x]} and \\spad{F} is a function field over \\spad{R}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) |#1|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) NIL NIL (-361 |p| |extdeg|) ((|constructor| (NIL "FiniteFieldNormalBasis(\\spad{p},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the prime field with \\spad{p} elements. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial created by \\spadfunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}.")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: The time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| (|PrimeField| |#1|))) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| (|PrimeField| |#1|)) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| (-925 |#1|) (QUOTE (-146))) (|HasCategory| (-925 |#1|) (QUOTE (-378)))) (|HasCategory| (-925 |#1|) (QUOTE (-148))) (|HasCategory| (-925 |#1|) (QUOTE (-378))) (|HasCategory| (-925 |#1|) (QUOTE (-146)))) +((-3763 (|HasCategory| (-925 |#1|) (QUOTE (-146))) (|HasCategory| (-925 |#1|) (QUOTE (-378)))) (|HasCategory| (-925 |#1|) (QUOTE (-148))) (|HasCategory| (-925 |#1|) (QUOTE (-378))) (|HasCategory| (-925 |#1|) (QUOTE (-146)))) (-362 GF |uni|) ((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(\\spad{GF},{}uni) implements a finite extension of the ground field {\\em GF}. The elements are represented by coordinate vectors with respect to. a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element,{} where \\spad{q} is the size of {\\em GF}. The normal element is chosen as a root of the extension polynomial,{} which MUST be normal over {\\em GF} (user responsibility)")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) +((-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) (-363 GF |extdeg|) ((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(\\spad{GF},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the ground field {\\em GF}. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial,{} created by {\\em createNormalPoly} from \\spadtype{FiniteFieldPolynomialPackage}")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) +((-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) (-364 |p| |n|) ((|constructor| (NIL "FiniteField(\\spad{p},{}\\spad{n}) implements finite fields with p**n elements. This packages checks that \\spad{p} is prime. For a non-checking version,{} see \\spadtype{InnerFiniteField}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| (-925 |#1|) (QUOTE (-146))) (|HasCategory| (-925 |#1|) (QUOTE (-378)))) (|HasCategory| (-925 |#1|) (QUOTE (-148))) (|HasCategory| (-925 |#1|) (QUOTE (-378))) (|HasCategory| (-925 |#1|) (QUOTE (-146)))) +((-3763 (|HasCategory| (-925 |#1|) (QUOTE (-146))) (|HasCategory| (-925 |#1|) (QUOTE (-378)))) (|HasCategory| (-925 |#1|) (QUOTE (-148))) (|HasCategory| (-925 |#1|) (QUOTE (-378))) (|HasCategory| (-925 |#1|) (QUOTE (-146)))) (-365 GF |defpol|) ((|constructor| (NIL "FiniteFieldExtensionByPolynomial(\\spad{GF},{} defpol) implements the extension of the finite field {\\em GF} generated by the extension polynomial {\\em defpol} which MUST be irreducible. Note: the user has the responsibility to ensure that {\\em defpol} is irreducible."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) -(-366 -3029 GF) +((-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) +(-366 -3027 GF) ((|constructor| (NIL "FiniteFieldPolynomialPackage2(\\spad{F},{}\\spad{GF}) exports some functions concerning finite fields,{} which depend on a finite field {\\em GF} and an algebraic extension \\spad{F} of {\\em GF},{} \\spadignore{e.g.} a zero of a polynomial over {\\em GF} in \\spad{F}.")) (|rootOfIrreduciblePoly| ((|#1| (|SparseUnivariatePolynomial| |#2|)) "\\spad{rootOfIrreduciblePoly(f)} computes one root of the monic,{} irreducible polynomial \\spad{f},{} which degree must divide the extension degree of {\\em F} over {\\em GF},{} \\spadignore{i.e.} \\spad{f} splits into linear factors over {\\em F}.")) (|Frobenius| ((|#1| |#1|) "\\spad{Frobenius(x)} \\undocumented{}")) (|basis| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{basis(n)} \\undocumented{}")) (|lookup| (((|PositiveInteger|) |#1|) "\\spad{lookup(x)} \\undocumented{}")) (|coerce| ((|#1| |#2|) "\\spad{coerce(x)} \\undocumented{}"))) NIL NIL @@ -1400,14 +1400,14 @@ NIL ((|constructor| (NIL "This package provides a number of functions for generating,{} counting and testing irreducible,{} normal,{} primitive,{} random polynomials over finite fields.")) (|reducedQPowers| (((|PrimitiveArray| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{reducedQPowers(f)} generates \\spad{[x,x**q,x**(q**2),...,x**(q**(n-1))]} reduced modulo \\spad{f} where \\spad{q = size()\\$GF} and \\spad{n = degree f}.")) (|leastAffineMultiple| (((|SparseUnivariatePolynomial| |#1|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{leastAffineMultiple(f)} computes the least affine polynomial which is divisible by the polynomial \\spad{f} over the finite field {\\em GF},{} \\spadignore{i.e.} a polynomial whose exponents are 0 or a power of \\spad{q},{} the size of {\\em GF}.")) (|random| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{random(m,n)}\\$FFPOLY(\\spad{GF}) generates a random monic polynomial of degree \\spad{d} over the finite field {\\em GF},{} \\spad{d} between \\spad{m} and \\spad{n}.") (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{random(n)}\\$FFPOLY(\\spad{GF}) generates a random monic polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|nextPrimitiveNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitiveNormalPoly(f)} yields the next primitive normal polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g} or,{} in case these numbers are equal,{} if the {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than this number for \\spad{g}. If these numbers are equals,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g},{} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are coefficients according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}. This operation is equivalent to nextNormalPrimitivePoly(\\spad{f}).")) (|nextNormalPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPrimitivePoly(f)} yields the next normal primitive polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g} or if {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than this number for \\spad{g}. Otherwise,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}. This operation is equivalent to nextPrimitiveNormalPoly(\\spad{f}).")) (|nextNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPoly(f)} yields the next normal polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than that for \\spad{g}. In case these numbers are equal,{} \\spad{f < g} if if the number of monomials of \\spad{f} is less that for \\spad{g} or if the list of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|nextPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitivePoly(f)} yields the next primitive polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g}. If these values are equal,{} then \\spad{f < g} if if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|nextIrreduciblePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextIrreduciblePoly(f)} yields the next monic irreducible polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than this number for \\spad{g}. If \\spad{f} and \\spad{g} have the same number of monomials,{} the lists of exponents are compared lexicographically. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|createPrimitiveNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitiveNormalPoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal and primitive polynomial of degree \\spad{n} over the field {\\em GF}. polynomial of degree \\spad{n} over the field {\\em GF}.")) (|createNormalPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal and primitive polynomial of degree \\spad{n} over the field {\\em GF}. Note: this function is equivalent to createPrimitiveNormalPoly(\\spad{n})")) (|createNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|createPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) generates a primitive polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|createIrreduciblePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createIrreduciblePoly(n)}\\$FFPOLY(\\spad{GF}) generates a monic irreducible univariate polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfNormalPoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfNormalPoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of normal polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfPrimitivePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of primitive polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfIrreduciblePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfIrreduciblePoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of monic irreducible univariate polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|normal?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{normal?(f)} tests whether the polynomial \\spad{f} over a finite field is normal,{} \\spadignore{i.e.} its roots are linearly independent over the field.")) (|primitive?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{primitive?(f)} tests whether the polynomial \\spad{f} over a finite field is primitive,{} \\spadignore{i.e.} all its roots are primitive."))) NIL NIL -(-368 -3029 FP FPP) +(-368 -3027 FP FPP) ((|constructor| (NIL "This package solves linear diophantine equations for Bivariate polynomials over finite fields")) (|solveLinearPolynomialEquation| (((|Union| (|List| |#3|) "failed") (|List| |#3|) |#3|) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists."))) NIL NIL (-369 GF |n|) ((|constructor| (NIL "FiniteFieldExtensionByPolynomial(\\spad{GF},{} \\spad{n}) implements an extension of the finite field {\\em GF} of degree \\spad{n} generated by the extension polynomial constructed by \\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage} from \\spadtype{FiniteFieldPolynomialPackage}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) +((-3763 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-378)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-146)))) (-370 R |ls|) ((|constructor| (NIL "This is just an interface between several packages and domains. The goal is to compute lexicographical Groebner bases of sets of polynomial with type \\spadtype{Polynomial R} by the {\\em FGLM} algorithm if this is possible (\\spadignore{i.e.} if the input system generates a zero-dimensional ideal).")) (|groebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|))) "\\axiom{groebner(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}}. If \\axiom{\\spad{lq1}} generates a zero-dimensional ideal then the {\\em FGLM} strategy is used,{} otherwise the {\\em Sugar} strategy is used.")) (|fglmIfCan| (((|Union| (|List| (|Polynomial| |#1|)) "failed") (|List| (|Polynomial| |#1|))) "\\axiom{fglmIfCan(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}} by using the {\\em FGLM} strategy,{} if \\axiom{zeroDimensional?(\\spad{lq1})} holds.")) (|zeroDimensional?| (((|Boolean|) (|List| (|Polynomial| |#1|))) "\\axiom{zeroDimensional?(\\spad{lq1})} returns \\spad{true} iff \\axiom{\\spad{lq1}} generates a zero-dimensional ideal \\spad{w}.\\spad{r}.\\spad{t}. the variables of \\axiom{\\spad{ls}}."))) NIL @@ -1486,7 +1486,7 @@ NIL NIL (-389) ((|constructor| (NIL "\\spadtype{Float} implements arbitrary precision floating point arithmetic. The number of significant digits of each operation can be set to an arbitrary value (the default is 20 decimal digits). The operation \\spad{float(mantissa,exponent,\\spadfunFrom{base}{FloatingPointSystem})} for integer \\spad{mantissa},{} \\spad{exponent} specifies the number \\spad{mantissa * \\spadfunFrom{base}{FloatingPointSystem} ** exponent} The underlying representation for floats is binary not decimal. The implications of this are described below. \\blankline The model adopted is that arithmetic operations are rounded to to nearest unit in the last place,{} that is,{} accurate to within \\spad{2**(-\\spadfunFrom{bits}{FloatingPointSystem})}. Also,{} the elementary functions and constants are accurate to one unit in the last place. A float is represented as a record of two integers,{} the mantissa and the exponent. The \\spadfunFrom{base}{FloatingPointSystem} of the representation is binary,{} hence a \\spad{Record(m:mantissa,e:exponent)} represents the number \\spad{m * 2 ** e}. Though it is not assumed that the underlying integers are represented with a binary \\spadfunFrom{base}{FloatingPointSystem},{} the code will be most efficient when this is the the case (this is \\spad{true} in most implementations of Lisp). The decision to choose the \\spadfunFrom{base}{FloatingPointSystem} to be binary has some unfortunate consequences. First,{} decimal numbers like 0.3 cannot be represented exactly. Second,{} there is a further loss of accuracy during conversion to decimal for output. To compensate for this,{} if \\spad{d} digits of precision are specified,{} \\spad{1 + ceiling(log2 d)} bits are used. Two numbers that are displayed identically may therefore be not equal. On the other hand,{} a significant efficiency loss would be incurred if we chose to use a decimal \\spadfunFrom{base}{FloatingPointSystem} when the underlying integer base is binary. \\blankline Algorithms used: For the elementary functions,{} the general approach is to apply identities so that the taylor series can be used,{} and,{} so that it will converge within \\spad{O( sqrt n )} steps. For example,{} using the identity \\spad{exp(x) = exp(x/2)**2},{} we can compute \\spad{exp(1/3)} to \\spad{n} digits of precision as follows. We have \\spad{exp(1/3) = exp(2 ** (-sqrt s) / 3) ** (2 ** sqrt s)}. The taylor series will converge in less than sqrt \\spad{n} steps and the exponentiation requires sqrt \\spad{n} multiplications for a total of \\spad{2 sqrt n} multiplications. Assuming integer multiplication costs \\spad{O( n**2 )} the overall running time is \\spad{O( sqrt(n) n**2 )}. This approach is the best known approach for precisions up to about 10,{}000 digits at which point the methods of Brent which are \\spad{O( log(n) n**2 )} become competitive. Note also that summing the terms of the taylor series for the elementary functions is done using integer operations. This avoids the overhead of floating point operations and results in efficient code at low precisions. This implementation makes no attempt to reuse storage,{} relying on the underlying system to do \\spadgloss{garbage collection}. \\spad{I} estimate that the efficiency of this package at low precisions could be improved by a factor of 2 if in-place operations were available. \\blankline Running times: in the following,{} \\spad{n} is the number of bits of precision \\indented{5}{\\spad{*},{} \\spad{/},{} \\spad{sqrt},{} \\spad{pi},{} \\spad{exp1},{} \\spad{log2},{} \\spad{log10}: \\spad{ O( n**2 )}} \\indented{5}{\\spad{exp},{} \\spad{log},{} \\spad{sin},{} \\spad{atan}:\\space{2}\\spad{ O( sqrt(n) n**2 )}} The other elementary functions are coded in terms of the ones above.")) (|outputSpacing| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputSpacing(n)} inserts a space after \\spad{n} (default 10) digits on output; outputSpacing(0) means no spaces are inserted.")) (|outputGeneral| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputGeneral(n)} sets the output mode to general notation with \\spad{n} significant digits displayed.") (((|Void|)) "\\spad{outputGeneral()} sets the output mode (default mode) to general notation; numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude.")) (|outputFixed| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFixed(n)} sets the output mode to fixed point notation,{} with \\spad{n} digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFixed()} sets the output mode to fixed point notation; the output will contain a decimal point.")) (|outputFloating| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFloating(n)} sets the output mode to floating (scientific) notation with \\spad{n} significant digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFloating()} sets the output mode to floating (scientific) notation,{} \\spadignore{i.e.} \\spad{mantissa * 10 exponent} is displayed as \\spad{0.mantissa E exponent}.")) (|atan| (($ $ $) "\\spad{atan(x,y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|exp1| (($) "\\spad{exp1()} returns exp 1: \\spad{2.7182818284...}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm for \\spad{x} to base 10.") (($) "\\spad{log10()} returns \\spad{ln 10}: \\spad{2.3025809299...}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm for \\spad{x} to base 2.") (($) "\\spad{log2()} returns \\spad{ln 2},{} \\spadignore{i.e.} \\spad{0.6931471805...}.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n, b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)},{} that is \\spad{|(r-f)/f| < b**(-n)}.") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(x,n)} adds \\spad{n} to the exponent of float \\spad{x}.")) (|relerror| (((|Integer|) $ $) "\\spad{relerror(x,y)} computes the absolute value of \\spad{x - y} divided by \\spad{y},{} when \\spad{y \\~= 0}.")) (|normalize| (($ $) "\\spad{normalize(x)} normalizes \\spad{x} at current precision.")) (** (($ $ $) "\\spad{x ** y} computes \\spad{exp(y log x)} where \\spad{x >= 0}.")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) -((-4443 . T) (-4451 . T) (-3494 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-4443 . T) (-4451 . T) (-3493 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-390 |Par|) ((|constructor| (NIL "\\indented{3}{This is a package for the approximation of real solutions for} systems of polynomial equations over the rational numbers. The results are expressed as either rational numbers or floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|realRoots| (((|List| |#1|) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{realRoots(rf, eps)} finds the real zeros of a univariate rational function with precision given by eps.") (((|List| (|List| |#1|)) (|List| (|Fraction| (|Polynomial| (|Integer|)))) (|List| (|Symbol|)) |#1|) "\\spad{realRoots(lp,lv,eps)} computes the list of the real solutions of the list \\spad{lp} of rational functions with rational coefficients with respect to the variables in \\spad{lv},{} with precision \\spad{eps}. Each solution is expressed as a list of numbers in order corresponding to the variables in \\spad{lv}.")) (|solve| (((|List| (|Equation| (|Polynomial| |#1|))) (|Equation| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(eq,eps)} finds all of the real solutions of the univariate equation \\spad{eq} of rational functions with respect to the unique variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{solve(p,eps)} finds all of the real solutions of the univariate rational function \\spad{p} with rational coefficients with respect to the unique variable appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Integer|))))) |#1|) "\\spad{solve(leq,eps)} finds all of the real solutions of the system \\spad{leq} of equationas of rational functions with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(lp,eps)} finds all of the real solutions of the system \\spad{lp} of rational functions over the rational numbers with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}."))) @@ -1540,7 +1540,7 @@ NIL ((|constructor| (NIL "Code to manipulate Fortran Output Stack")) (|topFortranOutputStack| (((|String|)) "\\spad{topFortranOutputStack()} returns the top element of the Fortran output stack")) (|pushFortranOutputStack| (((|Void|) (|String|)) "\\spad{pushFortranOutputStack(f)} pushes \\spad{f} onto the Fortran output stack") (((|Void|) (|FileName|)) "\\spad{pushFortranOutputStack(f)} pushes \\spad{f} onto the Fortran output stack")) (|popFortranOutputStack| (((|Void|)) "\\spad{popFortranOutputStack()} pops the Fortran output stack")) (|showFortranOutputStack| (((|Stack| (|String|))) "\\spad{showFortranOutputStack()} returns the Fortran output stack")) (|clearFortranOutputStack| (((|Stack| (|String|))) "\\spad{clearFortranOutputStack()} clears the Fortran output stack"))) NIL NIL -(-403 -3029 UP UPUP R) +(-403 -3027 UP UPUP R) ((|constructor| (NIL "\\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 11 Jul 1990")) (|order| (((|NonNegativeInteger|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{order(x)} \\undocumented"))) NIL NIL @@ -1568,8 +1568,8 @@ NIL ((|constructor| (NIL "\\axiomType{FortranProgram} allows the user to build and manipulate simple models of FORTRAN subprograms. These can then be transformed into actual FORTRAN notation.")) (|coerce| (($ (|Equation| (|Expression| (|Complex| (|Float|))))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|Float|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|Integer|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Expression| (|Complex| (|Float|)))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|Float|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|Integer|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Equation| (|Expression| (|MachineComplex|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|MachineFloat|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|MachineInteger|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Expression| (|MachineComplex|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|MachineFloat|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|MachineInteger|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(r)} \\undocumented{}") (($ (|List| (|FortranCode|))) "\\spad{coerce(lfc)} \\undocumented{}") (($ (|FortranCode|)) "\\spad{coerce(fc)} \\undocumented{}"))) NIL NIL -(-410 -3029 UP) -((|constructor| (NIL "\\indented{1}{Full partial fraction expansion of rational functions} Author: Manuel Bronstein Date Created: 9 December 1992 Date Last Updated: 6 October 1993 References: \\spad{M}.Bronstein & \\spad{B}.Salvy,{} \\indented{12}{Full Partial Fraction Decomposition of Rational Functions,{}} \\indented{12}{in Proceedings of ISSAC'93,{} Kiev,{} ACM Press.}")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(f, n)} returns the \\spad{n}-th derivative of \\spad{f}.") (($ $) "\\spad{D(f)} returns the derivative of \\spad{f}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(f, n)} returns the \\spad{n}-th derivative of \\spad{f}.") (($ $) "\\spad{differentiate(f)} returns the derivative of \\spad{f}.")) (|construct| (($ (|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|)))) "\\spad{construct(l)} is the inverse of fracPart.")) (|fracPart| (((|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|))) $) "\\spad{fracPart(f)} returns the list of summands of the fractional part of \\spad{f}.")) (|polyPart| ((|#2| $) "\\spad{polyPart(f)} returns the polynomial part of \\spad{f}.")) (|fullPartialFraction| (($ (|Fraction| |#2|)) "\\spad{fullPartialFraction(f)} returns \\spad{[p, [[j, Dj, Hj]...]]} such that \\spad{f = p(x) + \\sum_{[j,Dj,Hj] in l} \\sum_{Dj(a)=0} Hj(a)/(x - a)\\^j}.")) (+ (($ |#2| $) "\\spad{p + x} returns the sum of \\spad{p} and \\spad{x}"))) +(-410 -3027 UP) +((|constructor| (NIL "\\indented{1}{Full partial fraction expansion of rational functions} Author: Manuel Bronstein Date Created: 9 December 1992 Date Last Updated: 6 October 1993 References: \\spad{M}.Bronstein & \\spad{B}.Salvy,{} \\indented{12}{Full Partial Fraction Decomposition of Rational Functions,{}} \\indented{12}{in Proceedings of ISSAC'93,{} Kiev,{} ACM Press.}")) (|construct| (($ (|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|)))) "\\spad{construct(l)} is the inverse of fracPart.")) (|fracPart| (((|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|))) $) "\\spad{fracPart(f)} returns the list of summands of the fractional part of \\spad{f}.")) (|polyPart| ((|#2| $) "\\spad{polyPart(f)} returns the polynomial part of \\spad{f}.")) (|fullPartialFraction| (($ (|Fraction| |#2|)) "\\spad{fullPartialFraction(f)} returns \\spad{[p, [[j, Dj, Hj]...]]} such that \\spad{f = p(x) + \\sum_{[j,Dj,Hj] in l} \\sum_{Dj(a)=0} Hj(a)/(x - a)\\^j}.")) (+ (($ |#2| $) "\\spad{p + x} returns the sum of \\spad{p} and \\spad{x}"))) NIL NIL (-411 R) @@ -1590,7 +1590,7 @@ NIL ((|HasAttribute| |#1| (QUOTE -4443)) (|HasAttribute| |#1| (QUOTE -4451))) (-415) ((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline \\indented{2}{1: \\spadfunFrom{base}{FloatingPointSystem} of the \\spadfunFrom{exponent}{FloatingPointSystem}.} \\indented{9}{(actual implemenations are usually binary or decimal)} \\indented{2}{2: \\spadfunFrom{precision}{FloatingPointSystem} of the \\spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \\indented{2}{3: rounding error for operations} \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\\spad{\"+\"}) does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling\\spad{'s} precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling\\spad{'s} precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\spad{base()} returns the base of the \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note: \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,e,b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) -((-3494 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-3493 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-416 R S) ((|constructor| (NIL "\\spadtype{FactoredFunctions2} contains functions that involve factored objects whose underlying domains may not be the same. For example,{} \\spadfun{map} might be used to coerce an object of type \\spadtype{Factored(Integer)} to \\spadtype{Factored(Complex(Integer))}.")) (|map| (((|Factored| |#2|) (|Mapping| |#2| |#1|) (|Factored| |#1|)) "\\spad{map(fn,u)} is used to apply the function \\userfun{\\spad{fn}} to every factor of \\spadvar{\\spad{u}}. The new factored object will have all its information flags set to \"nil\". This function is used,{} for example,{} to coerce every factor base to another type."))) @@ -1603,7 +1603,7 @@ NIL (-418 S) ((|constructor| (NIL "Fraction takes an IntegralDomain \\spad{S} and produces the domain of Fractions with numerators and denominators from \\spad{S}. If \\spad{S} is also a GcdDomain,{} then \\spad{gcd}\\spad{'s} between numerator and denominator will be cancelled during all operations.")) (|canonical| ((|attribute|) "\\spad{canonical} means that equal elements are in fact identical."))) ((-4447 -12 (|has| |#1| (-6 -4458)) (|has| |#1| (-463)) (|has| |#1| (-6 -4447))) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (QUOTE (-1039))) (|HasCategory| |#1| (QUOTE (-831))) (-3765 (|HasCategory| |#1| (QUOTE (-831))) (|HasCategory| |#1| (QUOTE (-861)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-1169))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839))))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839))))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-556))) (-12 (|HasAttribute| |#1| (QUOTE -4458)) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-463)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) +((|HasCategory| |#1| (QUOTE (-924))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (QUOTE (-1039))) (|HasCategory| |#1| (QUOTE (-831))) (-3763 (|HasCategory| |#1| (QUOTE (-831))) (|HasCategory| |#1| (QUOTE (-861)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-1169))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839))))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839))))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-839)))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-556))) (-12 (|HasAttribute| |#1| (QUOTE -4458)) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-463)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-419 S R UP) ((|constructor| (NIL "A \\spadtype{FramedAlgebra} is a \\spadtype{FiniteRankAlgebra} together with a fixed \\spad{R}-module basis.")) (|regularRepresentation| (((|Matrix| |#2|) $) "\\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed basis.")) (|discriminant| ((|#2|) "\\spad{discriminant()} = determinant(traceMatrix()).")) (|traceMatrix| (((|Matrix| |#2|)) "\\spad{traceMatrix()} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr(vi * vj)} ),{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|convert| (($ (|Vector| |#2|)) "\\spad{convert([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.") (((|Vector| |#2|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) NIL @@ -1624,11 +1624,11 @@ NIL ((|constructor| (NIL "\\indented{1}{Lifting of morphisms to fractional ideals.} Author: Manuel Bronstein Date Created: 1 Feb 1989 Date Last Updated: 27 Feb 1990 Keywords: ideal,{} algebra,{} module.")) (|map| (((|FractionalIdeal| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{map(f,i)} \\undocumented{}"))) NIL NIL -(-424 R -3029 UP A) +(-424 R -3027 UP A) ((|constructor| (NIL "Fractional ideals in a framed algebra.")) (|randomLC| ((|#4| (|NonNegativeInteger|) (|Vector| |#4|)) "\\spad{randomLC(n,x)} should be local but conditional.")) (|minimize| (($ $) "\\spad{minimize(I)} returns a reduced set of generators for \\spad{I}.")) (|denom| ((|#1| $) "\\spad{denom(1/d * (f1,...,fn))} returns \\spad{d}.")) (|numer| (((|Vector| |#4|) $) "\\spad{numer(1/d * (f1,...,fn))} = the vector \\spad{[f1,...,fn]}.")) (|norm| ((|#2| $) "\\spad{norm(I)} returns the norm of the ideal \\spad{I}.")) (|basis| (((|Vector| |#4|) $) "\\spad{basis((f1,...,fn))} returns the vector \\spad{[f1,...,fn]}.")) (|ideal| (($ (|Vector| |#4|)) "\\spad{ideal([f1,...,fn])} returns the ideal \\spad{(f1,...,fn)}."))) ((-4457 . T)) NIL -(-425 R -3029 UP A |ibasis|) +(-425 R -3027 UP A |ibasis|) ((|constructor| (NIL "Module representation of fractional ideals.")) (|module| (($ (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{module(I)} returns \\spad{I} viewed has a module over \\spad{R}.") (($ (|Vector| |#4|)) "\\spad{module([f1,...,fn])} = the module generated by \\spad{(f1,...,fn)} over \\spad{R}.")) (|norm| ((|#2| $) "\\spad{norm(f)} returns the norm of the module \\spad{f}.")) (|basis| (((|Vector| |#4|) $) "\\spad{basis((f1,...,fn))} = the vector \\spad{[f1,...,fn]}."))) NIL ((|HasCategory| |#4| (LIST (QUOTE -1055) (|devaluate| |#2|)))) @@ -1647,7 +1647,7 @@ NIL (-429 R) ((|constructor| (NIL "\\spadtype{Factored} creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like multiplication and \\spad{gcd} are relatively easy to do. Others,{} like addition require somewhat more work,{} and unless the argument domain provides a factor function,{} the result may not be completely factored. Each object consists of a unit and a list of factors,{} where a factor has a member of \\spad{R} (the \"base\"),{} and exponent and a flag indicating what is known about the base. A flag may be one of \"nil\",{} \"sqfr\",{} \"irred\" or \"prime\",{} which respectively mean that nothing is known about the base,{} it is square-free,{} it is irreducible,{} or it is prime. The current restriction to integral domains allows simplification to be performed without worrying about multiplication order.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(u)} returns a rational number if \\spad{u} really is one,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(u)} assumes spadvar{\\spad{u}} is actually a rational number and does the conversion to rational number (see \\spadtype{Fraction Integer}).")) (|rational?| (((|Boolean|) $) "\\spad{rational?(u)} tests if \\spadvar{\\spad{u}} is actually a rational number (see \\spadtype{Fraction Integer}).")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps the function \\userfun{\\spad{fn}} across the factors of \\spadvar{\\spad{u}} and creates a new factored object. Note: this clears the information flags (sets them to \"nil\") because the effect of \\userfun{\\spad{fn}} is clearly not known in general.")) (|unitNormalize| (($ $) "\\spad{unitNormalize(u)} normalizes the unit part of the factorization. For example,{} when working with factored integers,{} this operation will ensure that the bases are all positive integers.")) (|unit| ((|#1| $) "\\spad{unit(u)} extracts the unit part of the factorization.")) (|flagFactor| (($ |#1| (|Integer|) (|Union| "nil" "sqfr" "irred" "prime")) "\\spad{flagFactor(base,exponent,flag)} creates a factored object with a single factor whose \\spad{base} is asserted to be properly described by the information \\spad{flag}.")) (|sqfrFactor| (($ |#1| (|Integer|)) "\\spad{sqfrFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be square-free (flag = \"sqfr\").")) (|primeFactor| (($ |#1| (|Integer|)) "\\spad{primeFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be prime (flag = \"prime\").")) (|numberOfFactors| (((|NonNegativeInteger|) $) "\\spad{numberOfFactors(u)} returns the number of factors in \\spadvar{\\spad{u}}.")) (|nthFlag| (((|Union| "nil" "sqfr" "irred" "prime") $ (|Integer|)) "\\spad{nthFlag(u,n)} returns the information flag of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} \"nil\" is returned.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(u,n)} returns the base of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 1 is returned. If \\spadvar{\\spad{u}} consists only of a unit,{} the unit is returned.")) (|nthExponent| (((|Integer|) $ (|Integer|)) "\\spad{nthExponent(u,n)} returns the exponent of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 0 is returned.")) (|irreducibleFactor| (($ |#1| (|Integer|)) "\\spad{irreducibleFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be irreducible (flag = \"irred\").")) (|factors| (((|List| (|Record| (|:| |factor| |#1|) (|:| |exponent| (|Integer|)))) $) "\\spad{factors(u)} returns a list of the factors in a form suitable for iteration. That is,{} it returns a list where each element is a record containing a base and exponent. The original object is the product of all the factors and the unit (which can be extracted by \\axiom{unit(\\spad{u})}).")) (|nilFactor| (($ |#1| (|Integer|)) "\\spad{nilFactor(base,exponent)} creates a factored object with a single factor with no information about the kind of \\spad{base} (flag = \"nil\").")) (|factorList| (((|List| (|Record| (|:| |flg| (|Union| "nil" "sqfr" "irred" "prime")) (|:| |fctr| |#1|) (|:| |xpnt| (|Integer|)))) $) "\\spad{factorList(u)} returns the list of factors with flags (for use by factoring code).")) (|makeFR| (($ |#1| (|List| (|Record| (|:| |flg| (|Union| "nil" "sqfr" "irred" "prime")) (|:| |fctr| |#1|) (|:| |xpnt| (|Integer|))))) "\\spad{makeFR(unit,listOfFactors)} creates a factored object (for use by factoring code).")) (|exponent| (((|Integer|) $) "\\spad{exponent(u)} returns the exponent of the first factor of \\spadvar{\\spad{u}},{} or 0 if the factored form consists solely of a unit.")) (|expand| ((|#1| $) "\\spad{expand(f)} multiplies the unit and factors together,{} yielding an \"unfactored\" object. Note: this is purposely not called \\spadfun{coerce} which would cause the interpreter to do this automatically."))) ((-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -318) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -295) (QUOTE $) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-1239))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-1239)))) (|HasCategory| |#1| (QUOTE (-1039))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-463)))) +((|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -318) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -295) (QUOTE $) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-1239))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-1239)))) (|HasCategory| |#1| (QUOTE (-1039))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-463)))) (-430 R) ((|constructor| (NIL "\\spadtype{FactoredFunctionUtilities} implements some utility functions for manipulating factored objects.")) (|mergeFactors| (((|Factored| |#1|) (|Factored| |#1|) (|Factored| |#1|)) "\\spad{mergeFactors(u,v)} is used when the factorizations of \\spadvar{\\spad{u}} and \\spadvar{\\spad{v}} are known to be disjoint,{} \\spadignore{e.g.} resulting from a content/primitive part split. Essentially,{} it creates a new factored object by multiplying the units together and appending the lists of factors.")) (|refine| (((|Factored| |#1|) (|Factored| |#1|) (|Mapping| (|Factored| |#1|) |#1|)) "\\spad{refine(u,fn)} is used to apply the function \\userfun{\\spad{fn}} to each factor of \\spadvar{\\spad{u}} and then build a new factored object from the results. For example,{} if \\spadvar{\\spad{u}} were created by calling \\spad{nilFactor(10,2)} then \\spad{refine(u,factor)} would create a factored object equal to that created by \\spad{factor(100)} or \\spad{primeFactor(2,2) * primeFactor(5,2)}."))) NIL @@ -1676,7 +1676,7 @@ NIL ((|constructor| (NIL "A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \\spadtype{Set} for an example.")) (|min| ((|#1| $) "\\spad{min(u)} returns the smallest element of aggregate \\spad{u}.")) (|max| ((|#1| $) "\\spad{max(u)} returns the largest element of aggregate \\spad{u}.")) (|universe| (($) "\\spad{universe()}\\$\\spad{D} returns the universal set for finite set aggregate \\spad{D}.")) (|complement| (($ $) "\\spad{complement(u)} returns the complement of the set \\spad{u},{} \\spadignore{i.e.} the set of all values not in \\spad{u}.")) (|cardinality| (((|NonNegativeInteger|) $) "\\spad{cardinality(u)} returns the number of elements of \\spad{u}. Note: \\axiom{cardinality(\\spad{u}) = \\#u}."))) ((-4460 . T) (-4450 . T) (-4461 . T)) NIL -(-437 R -3029) +(-437 R -3027) ((|constructor| (NIL "\\spadtype{FunctionSpaceComplexIntegration} provides functions for the indefinite integration of complex-valued functions.")) (|complexIntegrate| ((|#2| |#2| (|Symbol|)) "\\spad{complexIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable.")) (|internalIntegrate0| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate0 should} be a local function,{} but is conditional.")) (|internalIntegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable."))) NIL NIL @@ -1684,7 +1684,7 @@ NIL ((|constructor| (NIL "\\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:")) (|makeCos| (($ |#2| |#1|) "\\spad{makeCos(e,r)} makes a sin expression with given argument and coefficient")) (|makeSin| (($ |#2| |#1|) "\\spad{makeSin(e,r)} makes a sin expression with given argument and coefficient")) (|coerce| (($ (|FourierComponent| |#2|)) "\\spad{coerce(c)} converts sin/cos terms into Fourier Series") (($ |#1|) "\\spad{coerce(r)} converts coefficients into Fourier Series"))) ((-4447 -12 (|has| |#1| (-6 -4447)) (|has| |#2| (-6 -4447))) (-4454 . T) (-4455 . T) (-4457 . T)) ((-12 (|HasAttribute| |#1| (QUOTE -4447)) (|HasAttribute| |#2| (QUOTE -4447)))) -(-439 R -3029) +(-439 R -3027) ((|constructor| (NIL "\\spadtype{FunctionSpaceIntegration} provides functions for the indefinite integration of real-valued functions.")) (|integrate| (((|Union| |#2| (|List| |#2|)) |#2| (|Symbol|)) "\\spad{integrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a real variable."))) NIL NIL @@ -1694,17 +1694,17 @@ NIL ((|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-1129))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (-441 R) ((|constructor| (NIL "A space of formal functions with arguments in an arbitrary ordered set.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| $)) $ (|Kernel| $)) "\\spad{univariate(f, k)} returns \\spad{f} viewed as a univariate fraction in \\spad{k}.")) (/ (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $)) (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{p1/p2} returns the quotient of \\spad{p1} and \\spad{p2} as an element of \\%.")) (|denominator| (($ $) "\\spad{denominator(f)} returns the denominator of \\spad{f} converted to \\%.")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|convert| (($ (|Factored| $)) "\\spad{convert(f1\\^e1 ... fm\\^em)} returns \\spad{(f1)\\^e1 ... (fm)\\^em} as an element of \\%,{} using formal kernels created using a \\spadfunFrom{paren}{ExpressionSpace}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|numerator| (($ $) "\\spad{numerator(f)} returns the numerator of \\spad{f} converted to \\%.")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R} if \\spad{R} is an integral domain. If not,{} then numer(\\spad{f}) = \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|coerce| (($ (|Fraction| (|Polynomial| (|Fraction| |#1|)))) "\\spad{coerce(f)} returns \\spad{f} as an element of \\%.") (($ (|Polynomial| (|Fraction| |#1|))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.") (($ (|Fraction| |#1|)) "\\spad{coerce(q)} returns \\spad{q} as an element of \\%.") (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.")) (|isMult| (((|Union| (|Record| (|:| |coef| (|Integer|)) (|:| |var| (|Kernel| $))) "failed") $) "\\spad{isMult(p)} returns \\spad{[n, x]} if \\spad{p = n * x} and \\spad{n <> 0}.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if \\spad{p = m1 +...+ mn} and \\spad{n > 1}.")) (|isExpt| (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|Symbol|)) "\\spad{isExpt(p,f)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = f(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|BasicOperator|)) "\\spad{isExpt(p,op)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = op(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1*...*an} and \\spad{n > 1}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns \\spad{x} * \\spad{x} * \\spad{x} * ... * \\spad{x} (\\spad{n} times).")) (|eval| (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ $)) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a)**n} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a1,...,am)**n} in \\spad{x} by \\spad{f(a1,...,am)} for any a1,{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)**ni} in \\spad{x} by \\spad{fi(a1,...,an)} for any a1,{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a)**ni} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|List| (|BasicOperator|)) (|List| $) (|Symbol|)) "\\spad{eval(x, [s1,...,sm], [f1,...,fm], y)} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $ (|BasicOperator|) $ (|Symbol|)) "\\spad{eval(x, s, f, y)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $) "\\spad{eval(f)} unquotes all the quoted operators in \\spad{f}.") (($ $ (|List| (|Symbol|))) "\\spad{eval(f, [foo1,...,foon])} unquotes all the \\spad{fooi}\\spad{'s} in \\spad{f}.") (($ $ (|Symbol|)) "\\spad{eval(f, foo)} unquotes all the foo\\spad{'s} in \\spad{f}.")) (|applyQuote| (($ (|Symbol|) (|List| $)) "\\spad{applyQuote(foo, [x1,...,xn])} returns \\spad{'foo(x1,...,xn)}.") (($ (|Symbol|) $ $ $ $) "\\spad{applyQuote(foo, x, y, z, t)} returns \\spad{'foo(x,y,z,t)}.") (($ (|Symbol|) $ $ $) "\\spad{applyQuote(foo, x, y, z)} returns \\spad{'foo(x,y,z)}.") (($ (|Symbol|) $ $) "\\spad{applyQuote(foo, x, y)} returns \\spad{'foo(x,y)}.") (($ (|Symbol|) $) "\\spad{applyQuote(foo, x)} returns \\spad{'foo(x)}.")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(f)} returns the list of all the variables of \\spad{f}.")) (|ground| ((|#1| $) "\\spad{ground(f)} returns \\spad{f} as an element of \\spad{R}. An error occurs if \\spad{f} is not an element of \\spad{R}.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(f)} tests if \\spad{f} is an element of \\spad{R}."))) -((-4457 -3765 (|has| |#1| (-1066)) (|has| |#1| (-484))) (-4455 |has| |#1| (-174)) (-4454 |has| |#1| (-174)) ((-4462 "*") |has| |#1| (-567)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-567)) (-4452 |has| |#1| (-567))) +((-4457 -3763 (|has| |#1| (-1066)) (|has| |#1| (-484))) (-4455 |has| |#1| (-174)) (-4454 |has| |#1| (-174)) ((-4462 "*") |has| |#1| (-567)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-567)) (-4452 |has| |#1| (-567))) NIL -(-442 R -3029) +(-442 R -3027) ((|constructor| (NIL "Provides some special functions over an integral domain.")) (|iiabs| ((|#2| |#2|) "\\spad{iiabs(x)} should be local but conditional.")) (|iiGamma| ((|#2| |#2|) "\\spad{iiGamma(x)} should be local but conditional.")) (|airyBi| ((|#2| |#2|) "\\spad{airyBi(x)} returns the airybi function applied to \\spad{x}")) (|airyAi| ((|#2| |#2|) "\\spad{airyAi(x)} returns the airyai function applied to \\spad{x}")) (|besselK| ((|#2| |#2| |#2|) "\\spad{besselK(x,y)} returns the besselk function applied to \\spad{x} and \\spad{y}")) (|besselI| ((|#2| |#2| |#2|) "\\spad{besselI(x,y)} returns the besseli function applied to \\spad{x} and \\spad{y}")) (|besselY| ((|#2| |#2| |#2|) "\\spad{besselY(x,y)} returns the bessely function applied to \\spad{x} and \\spad{y}")) (|besselJ| ((|#2| |#2| |#2|) "\\spad{besselJ(x,y)} returns the besselj function applied to \\spad{x} and \\spad{y}")) (|polygamma| ((|#2| |#2| |#2|) "\\spad{polygamma(x,y)} returns the polygamma function applied to \\spad{x} and \\spad{y}")) (|digamma| ((|#2| |#2|) "\\spad{digamma(x)} returns the digamma function applied to \\spad{x}")) (|Beta| ((|#2| |#2| |#2|) "\\spad{Beta(x,y)} returns the beta function applied to \\spad{x} and \\spad{y}")) (|Gamma| ((|#2| |#2| |#2|) "\\spad{Gamma(a,x)} returns the incomplete Gamma function applied to a and \\spad{x}") ((|#2| |#2|) "\\spad{Gamma(f)} returns the formal Gamma function applied to \\spad{f}")) (|abs| ((|#2| |#2|) "\\spad{abs(f)} returns the absolute value operator applied to \\spad{f}")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a special function operator")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a special function operator."))) NIL NIL -(-443 R -3029) +(-443 R -3027) ((|constructor| (NIL "FunctionsSpacePrimitiveElement provides functions to compute primitive elements in functions spaces.")) (|primitiveElement| (((|Record| (|:| |primelt| |#2|) (|:| |pol1| (|SparseUnivariatePolynomial| |#2|)) (|:| |pol2| (|SparseUnivariatePolynomial| |#2|)) (|:| |prim| (|SparseUnivariatePolynomial| |#2|))) |#2| |#2|) "\\spad{primitiveElement(a1, a2)} returns \\spad{[a, q1, q2, q]} such that \\spad{k(a1, a2) = k(a)},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The minimal polynomial for a2 may involve \\spad{a1},{} but the minimal polynomial for \\spad{a1} may not involve a2; This operations uses \\spadfun{resultant}.") (((|Record| (|:| |primelt| |#2|) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#2|))) (|:| |prim| (|SparseUnivariatePolynomial| |#2|))) (|List| |#2|)) "\\spad{primitiveElement([a1,...,an])} returns \\spad{[a, [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}."))) NIL ((|HasCategory| |#2| (QUOTE (-27)))) -(-444 R -3029) +(-444 R -3027) ((|constructor| (NIL "This package provides function which replaces transcendental kernels in a function space by random integers. The correspondence between the kernels and the integers is fixed between calls to new().")) (|newReduc| (((|Void|)) "\\spad{newReduc()} \\undocumented")) (|bringDown| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) |#2| (|Kernel| |#2|)) "\\spad{bringDown(f,k)} \\undocumented") (((|Fraction| (|Integer|)) |#2|) "\\spad{bringDown(f)} \\undocumented"))) NIL NIL @@ -1712,7 +1712,7 @@ NIL ((|constructor| (NIL "Creates and manipulates objects which correspond to the basic FORTRAN data types: REAL,{} INTEGER,{} COMPLEX,{} LOGICAL and CHARACTER")) (= (((|Boolean|) $ $) "\\spad{x=y} tests for equality")) (|logical?| (((|Boolean|) $) "\\spad{logical?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type LOGICAL.")) (|character?| (((|Boolean|) $) "\\spad{character?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type CHARACTER.")) (|doubleComplex?| (((|Boolean|) $) "\\spad{doubleComplex?(t)} tests whether \\spad{t} is equivalent to the (non-standard) FORTRAN type DOUBLE COMPLEX.")) (|complex?| (((|Boolean|) $) "\\spad{complex?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type COMPLEX.")) (|integer?| (((|Boolean|) $) "\\spad{integer?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type INTEGER.")) (|double?| (((|Boolean|) $) "\\spad{double?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type DOUBLE PRECISION")) (|real?| (((|Boolean|) $) "\\spad{real?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type REAL.")) (|coerce| (((|SExpression|) $) "\\spad{coerce(x)} returns the \\spad{s}-expression associated with \\spad{x}") (((|Symbol|) $) "\\spad{coerce(x)} returns the symbol associated with \\spad{x}") (($ (|Symbol|)) "\\spad{coerce(s)} transforms the symbol \\spad{s} into an element of FortranScalarType provided \\spad{s} is one of real,{} complex,{}double precision,{} logical,{} integer,{} character,{} REAL,{} COMPLEX,{} LOGICAL,{} INTEGER,{} CHARACTER,{} DOUBLE PRECISION") (($ (|String|)) "\\spad{coerce(s)} transforms the string \\spad{s} into an element of FortranScalarType provided \\spad{s} is one of \"real\",{} \"double precision\",{} \"complex\",{} \"logical\",{} \"integer\",{} \"character\",{} \"REAL\",{} \"COMPLEX\",{} \"LOGICAL\",{} \"INTEGER\",{} \"CHARACTER\",{} \"DOUBLE PRECISION\""))) NIL NIL -(-446 R -3029 UP) +(-446 R -3027 UP) ((|constructor| (NIL "\\indented{1}{Used internally by IR2F} Author: Manuel Bronstein Date Created: 12 May 1988 Date Last Updated: 22 September 1993 Keywords: function,{} space,{} polynomial,{} factoring")) (|anfactor| (((|Union| (|Factored| (|SparseUnivariatePolynomial| (|AlgebraicNumber|))) "failed") |#3|) "\\spad{anfactor(p)} tries to factor \\spad{p} over algebraic numbers,{} returning \"failed\" if it cannot")) (|UP2ifCan| (((|Union| (|:| |overq| (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) (|:| |overan| (|SparseUnivariatePolynomial| (|AlgebraicNumber|))) (|:| |failed| (|Boolean|))) |#3|) "\\spad{UP2ifCan(x)} should be local but conditional.")) (|qfactor| (((|Union| (|Factored| (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "failed") |#3|) "\\spad{qfactor(p)} tries to factor \\spad{p} over fractions of integers,{} returning \"failed\" if it cannot")) (|ffactor| (((|Factored| |#3|) |#3|) "\\spad{ffactor(p)} tries to factor a univariate polynomial \\spad{p} over \\spad{F}"))) NIL ((|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-48))))) @@ -1744,7 +1744,7 @@ NIL ((|constructor| (NIL "\\spadtype{GaloisGroupFactorizer} provides functions to factor resolvents.")) (|btwFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|) (|Set| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{btwFact(p,sqf,pd,r)} returns the factorization of \\spad{p},{} the result is a Record such that \\spad{contp=}content \\spad{p},{} \\spad{factors=}List of irreducible factors of \\spad{p} with exponent. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors). \\spad{pd} is the \\spadtype{Set} of possible degrees. \\spad{r} is a lower bound for the number of factors of \\spad{p}. Please do not use this function in your code because its design may change.")) (|henselFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|)) "\\spad{henselFact(p,sqf)} returns the factorization of \\spad{p},{} the result is a Record such that \\spad{contp=}content \\spad{p},{} \\spad{factors=}List of irreducible factors of \\spad{p} with exponent. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors).")) (|factorOfDegree| (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|) (|Boolean|)) "\\spad{factorOfDegree(d,p,listOfDegrees,r,sqf)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees},{} and that \\spad{p} has at least \\spad{r} factors. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors).") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factorOfDegree(d,p,listOfDegrees,r)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees},{} and that \\spad{p} has at least \\spad{r} factors.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factorOfDegree(d,p,listOfDegrees)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|NonNegativeInteger|)) "\\spad{factorOfDegree(d,p,r)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has at least \\spad{r} factors.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1|) "\\spad{factorOfDegree(d,p)} returns a factor of \\spad{p} of degree \\spad{d}.")) (|factorSquareFree| (((|Factored| |#1|) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,d,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{d} divides the degree of all factors of \\spad{p} and that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,listOfDegrees,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees} and that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factorSquareFree(p,listOfDegrees)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(p)} returns the factorization of \\spad{p} which is supposed not having any repeated factor (this is not checked).")) (|factor| (((|Factored| |#1|) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factor(p,d,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{d} divides the degree of all factors of \\spad{p} and that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factor(p,listOfDegrees,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees} and that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factor(p,listOfDegrees)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}.") (((|Factored| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{factor(p,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns the factorization of \\spad{p} over the integers.")) (|tryFunctionalDecomposition| (((|Boolean|) (|Boolean|)) "\\spad{tryFunctionalDecomposition(b)} chooses whether factorizers have to look for functional decomposition of polynomials (\\spad{true}) or not (\\spad{false}). Returns the previous value.")) (|tryFunctionalDecomposition?| (((|Boolean|)) "\\spad{tryFunctionalDecomposition?()} returns \\spad{true} if factorizers try functional decomposition of polynomials before factoring them.")) (|eisensteinIrreducible?| (((|Boolean|) |#1|) "\\spad{eisensteinIrreducible?(p)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by Eisenstein\\spad{'s} criterion,{} \\spad{false} is inconclusive.")) (|useEisensteinCriterion| (((|Boolean|) (|Boolean|)) "\\spad{useEisensteinCriterion(b)} chooses whether factorizers check Eisenstein\\spad{'s} criterion before factoring: \\spad{true} for using it,{} \\spad{false} else. Returns the previous value.")) (|useEisensteinCriterion?| (((|Boolean|)) "\\spad{useEisensteinCriterion?()} returns \\spad{true} if factorizers check Eisenstein\\spad{'s} criterion before factoring.")) (|useSingleFactorBound| (((|Boolean|) (|Boolean|)) "\\spad{useSingleFactorBound(b)} chooses the algorithm to be used by the factorizers: \\spad{true} for algorithm with single factor bound,{} \\spad{false} for algorithm with overall bound. Returns the previous value.")) (|useSingleFactorBound?| (((|Boolean|)) "\\spad{useSingleFactorBound?()} returns \\spad{true} if algorithm with single factor bound is used for factorization,{} \\spad{false} for algorithm with overall bound.")) (|modularFactor| (((|Record| (|:| |prime| (|Integer|)) (|:| |factors| (|List| |#1|))) |#1|) "\\spad{modularFactor(f)} chooses a \"good\" prime and returns the factorization of \\spad{f} modulo this prime in a form that may be used by \\spadfunFrom{completeHensel}{GeneralHenselPackage}. If prime is zero it means that \\spad{f} has been proved to be irreducible over the integers or that \\spad{f} is a unit (\\spadignore{i.e.} 1 or \\spad{-1}). \\spad{f} shall be primitive (\\spadignore{i.e.} content(\\spad{p})\\spad{=1}) and square free (\\spadignore{i.e.} without repeated factors).")) (|numberOfFactors| (((|NonNegativeInteger|) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|))))) "\\spad{numberOfFactors(ddfactorization)} returns the number of factors of the polynomial \\spad{f} modulo \\spad{p} where \\spad{ddfactorization} is the distinct degree factorization of \\spad{f} computed by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \\spad{p}.")) (|stopMusserTrials| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{stopMusserTrials(n)} sets to \\spad{n} the bound on the number of factors for which \\spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \\spad{2**n} trials. Returns the previous value.") (((|PositiveInteger|)) "\\spad{stopMusserTrials()} returns the bound on the number of factors for which \\spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \\spad{2**stopMusserTrials()} trials.")) (|musserTrials| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{musserTrials(n)} sets to \\spad{n} the number of primes to be tried in \\spadfun{modularFactor} and returns the previous value.") (((|PositiveInteger|)) "\\spad{musserTrials()} returns the number of primes that are tried in \\spadfun{modularFactor}.")) (|degreePartition| (((|Multiset| (|NonNegativeInteger|)) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|))))) "\\spad{degreePartition(ddfactorization)} returns the degree partition of the polynomial \\spad{f} modulo \\spad{p} where \\spad{ddfactorization} is the distinct degree factorization of \\spad{f} computed by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \\spad{p}.")) (|makeFR| (((|Factored| |#1|) (|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|))))))) "\\spad{makeFR(flist)} turns the final factorization of henselFact into a \\spadtype{Factored} object."))) NIL NIL -(-454 R UP -3029) +(-454 R UP -3027) ((|constructor| (NIL "\\spadtype{GaloisGroupFactorizationUtilities} provides functions that will be used by the factorizer.")) (|length| ((|#3| |#2|) "\\spad{length(p)} returns the sum of the absolute values of the coefficients of the polynomial \\spad{p}.")) (|height| ((|#3| |#2|) "\\spad{height(p)} returns the maximal absolute value of the coefficients of the polynomial \\spad{p}.")) (|infinityNorm| ((|#3| |#2|) "\\spad{infinityNorm(f)} returns the maximal absolute value of the coefficients of the polynomial \\spad{f}.")) (|quadraticNorm| ((|#3| |#2|) "\\spad{quadraticNorm(f)} returns the \\spad{l2} norm of the polynomial \\spad{f}.")) (|norm| ((|#3| |#2| (|PositiveInteger|)) "\\spad{norm(f,p)} returns the \\spad{lp} norm of the polynomial \\spad{f}.")) (|singleFactorBound| (((|Integer|) |#2|) "\\spad{singleFactorBound(p,r)} returns a bound on the infinite norm of the factor of \\spad{p} with smallest Bombieri\\spad{'s} norm. \\spad{p} shall be of degree higher or equal to 2.") (((|Integer|) |#2| (|NonNegativeInteger|)) "\\spad{singleFactorBound(p,r)} returns a bound on the infinite norm of the factor of \\spad{p} with smallest Bombieri\\spad{'s} norm. \\spad{r} is a lower bound for the number of factors of \\spad{p}. \\spad{p} shall be of degree higher or equal to 2.")) (|rootBound| (((|Integer|) |#2|) "\\spad{rootBound(p)} returns a bound on the largest norm of the complex roots of \\spad{p}.")) (|bombieriNorm| ((|#3| |#2| (|PositiveInteger|)) "\\spad{bombieriNorm(p,n)} returns the \\spad{n}th Bombieri\\spad{'s} norm of \\spad{p}.") ((|#3| |#2|) "\\spad{bombieriNorm(p)} returns quadratic Bombieri\\spad{'s} norm of \\spad{p}.")) (|beauzamyBound| (((|Integer|) |#2|) "\\spad{beauzamyBound(p)} returns a bound on the larger coefficient of any factor of \\spad{p}."))) NIL NIL @@ -1791,7 +1791,7 @@ NIL (-465 |vl| R E) ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is specified by its third parameter. Suggested types which define term orderings include: \\spadtype{DirectProduct},{} \\spadtype{HomogeneousDirectProduct},{} \\spadtype{SplitHomogeneousDirectProduct} and finally \\spadtype{OrderedDirectProduct} which accepts an arbitrary user function to define a term ordering.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) (((-4462 "*") |has| |#2| (-174)) (-4453 |has| |#2| (-567)) (-4458 |has| |#2| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#2| (QUOTE (-924))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) +((|HasCategory| |#2| (QUOTE (-924))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) (-466 R BP) ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni.} January 1990 The equation \\spad{Af+Bg=h} and its generalization to \\spad{n} polynomials is solved for solutions over the \\spad{R},{} euclidean domain. A table containing the solutions of \\spad{Af+Bg=x**k} is used. The operations are performed modulus a prime which are in principle big enough,{} but the solutions are tested and,{} in case of failure,{} a hensel lifting process is used to get to the right solutions. It will be used in the factorization of multivariate polynomials over finite field,{} with \\spad{R=F[x]}.")) (|testModulus| (((|Boolean|) |#1| (|List| |#2|)) "\\spad{testModulus(p,lp)} returns \\spad{true} if the the prime \\spad{p} is valid for the list of polynomials \\spad{lp},{} \\spadignore{i.e.} preserves the degree and they remain relatively prime.")) (|solveid| (((|Union| (|List| |#2|) "failed") |#2| |#1| (|Vector| (|List| |#2|))) "\\spad{solveid(h,table)} computes the coefficients of the extended euclidean algorithm for a list of polynomials whose tablePow is \\spad{table} and with right side \\spad{h}.")) (|tablePow| (((|Union| (|Vector| (|List| |#2|)) "failed") (|NonNegativeInteger|) |#1| (|List| |#2|)) "\\spad{tablePow(maxdeg,prime,lpol)} constructs the table with the coefficients of the Extended Euclidean Algorithm for \\spad{lpol}. Here the right side is \\spad{x**k},{} for \\spad{k} less or equal to \\spad{maxdeg}. The operation returns \"failed\" when the elements are not coprime modulo \\spad{prime}.")) (|compBound| (((|NonNegativeInteger|) |#2| (|List| |#2|)) "\\spad{compBound(p,lp)} computes a bound for the coefficients of the solution polynomials. Given a polynomial right hand side \\spad{p},{} and a list \\spad{lp} of left hand side polynomials. Exported because it depends on the valuation.")) (|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(p,prime)} reduces the polynomial \\spad{p} modulo \\spad{prime} of \\spad{R}. Note: this function is exported only because it\\spad{'s} conditional."))) NIL @@ -1856,7 +1856,7 @@ NIL ((|constructor| (NIL "GradedModule(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-module\\spad{''},{} \\spadignore{i.e.} collection of \\spad{R}-modules indexed by an abelian monoid \\spad{E}. An element \\spad{g} of \\spad{G[s]} for some specific \\spad{s} in \\spad{E} is said to be an element of \\spad{G} with {\\em degree} \\spad{s}. Sums are defined in each module \\spad{G[s]} so two elements of \\spad{G} have a sum if they have the same degree. \\blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.")) (+ (($ $ $) "\\spad{g+h} is the sum of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.")) (- (($ $ $) "\\spad{g-h} is the difference of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.") (($ $) "\\spad{-g} is the additive inverse of \\spad{g} in the module of elements of the same grade as \\spad{g}.")) (* (($ $ |#1|) "\\spad{g*r} is right module multiplication.") (($ |#1| $) "\\spad{r*g} is left module multiplication.")) ((|Zero|) (($) "0 denotes the zero of degree 0.")) (|degree| ((|#2| $) "\\spad{degree(g)} names the degree of \\spad{g}. The set of all elements of a given degree form an \\spad{R}-module."))) NIL NIL -(-482 |lv| -3029 R) +(-482 |lv| -3027 R) ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni,{} Summer \\spad{'88},{} revised November \\spad{'89}} Solve systems of polynomial equations using Groebner bases Total order Groebner bases are computed and then converted to lex ones This package is mostly intended for internal use.")) (|genericPosition| (((|Record| (|:| |dpolys| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |coords| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{genericPosition(lp,lv)} puts a radical zero dimensional ideal in general position,{} for system \\spad{lp} in variables \\spad{lv}.")) (|testDim| (((|Union| (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "failed") (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{testDim(lp,lv)} tests if the polynomial system \\spad{lp} in variables \\spad{lv} is zero dimensional.")) (|groebSolve| (((|List| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{groebSolve(lp,lv)} reduces the polynomial system \\spad{lp} in variables \\spad{lv} to triangular form. Algorithm based on groebner bases algorithm with linear algebra for change of ordering. Preprocessing for the general solver. The polynomials in input are of type \\spadtype{DMP}."))) NIL NIL @@ -1869,13 +1869,13 @@ NIL ((-4457 . T)) NIL (-485 |Coef| |var| |cen|) -((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x\\^r)}.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{coerce(f)} converts a Puiseux series to a general power series.") (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a Puiseux series."))) +((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x\\^r)}.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{coerce(f)} converts a Puiseux series to a general power series.") (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a Puiseux series."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4454 . T) (-4455 . T) (-4457 . 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T)) -((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-861))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117)))) +((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-861))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117)))) (-487 R E V P) ((|constructor| (NIL "A domain constructor of the category \\axiomType{TriangularSetCategory}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. Triangular sets are stored as sorted lists \\spad{w}.\\spad{r}.\\spad{t}. the main variables of their members but they are displayed in reverse order.\\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}"))) ((-4461 . T) (-4460 . T)) @@ -1891,7 +1891,7 @@ NIL (-490 |Key| |Entry| |hashfn|) ((|constructor| (NIL "This domain provides access to the underlying Lisp hash tables. By varying the hashfn parameter,{} tables suited for different purposes can be obtained."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) (-491) ((|constructor| (NIL "\\indented{1}{Author : Larry Lambe} Date Created : August 1988 Date Last Updated : March 9 1990 Related Constructors: OrderedSetInts,{} Commutator,{} FreeNilpotentLie AMS Classification: Primary 17B05,{} 17B30; Secondary 17A50 Keywords: free Lie algebra,{} Hall basis,{} basic commutators Description : Generate a basis for the free Lie algebra on \\spad{n} generators over a ring \\spad{R} with identity up to basic commutators of length \\spad{c} using the algorithm of \\spad{P}. Hall as given in Serre\\spad{'s} book Lie Groups \\spad{--} Lie Algebras")) (|generate| (((|Vector| (|List| (|Integer|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generate(numberOfGens, maximalWeight)} generates a vector of elements of the form [left,{}weight,{}right] which represents a \\spad{P}. Hall basis element for the free lie algebra on \\spad{numberOfGens} generators. We only generate those basis elements of weight less than or equal to maximalWeight")) (|inHallBasis?| (((|Boolean|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{inHallBasis?(numberOfGens, leftCandidate, rightCandidate, left)} tests to see if a new element should be added to the \\spad{P}. Hall basis being constructed. The list \\spad{[leftCandidate,wt,rightCandidate]} is included in the basis if in the unique factorization of \\spad{rightCandidate},{} we have left factor leftOfRight,{} and leftOfRight \\spad{<=} \\spad{leftCandidate}")) (|lfunc| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{lfunc(d,n)} computes the rank of the \\spad{n}th factor in the lower central series of the free \\spad{d}-generated free Lie algebra; This rank is \\spad{d} if \\spad{n} = 1 and binom(\\spad{d},{}2) if \\spad{n} = 2"))) NIL @@ -1899,11 +1899,11 @@ NIL (-492 |vl| R) ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is total degree ordering refined by reverse lexicographic ordering with respect to the position that the variables appear in the list of variables parameter.")) 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(|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))))) (|HasCategory| (-575) (QUOTE (-861))) (-12 (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (QUOTE (-237))) (|HasCategory| |#2| (QUOTE (-1066)))) (-12 (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194))))) (-3763 (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-737)))) (-3763 (|HasCategory| |#2| (QUOTE (-1066))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-1117)))) (|HasAttribute| |#2| (QUOTE -4457)) (-12 (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (QUOTE (-1066)))) (-12 (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194))))) (|HasCategory| |#2| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))))) (-494) ((|constructor| (NIL "This domain represents the header of a definition.")) (|parameters| (((|List| (|ParameterAst|)) $) "\\spad{parameters(h)} gives the parameters specified in the definition header \\spad{`h'}.")) (|name| (((|Identifier|) $) "\\spad{name(h)} returns the name of the operation defined defined.")) (|headAst| (($ (|Identifier|) (|List| (|ParameterAst|))) "\\spad{headAst(f,[x1,..,xn])} constructs a function definition header."))) NIL @@ -1911,8 +1911,8 @@ NIL (-495 S) ((|constructor| (NIL "Heap implemented in a flexible array to allow for insertions")) (|heap| (($ (|List| |#1|)) "\\spad{heap(ls)} creates a heap of elements consisting of the elements of \\spad{ls}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) -(-496 -3029 UP UPUP R) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +(-496 -3027 UP UPUP R) ((|constructor| (NIL "This domains implements finite rational divisors on an hyperelliptic curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve. The equation of the curve must be \\spad{y^2} = \\spad{f}(\\spad{x}) and \\spad{f} must have odd degree."))) NIL NIL @@ -1923,7 +1923,7 @@ NIL (-498) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating hexadecimal expansions.")) (|hex| (($ (|Fraction| (|Integer|))) "\\spad{hex(r)} converts a rational number to a hexadecimal expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(h)} returns the fractional part of a hexadecimal expansion."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3765 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) +((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3763 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) (-499 A S) ((|constructor| (NIL "A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \\spadatt{finiteAggregate} have a finite number of members. Those with attribute \\spadatt{shallowlyMutable} allow an element to be modified or updated without changing its overall value.")) (|member?| (((|Boolean|) |#2| $) "\\spad{member?(x,u)} tests if \\spad{x} is a member of \\spad{u}. For collections,{} \\axiom{member?(\\spad{x},{}\\spad{u}) = reduce(or,{}[x=y for \\spad{y} in \\spad{u}],{}\\spad{false})}.")) (|members| (((|List| |#2|) $) "\\spad{members(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|parts| (((|List| |#2|) $) "\\spad{parts(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|count| (((|NonNegativeInteger|) |#2| $) "\\spad{count(x,u)} returns the number of occurrences of \\spad{x} in \\spad{u}. For collections,{} \\axiom{count(\\spad{x},{}\\spad{u}) = reduce(+,{}[x=y for \\spad{y} in \\spad{u}],{}0)}.") (((|NonNegativeInteger|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{count(p,u)} returns the number of elements \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. For collections,{} \\axiom{count(\\spad{p},{}\\spad{u}) = reduce(+,{}[1 for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})],{}0)}.")) (|every?| (((|Boolean|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{every?(f,u)} tests if \\spad{p}(\\spad{x}) is \\spad{true} for all elements \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{every?(\\spad{p},{}\\spad{u}) = reduce(and,{}map(\\spad{f},{}\\spad{u}),{}\\spad{true},{}\\spad{false})}.")) (|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{any?(p,u)} tests if \\axiom{\\spad{p}(\\spad{x})} is \\spad{true} for any element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{any?(\\spad{p},{}\\spad{u}) = reduce(or,{}map(\\spad{f},{}\\spad{u}),{}\\spad{false},{}\\spad{true})}.")) (|map!| (($ (|Mapping| |#2| |#2|) $) "\\spad{map!(f,u)} destructively replaces each element \\spad{x} of \\spad{u} by \\axiom{\\spad{f}(\\spad{x})}.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(f,u)} returns a copy of \\spad{u} with each element \\spad{x} replaced by \\spad{f}(\\spad{x}). For collections,{} \\axiom{map(\\spad{f},{}\\spad{u}) = [\\spad{f}(\\spad{x}) for \\spad{x} in \\spad{u}]}."))) NIL @@ -1948,7 +1948,7 @@ NIL ((|constructor| (NIL "Category for the hyperbolic trigonometric functions.")) (|tanh| (($ $) "\\spad{tanh(x)} returns the hyperbolic tangent of \\spad{x}.")) (|sinh| (($ $) "\\spad{sinh(x)} returns the hyperbolic sine of \\spad{x}.")) (|sech| (($ $) "\\spad{sech(x)} returns the hyperbolic secant of \\spad{x}.")) (|csch| (($ $) "\\spad{csch(x)} returns the hyperbolic cosecant of \\spad{x}.")) (|coth| (($ $) "\\spad{coth(x)} returns the hyperbolic cotangent of \\spad{x}.")) (|cosh| (($ $) "\\spad{cosh(x)} returns the hyperbolic cosine of \\spad{x}."))) NIL NIL -(-505 -3029 UP |AlExt| |AlPol|) +(-505 -3027 UP |AlExt| |AlPol|) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of a field over which we can factor UP\\spad{'s}.")) (|factor| (((|Factored| |#4|) |#4| (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{factor(p, f)} returns a prime factorisation of \\spad{p}; \\spad{f} is a factorisation map for elements of UP."))) NIL NIL @@ -1959,16 +1959,16 @@ NIL (-507 S |mn|) ((|constructor| (NIL "\\indented{1}{Author Micheal Monagan Aug/87} This is the basic one dimensional array data type."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-508 R |mnRow| |mnCol|) ((|constructor| (NIL "\\indented{1}{An IndexedTwoDimensionalArray is a 2-dimensional array where} the minimal row and column indices are parameters of the type. Rows and columns are returned as IndexedOneDimensionalArray\\spad{'s} with minimal indices matching those of the IndexedTwoDimensionalArray. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-509 K R UP) ((|constructor| (NIL "\\indented{1}{Author: Clifton Williamson} Date Created: 9 August 1993 Date Last Updated: 3 December 1993 Basic Operations: chineseRemainder,{} factorList Related Domains: PAdicWildFunctionFieldIntegralBasis(\\spad{K},{}\\spad{R},{}UP,{}\\spad{F}) Also See: WildFunctionFieldIntegralBasis,{} FunctionFieldIntegralBasis AMS Classifications: Keywords: function field,{} finite field,{} integral basis Examples: References: Description:")) (|chineseRemainder| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) (|List| |#3|) (|List| (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) (|NonNegativeInteger|)) "\\spad{chineseRemainder(lu,lr,n)} \\undocumented")) (|listConjugateBases| (((|List| (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{listConjugateBases(bas,q,n)} returns the list \\spad{[bas,bas^Frob,bas^(Frob^2),...bas^(Frob^(n-1))]},{} where \\spad{Frob} raises the coefficients of all polynomials appearing in the basis \\spad{bas} to the \\spad{q}th power.")) (|factorList| (((|List| (|SparseUnivariatePolynomial| |#1|)) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factorList(k,n,m,j)} \\undocumented"))) NIL NIL -(-510 R UP -3029) +(-510 R UP -3027) ((|constructor| (NIL "This package contains functions used in the packages FunctionFieldIntegralBasis and NumberFieldIntegralBasis.")) (|moduleSum| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) (|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) (|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{moduleSum(m1,m2)} returns the sum of two modules in the framed algebra \\spad{F}. Each module \\spad{mi} is represented as follows: \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn} and \\spad{mi} is a record \\spad{[basis,basisDen,basisInv]}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then a basis \\spad{v1,...,vn} for \\spad{mi} is given by \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|idealiserMatrix| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{idealiserMatrix(m1, m2)} returns the matrix representing the linear conditions on the Ring associatied with an ideal defined by \\spad{m1} and \\spad{m2}.")) (|idealiser| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{idealiser(m1,m2,d)} computes the order of an ideal defined by \\spad{m1} and \\spad{m2} where \\spad{d} is the known part of the denominator") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{idealiser(m1,m2)} computes the order of an ideal defined by \\spad{m1} and \\spad{m2}")) (|leastPower| (((|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{leastPower(p,n)} returns \\spad{e},{} where \\spad{e} is the smallest integer such that \\spad{p **e >= n}")) (|divideIfCan!| ((|#1| (|Matrix| |#1|) (|Matrix| |#1|) |#1| (|Integer|)) "\\spad{divideIfCan!(matrix,matrixOut,prime,n)} attempts to divide the entries of \\spad{matrix} by \\spad{prime} and store the result in \\spad{matrixOut}. If it is successful,{} 1 is returned and if not,{} \\spad{prime} is returned. Here both \\spad{matrix} and \\spad{matrixOut} are \\spad{n}-by-\\spad{n} upper triangular matrices.")) (|matrixGcd| ((|#1| (|Matrix| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{matrixGcd(mat,sing,n)} is \\spad{gcd(sing,g)} where \\spad{g} is the \\spad{gcd} of the entries of the \\spad{n}-by-\\spad{n} upper-triangular matrix \\spad{mat}.")) (|diagonalProduct| ((|#1| (|Matrix| |#1|)) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) NIL NIL @@ -1988,7 +1988,7 @@ NIL ((|constructor| (NIL "InnerCommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#4|) "\\spad{splitDenominator([q1,...,qn])} returns \\spad{[[p1,...,pn], d]} such that \\spad{qi = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|clearDenominator| ((|#3| |#4|) "\\spad{clearDenominator([q1,...,qn])} returns \\spad{[p1,...,pn]} such that \\spad{qi = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|commonDenominator| ((|#1| |#4|) "\\spad{commonDenominator([q1,...,qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}\\spad{qn}."))) NIL NIL -(-515 -3029 |Expon| |VarSet| |DPoly|) +(-515 -3027 |Expon| |VarSet| |DPoly|) ((|constructor| (NIL "This domain represents polynomial ideals with coefficients in any field and supports the basic ideal operations,{} including intersection sum and quotient. An ideal is represented by a list of polynomials (the generators of the ideal) and a boolean that is \\spad{true} if the generators are a Groebner basis. The algorithms used are based on Groebner basis computations. The ordering is determined by the datatype of the input polynomials. Users may use refinements of total degree orderings.")) (|relationsIdeal| (((|SuchThat| (|List| (|Polynomial| |#1|)) (|List| (|Equation| (|Polynomial| |#1|)))) (|List| |#4|)) "\\spad{relationsIdeal(polyList)} returns the ideal of relations among the polynomials in \\spad{polyList}.")) (|saturate| (($ $ |#4| (|List| |#3|)) "\\spad{saturate(I,f,lvar)} is the saturation with respect to the prime principal ideal which is generated by \\spad{f} in the polynomial ring \\spad{F[lvar]}.") (($ $ |#4|) "\\spad{saturate(I,f)} is the saturation of the ideal \\spad{I} with respect to the multiplicative set generated by the polynomial \\spad{f}.")) (|coerce| (($ (|List| |#4|)) "\\spad{coerce(polyList)} converts the list of polynomials \\spad{polyList} to an ideal.")) (|generators| (((|List| |#4|) $) "\\spad{generators(I)} returns a list of generators for the ideal \\spad{I}.")) (|groebner?| (((|Boolean|) $) "\\spad{groebner?(I)} tests if the generators of the ideal \\spad{I} are a Groebner basis.")) (|groebnerIdeal| (($ (|List| |#4|)) "\\spad{groebnerIdeal(polyList)} constructs the ideal generated by the list of polynomials \\spad{polyList} which are assumed to be a Groebner basis. Note: this operation avoids a Groebner basis computation.")) (|ideal| (($ (|List| |#4|)) "\\spad{ideal(polyList)} constructs the ideal generated by the list of polynomials \\spad{polyList}.")) (|leadingIdeal| (($ $) "\\spad{leadingIdeal(I)} is the ideal generated by the leading terms of the elements of the ideal \\spad{I}.")) (|dimension| (((|Integer|) $) "\\spad{dimension(I)} gives the dimension of the ideal \\spad{I}. in the ring \\spad{F[lvar]},{} where lvar are the variables appearing in \\spad{I}") (((|Integer|) $ (|List| |#3|)) "\\spad{dimension(I,lvar)} gives the dimension of the ideal \\spad{I},{} in the ring \\spad{F[lvar]}")) (|backOldPos| (($ (|Record| (|:| |mval| (|Matrix| |#1|)) (|:| |invmval| (|Matrix| |#1|)) (|:| |genIdeal| $))) "\\spad{backOldPos(genPos)} takes the result produced by \\spadfunFrom{generalPosition}{PolynomialIdeals} and performs the inverse transformation,{} returning the original ideal \\spad{backOldPos(generalPosition(I,listvar))} = \\spad{I}.")) (|generalPosition| (((|Record| (|:| |mval| (|Matrix| |#1|)) (|:| |invmval| (|Matrix| |#1|)) (|:| |genIdeal| $)) $ (|List| |#3|)) "\\spad{generalPosition(I,listvar)} perform a random linear transformation on the variables in \\spad{listvar} and returns the transformed ideal along with the change of basis matrix.")) (|groebner| (($ $) "\\spad{groebner(I)} returns a set of generators of \\spad{I} that are a Groebner basis for \\spad{I}.")) (|quotient| (($ $ |#4|) "\\spad{quotient(I,f)} computes the quotient of the ideal \\spad{I} by the principal ideal generated by the polynomial \\spad{f},{} \\spad{(I:(f))}.") (($ $ $) "\\spad{quotient(I,J)} computes the quotient of the ideals \\spad{I} and \\spad{J},{} \\spad{(I:J)}.")) (|intersect| (($ (|List| $)) "\\spad{intersect(LI)} computes the intersection of the list of ideals \\spad{LI}.") (($ $ $) "\\spad{intersect(I,J)} computes the intersection of the ideals \\spad{I} and \\spad{J}.")) (|zeroDim?| (((|Boolean|) $) "\\spad{zeroDim?(I)} tests if the ideal \\spad{I} is zero dimensional,{} \\spadignore{i.e.} all its associated primes are maximal,{} in the ring \\spad{F[lvar]},{} where lvar are the variables appearing in \\spad{I}") (((|Boolean|) $ (|List| |#3|)) "\\spad{zeroDim?(I,lvar)} tests if the ideal \\spad{I} is zero dimensional,{} \\spadignore{i.e.} all its associated primes are maximal,{} in the ring \\spad{F[lvar]}")) (|inRadical?| (((|Boolean|) |#4| $) "\\spad{inRadical?(f,I)} tests if some power of the polynomial \\spad{f} belongs to the ideal \\spad{I}.")) (|in?| (((|Boolean|) $ $) "\\spad{in?(I,J)} tests if the ideal \\spad{I} is contained in the ideal \\spad{J}.")) (|element?| (((|Boolean|) |#4| $) "\\spad{element?(f,I)} tests whether the polynomial \\spad{f} belongs to the ideal \\spad{I}.")) (|zero?| (((|Boolean|) $) "\\spad{zero?(I)} tests whether the ideal \\spad{I} is the zero ideal")) (|one?| (((|Boolean|) $) "\\spad{one?(I)} tests whether the ideal \\spad{I} is the unit ideal,{} \\spadignore{i.e.} contains 1.")) (+ (($ $ $) "\\spad{I+J} computes the ideal generated by the union of \\spad{I} and \\spad{J}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{I**n} computes the \\spad{n}th power of the ideal \\spad{I}.")) (* (($ $ $) "\\spad{I*J} computes the product of the ideal \\spad{I} and \\spad{J}."))) NIL ((|HasCategory| |#3| (LIST (QUOTE -625) (QUOTE (-1194))))) @@ -2039,7 +2039,7 @@ NIL (-527 S |mn|) ((|constructor| (NIL "\\indented{1}{Author: Michael Monagan July/87,{} modified \\spad{SMW} June/91} A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets.")) (|shrinkable| (((|Boolean|) (|Boolean|)) "\\spad{shrinkable(b)} sets the shrinkable attribute of flexible arrays to \\spad{b} and returns the previous value")) (|physicalLength!| (($ $ (|Integer|)) "\\spad{physicalLength!(x,n)} changes the physical length of \\spad{x} to be \\spad{n} and returns the new array.")) (|physicalLength| (((|NonNegativeInteger|) $) "\\spad{physicalLength(x)} returns the number of elements \\spad{x} can accomodate before growing")) (|flexibleArray| (($ (|List| |#1|)) "\\spad{flexibleArray(l)} creates a flexible array from the list of elements \\spad{l}"))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-528) ((|constructor| (NIL "This domain represents AST for conditional expressions.")) (|elseBranch| (((|SpadAst|) $) "thenBranch(\\spad{e}) returns the `else-branch' of `e'.")) (|thenBranch| (((|SpadAst|) $) "\\spad{thenBranch(e)} returns the `then-branch' of `e'.")) (|condition| (((|SpadAst|) $) "\\spad{condition(e)} returns the condition of the if-expression `e'."))) NIL @@ -2047,15 +2047,15 @@ NIL (-529 |p| |n|) ((|constructor| (NIL "InnerFiniteField(\\spad{p},{}\\spad{n}) implements finite fields with \\spad{p**n} elements where \\spad{p} is assumed prime but does not check. For a version which checks that \\spad{p} is prime,{} see \\spadtype{FiniteField}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((-3765 (|HasCategory| (-592 |#1|) (QUOTE (-146))) (|HasCategory| (-592 |#1|) (QUOTE (-378)))) (|HasCategory| (-592 |#1|) (QUOTE (-148))) (|HasCategory| (-592 |#1|) (QUOTE (-378))) (|HasCategory| (-592 |#1|) (QUOTE (-146)))) +((-3763 (|HasCategory| (-592 |#1|) (QUOTE (-146))) (|HasCategory| (-592 |#1|) (QUOTE (-378)))) (|HasCategory| (-592 |#1|) (QUOTE (-148))) (|HasCategory| (-592 |#1|) (QUOTE (-378))) (|HasCategory| (-592 |#1|) (QUOTE (-146)))) (-530 R |mnRow| |mnCol| |Row| |Col|) ((|constructor| (NIL "\\indented{1}{This is an internal type which provides an implementation of} 2-dimensional arrays as PrimitiveArray\\spad{'s} of PrimitiveArray\\spad{'s}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-531 S |mn|) ((|constructor| (NIL "\\spadtype{IndexedList} is a basic implementation of the functions in \\spadtype{ListAggregate},{} often using functions in the underlying LISP system. The second parameter to the constructor (\\spad{mn}) is the beginning index of the list. That is,{} if \\spad{l} is a list,{} then \\spad{elt(l,mn)} is the first value. This constructor is probably best viewed as the implementation of singly-linked lists that are addressable by index rather than as a mere wrapper for LISP lists."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-532 R |Row| |Col| M) ((|constructor| (NIL "\\spadtype{InnerMatrixLinearAlgebraFunctions} is an internal package which provides standard linear algebra functions on domains in \\spad{MatrixCategory}")) (|inverse| (((|Union| |#4| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|generalizedInverse| ((|#4| |#4|) "\\spad{generalizedInverse(m)} returns the generalized (Moore--Penrose) inverse of the matrix \\spad{m},{} \\spadignore{i.e.} the matrix \\spad{h} such that m*h*m=h,{} h*m*h=m,{} \\spad{m*h} and \\spad{h*m} are both symmetric matrices.")) (|determinant| ((|#1| |#4|) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. an error message is returned if the matrix is not square.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) |#4|) "\\spad{nullity(m)} returns the mullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) |#4|) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| ((|#4| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}."))) NIL @@ -2067,7 +2067,7 @@ NIL (-534 R |mnRow| |mnCol|) ((|constructor| (NIL "An \\spad{IndexedMatrix} is a matrix where the minimal row and column indices are parameters of the type. The domains Row and Col are both IndexedVectors. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a 'Row' is the same as the index of the first column in a matrix and vice versa."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-567))) (|HasAttribute| |#1| (QUOTE (-4462 "*"))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-567))) (|HasAttribute| |#1| (QUOTE (-4462 "*"))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-535) ((|constructor| (NIL "This domain represents an `import' of types.")) (|imports| (((|List| (|TypeAst|)) $) "\\spad{imports(x)} returns the list of imported types.")) (|coerce| (($ (|List| (|TypeAst|))) "ts::ImportAst constructs an ImportAst for the list if types `ts'."))) NIL @@ -2100,7 +2100,7 @@ NIL ((|constructor| (NIL "\\indented{2}{IndexedExponents of an ordered set of variables gives a representation} for the degree of polynomials in commuting variables. It gives an ordered pairing of non negative integer exponents with variables"))) NIL NIL -(-543 K -3029 |Par|) +(-543 K -3027 |Par|) ((|constructor| (NIL "This package is the inner package to be used by NumericRealEigenPackage and NumericComplexEigenPackage for the computation of numeric eigenvalues and eigenvectors.")) (|innerEigenvectors| (((|List| (|Record| (|:| |outval| |#2|) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| |#2|))))) (|Matrix| |#1|) |#3| (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|))) "\\spad{innerEigenvectors(m,eps,factor)} computes explicitly the eigenvalues and the correspondent eigenvectors of the matrix \\spad{m}. The parameter \\spad{eps} determines the type of the output,{} \\spad{factor} is the univariate factorizer to \\spad{br} used to reduce the characteristic polynomial into irreducible factors.")) (|solve1| (((|List| |#2|) (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{solve1(pol, eps)} finds the roots of the univariate polynomial polynomial \\spad{pol} to precision eps. If \\spad{K} is \\spad{Fraction Integer} then only the real roots are returned,{} if \\spad{K} is \\spad{Complex Fraction Integer} then all roots are found.")) (|charpol| (((|SparseUnivariatePolynomial| |#1|) (|Matrix| |#1|)) "\\spad{charpol(m)} computes the characteristic polynomial of a matrix \\spad{m} with entries in \\spad{K}. This function returns a polynomial over \\spad{K},{} while the general one (that is in EiegenPackage) returns Fraction \\spad{P} \\spad{K}"))) NIL NIL @@ -2124,7 +2124,7 @@ NIL ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over an integral domain of characteristic 0.")) (|generalInfiniteProduct| ((|#2| |#2| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#2| |#2|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#2| |#2|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#2| |#2|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) NIL NIL -(-549 K -3029 |Par|) +(-549 K -3027 |Par|) ((|constructor| (NIL "This is an internal package for computing approximate solutions to systems of polynomial equations. The parameter \\spad{K} specifies the coefficient field of the input polynomials and must be either \\spad{Fraction(Integer)} or \\spad{Complex(Fraction Integer)}. The parameter \\spad{F} specifies where the solutions must lie and can be one of the following: \\spad{Float},{} \\spad{Fraction(Integer)},{} \\spad{Complex(Float)},{} \\spad{Complex(Fraction Integer)}. The last parameter specifies the type of the precision operand and must be either \\spad{Fraction(Integer)} or \\spad{Float}.")) (|makeEq| (((|List| (|Equation| (|Polynomial| |#2|))) (|List| |#2|) (|List| (|Symbol|))) "\\spad{makeEq(lsol,lvar)} returns a list of equations formed by corresponding members of \\spad{lvar} and \\spad{lsol}.")) (|innerSolve| (((|List| (|List| |#2|)) (|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|)) |#3|) "\\spad{innerSolve(lnum,lden,lvar,eps)} returns a list of solutions of the system of polynomials \\spad{lnum},{} with the side condition that none of the members of \\spad{lden} vanish identically on any solution. Each solution is expressed as a list corresponding to the list of variables in \\spad{lvar} and with precision specified by \\spad{eps}.")) (|innerSolve1| (((|List| |#2|) (|Polynomial| |#1|) |#3|) "\\spad{innerSolve1(p,eps)} returns the list of the zeros of the polynomial \\spad{p} with precision \\spad{eps}.") (((|List| |#2|) (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{innerSolve1(up,eps)} returns the list of the zeros of the univariate polynomial \\spad{up} with precision \\spad{eps}."))) NIL NIL @@ -2175,12 +2175,12 @@ NIL (-561 |Key| |Entry| |addDom|) ((|constructor| (NIL "This domain is used to provide a conditional \"add\" domain for the implementation of \\spadtype{Table}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) -(-562 R -3029) +((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) +(-562 R -3027) ((|constructor| (NIL "This package provides functions for the integration of algebraic integrands over transcendental functions.")) (|algint| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|SparseUnivariatePolynomial| |#2|) (|SparseUnivariatePolynomial| |#2|))) "\\spad{algint(f, x, y, d)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x}; \\spad{d} is the derivation to use on \\spad{k[x]}."))) NIL NIL -(-563 R0 -3029 UP UPUP R) +(-563 R0 -3027 UP UPUP R) ((|constructor| (NIL "This package provides functions for integrating a function on an algebraic curve.")) (|palginfieldint| (((|Union| |#5| "failed") |#5| (|Mapping| |#3| |#3|)) "\\spad{palginfieldint(f, d)} returns an algebraic function \\spad{g} such that \\spad{dg = f} if such a \\spad{g} exists,{} \"failed\" otherwise. Argument \\spad{f} must be a pure algebraic function.")) (|palgintegrate| (((|IntegrationResult| |#5|) |#5| (|Mapping| |#3| |#3|)) "\\spad{palgintegrate(f, d)} integrates \\spad{f} with respect to the derivation \\spad{d}. Argument \\spad{f} must be a pure algebraic function.")) (|algintegrate| (((|IntegrationResult| |#5|) |#5| (|Mapping| |#3| |#3|)) "\\spad{algintegrate(f, d)} integrates \\spad{f} with respect to the derivation \\spad{d}."))) NIL NIL @@ -2190,7 +2190,7 @@ NIL NIL (-565 R) ((|constructor| (NIL "\\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This category implements of interval arithmetic and transcendental + functions over intervals.")) (|contains?| (((|Boolean|) $ |#1|) "\\spad{contains?(i,f)} returns \\spad{true} if \\axiom{\\spad{f}} is contained within the interval \\axiom{\\spad{i}},{} \\spad{false} otherwise.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(u)} returns \\axiom{\\spad{true}} if every element of \\spad{u} is negative,{} \\axiom{\\spad{false}} otherwise.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(u)} returns \\axiom{\\spad{true}} if every element of \\spad{u} is positive,{} \\axiom{\\spad{false}} otherwise.")) (|width| ((|#1| $) "\\spad{width(u)} returns \\axiom{sup(\\spad{u}) - inf(\\spad{u})}.")) (|sup| ((|#1| $) "\\spad{sup(u)} returns the supremum of \\axiom{\\spad{u}}.")) (|inf| ((|#1| $) "\\spad{inf(u)} returns the infinum of \\axiom{\\spad{u}}.")) (|qinterval| (($ |#1| |#1|) "\\spad{qinterval(inf,sup)} creates a new interval \\axiom{[\\spad{inf},{}\\spad{sup}]},{} without checking the ordering on the elements.")) (|interval| (($ (|Fraction| (|Integer|))) "\\spad{interval(f)} creates a new interval around \\spad{f}.") (($ |#1|) "\\spad{interval(f)} creates a new interval around \\spad{f}.") (($ |#1| |#1|) "\\spad{interval(inf,sup)} creates a new interval,{} either \\axiom{[\\spad{inf},{}\\spad{sup}]} if \\axiom{\\spad{inf} \\spad{<=} \\spad{sup}} or \\axiom{[\\spad{sup},{}in]} otherwise."))) -((-3494 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-3493 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-566 S) ((|constructor| (NIL "The category of commutative integral domains,{} \\spadignore{i.e.} commutative rings with no zero divisors. \\blankline Conditional attributes: \\indented{2}{canonicalUnitNormal\\tab{20}the canonical field is the same for all associates} \\indented{2}{canonicalsClosed\\tab{20}the product of two canonicals is itself canonical}")) (|unit?| (((|Boolean|) $) "\\spad{unit?(x)} tests whether \\spad{x} is a unit,{} \\spadignore{i.e.} is invertible.")) (|associates?| (((|Boolean|) $ $) "\\spad{associates?(x,y)} tests whether \\spad{x} and \\spad{y} are associates,{} \\spadignore{i.e.} differ by a unit factor.")) (|unitCanonical| (($ $) "\\spad{unitCanonical(x)} returns \\spad{unitNormal(x).canonical}.")) (|unitNormal| (((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) "\\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \\spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the \"canonical\" element is the same across all associates of \\spad{x} if \\spad{unitNormal(x) = [u,c,a]} then \\spad{u*c = x},{} \\spad{a*u = 1}.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} either returns an element \\spad{c} such that \\spad{c*b=a} or \"failed\" if no such element can be found."))) @@ -2200,7 +2200,7 @@ NIL ((|constructor| (NIL "The category of commutative integral domains,{} \\spadignore{i.e.} commutative rings with no zero divisors. \\blankline Conditional attributes: \\indented{2}{canonicalUnitNormal\\tab{20}the canonical field is the same for all associates} \\indented{2}{canonicalsClosed\\tab{20}the product of two canonicals is itself canonical}")) (|unit?| (((|Boolean|) $) "\\spad{unit?(x)} tests whether \\spad{x} is a unit,{} \\spadignore{i.e.} is invertible.")) (|associates?| (((|Boolean|) $ $) "\\spad{associates?(x,y)} tests whether \\spad{x} and \\spad{y} are associates,{} \\spadignore{i.e.} differ by a unit factor.")) (|unitCanonical| (($ $) "\\spad{unitCanonical(x)} returns \\spad{unitNormal(x).canonical}.")) (|unitNormal| (((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) "\\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \\spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the \"canonical\" element is the same across all associates of \\spad{x} if \\spad{unitNormal(x) = [u,c,a]} then \\spad{u*c = x},{} \\spad{a*u = 1}.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} either returns an element \\spad{c} such that \\spad{c*b=a} or \"failed\" if no such element can be found."))) ((-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL -(-568 R -3029) +(-568 R -3027) ((|constructor| (NIL "This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for elemntary functions.")) (|lfextlimint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Symbol|) (|Kernel| |#2|) (|List| (|Kernel| |#2|))) "\\spad{lfextlimint(f,x,k,[k1,...,kn])} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f - c dk/dx}. Value \\spad{h} is looked for in a field containing \\spad{f} and \\spad{k1},{}...,{}\\spad{kn} (the \\spad{ki}\\spad{'s} must be logs).")) (|lfintegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{lfintegrate(f, x)} = \\spad{g} such that \\spad{dg/dx = f}.")) (|lfinfieldint| (((|Union| |#2| "failed") |#2| (|Symbol|)) "\\spad{lfinfieldint(f, x)} returns a function \\spad{g} such that \\spad{dg/dx = f} if \\spad{g} exists,{} \"failed\" otherwise.")) (|lflimitedint| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Symbol|) (|List| |#2|)) "\\spad{lflimitedint(f,x,[g1,...,gn])} returns functions \\spad{[h,[[ci, gi]]]} such that the \\spad{gi}\\spad{'s} are among \\spad{[g1,...,gn]},{} and \\spad{d(h+sum(ci log(gi)))/dx = f},{} if possible,{} \"failed\" otherwise.")) (|lfextendedint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Symbol|) |#2|) "\\spad{lfextendedint(f, x, g)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f - cg},{} if (\\spad{h},{} \\spad{c}) exist,{} \"failed\" otherwise."))) NIL NIL @@ -2212,7 +2212,7 @@ NIL ((|constructor| (NIL "\\blankline")) (|entry| (((|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")))) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{entry(n)} \\undocumented{}")) (|entries| (((|List| (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated"))))))) $) "\\spad{entries(x)} \\undocumented{}")) (|showAttributes| (((|Union| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")))) "failed") (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{showAttributes(x)} \\undocumented{}")) (|insert!| (($ (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated"))))))) "\\spad{insert!(r)} inserts an entry \\spad{r} into theIFTable")) (|fTable| (($ (|List| (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")))))))) "\\spad{fTable(l)} creates a functions table from the elements of \\spad{l}.")) (|keys| (((|List| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) $) "\\spad{keys(f)} returns the list of keys of \\spad{f}")) (|clearTheFTable| (((|Void|)) "\\spad{clearTheFTable()} clears the current table of functions.")) (|showTheFTable| (($) "\\spad{showTheFTable()} returns the current table of functions."))) NIL NIL -(-571 R -3029 L) +(-571 R -3027 L) ((|constructor| (NIL "This internal package rationalises integrands on curves of the form: \\indented{2}{\\spad{y\\^2 = a x\\^2 + b x + c}} \\indented{2}{\\spad{y\\^2 = (a x + b) / (c x + d)}} \\indented{2}{\\spad{f(x, y) = 0} where \\spad{f} has degree 1 in \\spad{x}} The rationalization is done for integration,{} limited integration,{} extended integration and the risch differential equation.")) (|palgLODE0| (((|Record| (|:| |particular| (|Union| |#2| "failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgLODE0(op,g,x,y,z,t,c)} returns the solution of \\spad{op f = g} Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}.") (((|Record| (|:| |particular| (|Union| |#2| "failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgLODE0(op, g, x, y, d, p)} returns the solution of \\spad{op f = g}. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}.")) (|lift| (((|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) (|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{lift(u,k)} \\undocumented")) (|multivariate| ((|#2| (|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) (|Kernel| |#2|) |#2|) "\\spad{multivariate(u,k,f)} \\undocumented")) (|univariate| (((|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|SparseUnivariatePolynomial| |#2|)) "\\spad{univariate(f,k,k,p)} \\undocumented")) (|palgRDE0| (((|Union| |#2| "failed") |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| "failed") |#2| |#2| (|Symbol|)) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgRDE0(f, g, x, y, foo, t, c)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{foo},{} called by \\spad{foo(a, b, x)},{} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}.") (((|Union| |#2| "failed") |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| "failed") |#2| |#2| (|Symbol|)) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgRDE0(f, g, x, y, foo, d, p)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}. Argument \\spad{foo},{} called by \\spad{foo(a, b, x)},{} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}.")) (|palglimint0| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palglimint0(f, x, y, [u1,...,un], z, t, c)} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}\\spad{'s} are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}.") (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palglimint0(f, x, y, [u1,...,un], d, p)} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}\\spad{'s} are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}.")) (|palgextint0| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgextint0(f, x, y, g, z, t, c)} returns functions \\spad{[h, d]} such that \\spad{dh/dx = f(x,y) - d g},{} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy},{} and \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{z} is a dummy variable not appearing in \\spad{f(x,y)}. The operation returns \"failed\" if no such functions exist.") (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgextint0(f, x, y, g, d, p)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f(x,y) - c g},{} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2 y(x)\\^2 = P(x)},{} or \"failed\" if no such functions exist.")) (|palgint0| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgint0(f, x, y, z, t, c)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{z} is a dummy variable not appearing in \\spad{f(x,y)}.") (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgint0(f, x, y, d, p)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2 y(x)\\^2 = P(x)}."))) NIL ((|HasCategory| |#3| (LIST (QUOTE -667) (|devaluate| |#2|)))) @@ -2220,11 +2220,11 @@ NIL ((|constructor| (NIL "This package provides various number theoretic functions on the integers.")) (|sumOfKthPowerDivisors| (((|Integer|) (|Integer|) (|NonNegativeInteger|)) "\\spad{sumOfKthPowerDivisors(n,k)} returns the sum of the \\spad{k}th powers of the integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. the sum of the \\spad{k}th powers of the divisors of \\spad{n} is often denoted by \\spad{sigma_k(n)}.")) (|sumOfDivisors| (((|Integer|) (|Integer|)) "\\spad{sumOfDivisors(n)} returns the sum of the integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. The sum of the divisors of \\spad{n} is often denoted by \\spad{sigma(n)}.")) (|numberOfDivisors| (((|Integer|) (|Integer|)) "\\spad{numberOfDivisors(n)} returns the number of integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. The number of divisors of \\spad{n} is often denoted by \\spad{tau(n)}.")) (|moebiusMu| (((|Integer|) (|Integer|)) "\\spad{moebiusMu(n)} returns the Moebius function \\spad{mu(n)}. \\spad{mu(n)} is either \\spad{-1},{}0 or 1 as follows: \\spad{mu(n) = 0} if \\spad{n} is divisible by a square > 1,{} \\spad{mu(n) = (-1)^k} if \\spad{n} is square-free and has \\spad{k} distinct prime divisors.")) (|legendre| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{legendre(a,p)} returns the Legendre symbol \\spad{L(a/p)}. \\spad{L(a/p) = (-1)**((p-1)/2) mod p} (\\spad{p} prime),{} which is 0 if \\spad{a} is 0,{} 1 if \\spad{a} is a quadratic residue \\spad{mod p} and \\spad{-1} otherwise. Note: because the primality test is expensive,{} if it is known that \\spad{p} is prime then use \\spad{jacobi(a,p)}.")) (|jacobi| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{jacobi(a,b)} returns the Jacobi symbol \\spad{J(a/b)}. When \\spad{b} is odd,{} \\spad{J(a/b) = product(L(a/p) for p in factor b )}. Note: by convention,{} 0 is returned if \\spad{gcd(a,b) ~= 1}. Iterative \\spad{O(log(b)^2)} version coded by Michael Monagan June 1987.")) (|harmonic| (((|Fraction| (|Integer|)) (|Integer|)) "\\spad{harmonic(n)} returns the \\spad{n}th harmonic number. This is \\spad{H[n] = sum(1/k,k=1..n)}.")) (|fibonacci| (((|Integer|) (|Integer|)) "\\spad{fibonacci(n)} returns the \\spad{n}th Fibonacci number. the Fibonacci numbers \\spad{F[n]} are defined by \\spad{F[0] = F[1] = 1} and \\spad{F[n] = F[n-1] + F[n-2]}. The algorithm has running time \\spad{O(log(n)^3)}. Reference: Knuth,{} The Art of Computer Programming Vol 2,{} Semi-Numerical Algorithms.")) (|eulerPhi| (((|Integer|) (|Integer|)) "\\spad{eulerPhi(n)} returns the number of integers between 1 and \\spad{n} (including 1) which are relatively prime to \\spad{n}. This is the Euler phi function \\spad{\\phi(n)} is also called the totient function.")) (|euler| (((|Integer|) (|Integer|)) "\\spad{euler(n)} returns the \\spad{n}th Euler number. This is \\spad{2^n E(n,1/2)},{} where \\spad{E(n,x)} is the \\spad{n}th Euler polynomial.")) (|divisors| (((|List| (|Integer|)) (|Integer|)) "\\spad{divisors(n)} returns a list of the divisors of \\spad{n}.")) (|chineseRemainder| (((|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{chineseRemainder(x1,m1,x2,m2)} returns \\spad{w},{} where \\spad{w} is such that \\spad{w = x1 mod m1} and \\spad{w = x2 mod m2}. Note: \\spad{m1} and \\spad{m2} must be relatively prime.")) (|bernoulli| (((|Fraction| (|Integer|)) (|Integer|)) "\\spad{bernoulli(n)} returns the \\spad{n}th Bernoulli number. this is \\spad{B(n,0)},{} where \\spad{B(n,x)} is the \\spad{n}th Bernoulli polynomial."))) NIL NIL -(-573 -3029 UP UPUP R) +(-573 -3027 UP UPUP R) ((|constructor| (NIL "algebraic Hermite redution.")) (|HermiteIntegrate| (((|Record| (|:| |answer| |#4|) (|:| |logpart| |#4|)) |#4| (|Mapping| |#2| |#2|)) "\\spad{HermiteIntegrate(f, ')} returns \\spad{[g,h]} such that \\spad{f = g' + h} and \\spad{h} has a only simple finite normal poles."))) NIL NIL -(-574 -3029 UP) +(-574 -3027 UP) ((|constructor| (NIL "Hermite integration,{} transcendental case.")) (|HermiteIntegrate| (((|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |logpart| (|Fraction| |#2|)) (|:| |specpart| (|Fraction| |#2|)) (|:| |polypart| |#2|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{HermiteIntegrate(f, D)} returns \\spad{[g, h, s, p]} such that \\spad{f = Dg + h + s + p},{} \\spad{h} has a squarefree denominator normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and all the squarefree factors of the denominator of \\spad{s} are special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D}. Furthermore,{} \\spad{h} and \\spad{s} have no polynomial parts. \\spad{D} is the derivation to use on \\spadtype{UP}."))) NIL NIL @@ -2236,15 +2236,15 @@ NIL ((|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))) (|:| |extra| (|Result|))) (|NumericalIntegrationProblem|) (|RoutinesTable|)) "\\spad{measure(prob,R)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical integration problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} listed in \\axiom{\\spad{R}} of \\axiom{category} \\axiomType{NumericalIntegrationCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information.") (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))) (|:| |extra| (|Result|))) (|NumericalIntegrationProblem|)) "\\spad{measure(prob)} is a top level ANNA function for identifying the most appropriate numerical routine for solving the numerical integration problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} of \\axiom{category} \\axiomType{NumericalIntegrationCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information.")) (|integrate| (((|Union| (|Result|) "failed") (|Expression| (|Float|)) (|SegmentBinding| (|OrderedCompletion| (|Float|))) (|Symbol|)) "\\spad{integrate(exp, x = a..b, numerical)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range,{} {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.\\newline \\blankline Default values for the absolute and relative error are used. \\blankline It is an error if the last argument is not {\\spad{\\tt} numerical}.") (((|Union| (|Result|) "failed") (|Expression| (|Float|)) (|SegmentBinding| (|OrderedCompletion| (|Float|))) (|String|)) "\\spad{integrate(exp, x = a..b, \"numerical\")} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range,{} {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.\\newline \\blankline Default values for the absolute and relative error are used. \\blankline It is an error of the last argument is not {\\spad{\\tt} \"numerical\"}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|)))) (|Float|) (|Float|) (|RoutinesTable|)) "\\spad{integrate(exp, [a..b,c..d,...], epsabs, epsrel, routines)} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges to the required absolute and relative accuracy,{} using the routines available in the RoutinesTable provided. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|)))) (|Float|) (|Float|)) "\\spad{integrate(exp, [a..b,c..d,...], epsabs, epsrel)} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges to the required absolute and relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|)))) (|Float|)) "\\spad{integrate(exp, [a..b,c..d,...], epsrel)} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges to the required relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline If epsrel = 0,{} a default absolute accuracy is used.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|))))) "\\spad{integrate(exp, [a..b,c..d,...])} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline Default values for the absolute and relative error are used.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|)))) "\\spad{integrate(exp, a..b)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline Default values for the absolute and relative error are used.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|))) (|Float|)) "\\spad{integrate(exp, a..b, epsrel)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}} to the required relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline If epsrel = 0,{} a default absolute accuracy is used.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|))) (|Float|) (|Float|)) "\\spad{integrate(exp, a..b, epsabs, epsrel)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}} to the required absolute and relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|NumericalIntegrationProblem|)) "\\spad{integrate(IntegrationProblem)} is a top level ANNA function to integrate an expression over a given range or ranges to the required absolute and relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|))) (|Float|) (|Float|) (|RoutinesTable|)) "\\spad{integrate(exp, a..b, epsrel, routines)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}} to the required absolute and relative accuracy using the routines available in the RoutinesTable provided. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}."))) NIL NIL -(-577 R -3029 L) +(-577 R -3027 L) ((|constructor| (NIL "This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for pure algebraic integrands.")) (|palgLODE| (((|Record| (|:| |particular| (|Union| |#2| "failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Symbol|)) "\\spad{palgLODE(op, g, kx, y, x)} returns the solution of \\spad{op f = g}. \\spad{y} is an algebraic function of \\spad{x}.")) (|palgRDE| (((|Union| |#2| "failed") |#2| |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| "failed") |#2| |#2| (|Symbol|))) "\\spad{palgRDE(nfp, f, g, x, y, foo)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} \"failed\" otherwise; \\spad{y} is an algebraic function of \\spad{x}; \\spad{foo(a, b, x)} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}. \\spad{nfp} is \\spad{n * df/dx}.")) (|palglimint| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|)) "\\spad{palglimint(f, x, y, [u1,...,un])} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}\\spad{'s} are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} \"failed\" otherwise; \\spad{y} is an algebraic function of \\spad{x}.")) (|palgextint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2|) "\\spad{palgextint(f, x, y, g)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f(x,y) - c g},{} where \\spad{y} is an algebraic function of \\spad{x}; returns \"failed\" if no such functions exist.")) (|palgint| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|)) "\\spad{palgint(f, x, y)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x}."))) NIL ((|HasCategory| |#3| (LIST (QUOTE -667) (|devaluate| |#2|)))) -(-578 R -3029) +(-578 R -3027) ((|constructor| (NIL "\\spadtype{PatternMatchIntegration} provides functions that use the pattern matcher to find some indefinite and definite integrals involving special functions and found in the litterature.")) (|pmintegrate| (((|Union| |#2| "failed") |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|)) "\\spad{pmintegrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b} if it can be found by the built-in pattern matching rules.") (((|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|)) "\\spad{pmintegrate(f, x)} returns either \"failed\" or \\spad{[g,h]} such that \\spad{integrate(f,x) = g + integrate(h,x)}.")) (|pmComplexintegrate| (((|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|)) "\\spad{pmComplexintegrate(f, x)} returns either \"failed\" or \\spad{[g,h]} such that \\spad{integrate(f,x) = g + integrate(h,x)}. It only looks for special complex integrals that pmintegrate does not return.")) (|splitConstant| (((|Record| (|:| |const| |#2|) (|:| |nconst| |#2|)) |#2| (|Symbol|)) "\\spad{splitConstant(f, x)} returns \\spad{[c, g]} such that \\spad{f = c * g} and \\spad{c} does not involve \\spad{t}."))) NIL ((-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-1156)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-640))))) -(-579 -3029 UP) +(-579 -3027 UP) ((|constructor| (NIL "This package provides functions for the base case of the Risch algorithm.")) (|limitedint| (((|Union| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|)))))) "failed") (|Fraction| |#2|) (|List| (|Fraction| |#2|))) "\\spad{limitedint(f, [g1,...,gn])} returns fractions \\spad{[h,[[ci, gi]]]} such that the \\spad{gi}\\spad{'s} are among \\spad{[g1,...,gn]},{} \\spad{ci' = 0},{} and \\spad{(h+sum(ci log(gi)))' = f},{} if possible,{} \"failed\" otherwise.")) (|extendedint| (((|Union| (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{extendedint(f, g)} returns fractions \\spad{[h, c]} such that \\spad{c' = 0} and \\spad{h' = f - cg},{} if \\spad{(h, c)} exist,{} \"failed\" otherwise.")) (|infieldint| (((|Union| (|Fraction| |#2|) "failed") (|Fraction| |#2|)) "\\spad{infieldint(f)} returns \\spad{g} such that \\spad{g' = f} or \"failed\" if the integral of \\spad{f} is not a rational function.")) (|integrate| (((|IntegrationResult| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{integrate(f)} returns \\spad{g} such that \\spad{g' = f}."))) NIL NIL @@ -2252,27 +2252,27 @@ NIL ((|constructor| (NIL "Provides integer testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991")) (|integerIfCan| (((|Union| (|Integer|) "failed") |#1|) "\\spad{integerIfCan(x)} returns \\spad{x} as an integer,{} \"failed\" if \\spad{x} is not an integer.")) (|integer?| (((|Boolean|) |#1|) "\\spad{integer?(x)} is \\spad{true} if \\spad{x} is an integer,{} \\spad{false} otherwise.")) (|integer| (((|Integer|) |#1|) "\\spad{integer(x)} returns \\spad{x} as an integer; error if \\spad{x} is not an integer."))) NIL NIL -(-581 -3029) +(-581 -3027) ((|constructor| (NIL "This package provides functions for the integration of rational functions.")) (|extendedIntegrate| (((|Union| (|Record| (|:| |ratpart| (|Fraction| (|Polynomial| |#1|))) (|:| |coeff| (|Fraction| (|Polynomial| |#1|)))) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|))) "\\spad{extendedIntegrate(f, x, g)} returns fractions \\spad{[h, c]} such that \\spad{dc/dx = 0} and \\spad{dh/dx = f - cg},{} if \\spad{(h, c)} exist,{} \"failed\" otherwise.")) (|limitedIntegrate| (((|Union| (|Record| (|:| |mainpart| (|Fraction| (|Polynomial| |#1|))) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| (|Polynomial| |#1|))) (|:| |logand| (|Fraction| (|Polynomial| |#1|))))))) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{limitedIntegrate(f, x, [g1,...,gn])} returns fractions \\spad{[h, [[ci,gi]]]} such that the \\spad{gi}\\spad{'s} are among \\spad{[g1,...,gn]},{} \\spad{dci/dx = 0},{} and \\spad{d(h + sum(ci log(gi)))/dx = f} if possible,{} \"failed\" otherwise.")) (|infieldIntegrate| (((|Union| (|Fraction| (|Polynomial| |#1|)) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{infieldIntegrate(f, x)} returns a fraction \\spad{g} such that \\spad{dg/dx = f} if \\spad{g} exists,{} \"failed\" otherwise.")) (|internalIntegrate| (((|IntegrationResult| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns \\spad{g} such that \\spad{dg/dx = f}."))) NIL NIL (-582 R) ((|constructor| (NIL "\\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This domain is an implementation of interval arithmetic and transcendental + functions over intervals."))) -((-3494 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-3493 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-583) ((|constructor| (NIL "This package provides the implementation for the \\spadfun{solveLinearPolynomialEquation} operation over the integers. It uses a lifting technique from the package GenExEuclid")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| (|Integer|))) "failed") (|List| (|SparseUnivariatePolynomial| (|Integer|))) (|SparseUnivariatePolynomial| (|Integer|))) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists."))) NIL NIL -(-584 R -3029) +(-584 R -3027) ((|constructor| (NIL "\\indented{1}{Tools for the integrator} Author: Manuel Bronstein Date Created: 25 April 1990 Date Last Updated: 9 June 1993 Keywords: elementary,{} function,{} integration.")) (|intPatternMatch| (((|IntegrationResult| |#2|) |#2| (|Symbol|) (|Mapping| (|IntegrationResult| |#2|) |#2| (|Symbol|)) (|Mapping| (|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|))) "\\spad{intPatternMatch(f, x, int, pmint)} tries to integrate \\spad{f} first by using the integration function \\spad{int},{} and then by using the pattern match intetgration function \\spad{pmint} on any remaining unintegrable part.")) (|mkPrim| ((|#2| |#2| (|Symbol|)) "\\spad{mkPrim(f, x)} makes the logs in \\spad{f} which are linear in \\spad{x} primitive with respect to \\spad{x}.")) (|removeConstantTerm| ((|#2| |#2| (|Symbol|)) "\\spad{removeConstantTerm(f, x)} returns \\spad{f} minus any additive constant with respect to \\spad{x}.")) (|vark| (((|List| (|Kernel| |#2|)) (|List| |#2|) (|Symbol|)) "\\spad{vark([f1,...,fn],x)} returns the set-theoretic union of \\spad{(varselect(f1,x),...,varselect(fn,x))}.")) (|union| (((|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|))) "\\spad{union(l1, l2)} returns set-theoretic union of \\spad{l1} and \\spad{l2}.")) (|ksec| (((|Kernel| |#2|) (|Kernel| |#2|) (|List| (|Kernel| |#2|)) (|Symbol|)) "\\spad{ksec(k, [k1,...,kn], x)} returns the second top-level \\spad{ki} after \\spad{k} involving \\spad{x}.")) (|kmax| (((|Kernel| |#2|) (|List| (|Kernel| |#2|))) "\\spad{kmax([k1,...,kn])} returns the top-level \\spad{ki} for integration.")) (|varselect| (((|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|)) (|Symbol|)) "\\spad{varselect([k1,...,kn], x)} returns the \\spad{ki} which involve \\spad{x}."))) NIL ((-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-293))) (|HasCategory| |#2| (QUOTE (-640))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-293)))) (|HasCategory| |#1| (QUOTE (-567)))) -(-585 -3029 UP) +(-585 -3027 UP) ((|constructor| (NIL "This package provides functions for the transcendental case of the Risch algorithm.")) (|monomialIntPoly| (((|Record| (|:| |answer| |#2|) (|:| |polypart| |#2|)) |#2| (|Mapping| |#2| |#2|)) "\\spad{monomialIntPoly(p, ')} returns [\\spad{q},{} \\spad{r}] such that \\spad{p = q' + r} and \\spad{degree(r) < degree(t')}. Error if \\spad{degree(t') < 2}.")) (|monomialIntegrate| (((|Record| (|:| |ir| (|IntegrationResult| (|Fraction| |#2|))) (|:| |specpart| (|Fraction| |#2|)) (|:| |polypart| |#2|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomialIntegrate(f, ')} returns \\spad{[ir, s, p]} such that \\spad{f = ir' + s + p} and all the squarefree factors of the denominator of \\spad{s} are special \\spad{w}.\\spad{r}.\\spad{t} the derivation '.")) (|expintfldpoly| (((|Union| (|LaurentPolynomial| |#1| |#2|) "failed") (|LaurentPolynomial| |#1| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|)) "\\spad{expintfldpoly(p, foo)} returns \\spad{q} such that \\spad{p' = q} or \"failed\" if no such \\spad{q} exists. Argument foo is a Risch differential equation function on \\spad{F}.")) (|primintfldpoly| (((|Union| |#2| "failed") |#2| (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed") |#1|) |#1|) "\\spad{primintfldpoly(p, ', t')} returns \\spad{q} such that \\spad{p' = q} or \"failed\" if no such \\spad{q} exists. Argument \\spad{t'} is the derivative of the primitive generating the extension.")) (|primlimintfrac| (((|Union| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|)))))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|List| (|Fraction| |#2|))) "\\spad{primlimintfrac(f, ', [u1,...,un])} returns \\spad{[v, [c1,...,cn]]} such that \\spad{ci' = 0} and \\spad{f = v' + +/[ci * ui'/ui]}. Error: if \\spad{degree numer f >= degree denom f}.")) (|primextintfrac| (((|Union| (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Fraction| |#2|)) "\\spad{primextintfrac(f, ', g)} returns \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0}. Error: if \\spad{degree numer f >= degree denom f} or if \\spad{degree numer g >= degree denom g} or if \\spad{denom g} is not squarefree.")) (|explimitedint| (((|Union| (|Record| (|:| |answer| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|))))))) (|:| |a0| |#1|)) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|) (|List| (|Fraction| |#2|))) "\\spad{explimitedint(f, ', foo, [u1,...,un])} returns \\spad{[v, [c1,...,cn], a]} such that \\spad{ci' = 0},{} \\spad{f = v' + a + reduce(+,[ci * ui'/ui])},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}. Returns \"failed\" if no such \\spad{v},{} \\spad{ci},{} a exist. Argument \\spad{foo} is a Risch differential equation function on \\spad{F}.")) (|primlimitedint| (((|Union| (|Record| (|:| |answer| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|))))))) (|:| |a0| |#1|)) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed") |#1|) (|List| (|Fraction| |#2|))) "\\spad{primlimitedint(f, ', foo, [u1,...,un])} returns \\spad{[v, [c1,...,cn], a]} such that \\spad{ci' = 0},{} \\spad{f = v' + a + reduce(+,[ci * ui'/ui])},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Returns \"failed\" if no such \\spad{v},{} \\spad{ci},{} a exist. Argument \\spad{foo} is an extended integration function on \\spad{F}.")) (|expextendedint| (((|Union| (|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |a0| |#1|)) (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|) (|Fraction| |#2|)) "\\spad{expextendedint(f, ', foo, g)} returns either \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0},{} or \\spad{[v, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}. Returns \"failed\" if neither case can hold. Argument \\spad{foo} is a Risch differential equation function on \\spad{F}.")) (|primextendedint| (((|Union| (|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |a0| |#1|)) (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed") |#1|) (|Fraction| |#2|)) "\\spad{primextendedint(f, ', foo, g)} returns either \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0},{} or \\spad{[v, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Returns \"failed\" if neither case can hold. Argument \\spad{foo} is an extended integration function on \\spad{F}.")) (|tanintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|List| |#1|) "failed") (|Integer|) |#1| |#1|)) "\\spad{tanintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}; Argument foo is a Risch differential system solver on \\spad{F}.")) (|expintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|)) "\\spad{expintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}; Argument foo is a Risch differential equation solver on \\spad{F}.")) (|primintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed") |#1|)) "\\spad{primintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Argument foo is an extended integration function on \\spad{F}."))) NIL NIL -(-586 R -3029) +(-586 R -3027) ((|constructor| (NIL "This package computes the inverse Laplace Transform.")) (|inverseLaplace| (((|Union| |#2| "failed") |#2| (|Symbol|) (|Symbol|)) "\\spad{inverseLaplace(f, s, t)} returns the Inverse Laplace transform of \\spad{f(s)} using \\spad{t} as the new variable or \"failed\" if unable to find a closed form."))) NIL NIL @@ -2304,11 +2304,11 @@ NIL ((|constructor| (NIL "A package to print strings without line-feed nor carriage-return.")) (|iprint| (((|Void|) (|String|)) "\\axiom{iprint(\\spad{s})} prints \\axiom{\\spad{s}} at the current position of the cursor."))) NIL NIL -(-594 R -3029) +(-594 R -3027) ((|constructor| (NIL "This package allows a sum of logs over the roots of a polynomial to be expressed as explicit logarithms and arc tangents,{} provided that the indexing polynomial can be factored into quadratics.")) (|complexExpand| ((|#2| (|IntegrationResult| |#2|)) "\\spad{complexExpand(i)} returns the expanded complex function corresponding to \\spad{i}.")) (|expand| (((|List| |#2|) (|IntegrationResult| |#2|)) "\\spad{expand(i)} returns the list of possible real functions corresponding to \\spad{i}.")) (|split| (((|IntegrationResult| |#2|) (|IntegrationResult| |#2|)) "\\spad{split(u(x) + sum_{P(a)=0} Q(a,x))} returns \\spad{u(x) + sum_{P1(a)=0} Q(a,x) + ... + sum_{Pn(a)=0} Q(a,x)} where \\spad{P1},{}...,{}\\spad{Pn} are the factors of \\spad{P}."))) NIL NIL -(-595 E -3029) +(-595 E -3027) ((|constructor| (NIL "\\indented{1}{Internally used by the integration packages} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 12 August 1992 Keywords: integration.")) (|map| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") (|Mapping| |#2| |#1|) (|Union| (|Record| (|:| |mainpart| |#1|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#1|) (|:| |logand| |#1|))))) "failed")) "\\spad{map(f,ufe)} \\undocumented") (((|Union| |#2| "failed") (|Mapping| |#2| |#1|) (|Union| |#1| "failed")) "\\spad{map(f,ue)} \\undocumented") (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") (|Mapping| |#2| |#1|) (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed")) "\\spad{map(f,ure)} \\undocumented") (((|IntegrationResult| |#2|) (|Mapping| |#2| |#1|) (|IntegrationResult| |#1|)) "\\spad{map(f,ire)} \\undocumented"))) NIL NIL @@ -2316,7 +2316,7 @@ NIL ((|constructor| (NIL "This domain provides representations for the intermediate form data structure used by the Spad elaborator.")) (|irDef| (($ (|Identifier|) (|InternalTypeForm|) $) "\\spad{irDef(f,ts,e)} returns an IR representation for a definition of a function named \\spad{f},{} with signature \\spad{ts} and body \\spad{e}.")) (|irCtor| (($ (|Identifier|) (|InternalTypeForm|)) "\\spad{irCtor(n,t)} returns an IR for a constructor reference of type designated by the type form \\spad{t}")) (|irVar| (($ (|Identifier|) (|InternalTypeForm|)) "\\spad{irVar(x,t)} returns an IR for a variable reference of type designated by the type form \\spad{t}"))) NIL NIL -(-597 -3029) +(-597 -3027) ((|constructor| (NIL "If a function \\spad{f} has an elementary integral \\spad{g},{} then \\spad{g} can be written in the form \\spad{g = h + c1 log(u1) + c2 log(u2) + ... + cn log(un)} where \\spad{h},{} which is in the same field than \\spad{f},{} is called the rational part of the integral,{} and \\spad{c1 log(u1) + ... cn log(un)} is called the logarithmic part of the integral. This domain manipulates integrals represented in that form,{} by keeping both parts separately. The logs are not explicitly computed.")) (|differentiate| ((|#1| $ (|Symbol|)) "\\spad{differentiate(ir,x)} differentiates \\spad{ir} with respect to \\spad{x}") ((|#1| $ (|Mapping| |#1| |#1|)) "\\spad{differentiate(ir,D)} differentiates \\spad{ir} with respect to the derivation \\spad{D}.")) (|integral| (($ |#1| (|Symbol|)) "\\spad{integral(f,x)} returns the formal integral of \\spad{f} with respect to \\spad{x}") (($ |#1| |#1|) "\\spad{integral(f,x)} returns the formal integral of \\spad{f} with respect to \\spad{x}")) (|elem?| (((|Boolean|) $) "\\spad{elem?(ir)} tests if an integration result is elementary over \\spad{F?}")) (|notelem| (((|List| (|Record| (|:| |integrand| |#1|) (|:| |intvar| |#1|))) $) "\\spad{notelem(ir)} returns the non-elementary part of an integration result")) (|logpart| (((|List| (|Record| (|:| |scalar| (|Fraction| (|Integer|))) (|:| |coeff| (|SparseUnivariatePolynomial| |#1|)) (|:| |logand| (|SparseUnivariatePolynomial| |#1|)))) $) "\\spad{logpart(ir)} returns the logarithmic part of an integration result")) (|ratpart| ((|#1| $) "\\spad{ratpart(ir)} returns the rational part of an integration result")) (|mkAnswer| (($ |#1| (|List| (|Record| (|:| |scalar| (|Fraction| (|Integer|))) (|:| |coeff| (|SparseUnivariatePolynomial| |#1|)) (|:| |logand| (|SparseUnivariatePolynomial| |#1|)))) (|List| (|Record| (|:| |integrand| |#1|) (|:| |intvar| |#1|)))) "\\spad{mkAnswer(r,l,ne)} creates an integration result from a rational part \\spad{r},{} a logarithmic part \\spad{l},{} and a non-elementary part \\spad{ne}."))) ((-4455 . T) (-4454 . T)) ((|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-1194))))) @@ -2347,7 +2347,7 @@ NIL (-604 |mn|) ((|constructor| (NIL "This domain implements low-level strings"))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (-3765 (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117)))) (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) +((-3763 (-12 (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (-3763 (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117)))) (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (-605 E V R P) ((|constructor| (NIL "tools for the summation packages.")) (|sum| (((|Record| (|:| |num| |#4|) (|:| |den| (|Integer|))) |#4| |#2|) "\\spad{sum(p(n), n)} returns \\spad{P(n)},{} the indefinite sum of \\spad{p(n)} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{P(n+1) - P(n) = a(n)}.") (((|Record| (|:| |num| |#4|) (|:| |den| (|Integer|))) |#4| |#2| (|Segment| |#4|)) "\\spad{sum(p(n), n = a..b)} returns \\spad{p(a) + p(a+1) + ... + p(b)}."))) NIL @@ -2355,7 +2355,7 @@ NIL (-606 |Coef|) ((|constructor| (NIL "InnerSparseUnivariatePowerSeries is an internal domain \\indented{2}{used for creating sparse Taylor and Laurent series.}")) (|cAcsch| (($ $) "\\spad{cAcsch(f)} computes the inverse hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsech| (($ $) "\\spad{cAsech(f)} computes the inverse hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcoth| (($ $) "\\spad{cAcoth(f)} computes the inverse hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtanh| (($ $) "\\spad{cAtanh(f)} computes the inverse hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcosh| (($ $) "\\spad{cAcosh(f)} computes the inverse hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsinh| (($ $) "\\spad{cAsinh(f)} computes the inverse hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsch| (($ $) "\\spad{cCsch(f)} computes the hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSech| (($ $) "\\spad{cSech(f)} computes the hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCoth| (($ $) "\\spad{cCoth(f)} computes the hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTanh| (($ $) "\\spad{cTanh(f)} computes the hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCosh| (($ $) "\\spad{cCosh(f)} computes the hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSinh| (($ $) "\\spad{cSinh(f)} computes the hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcsc| (($ $) "\\spad{cAcsc(f)} computes the arccosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsec| (($ $) "\\spad{cAsec(f)} computes the arcsecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcot| (($ $) "\\spad{cAcot(f)} computes the arccotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtan| (($ $) "\\spad{cAtan(f)} computes the arctangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcos| (($ $) "\\spad{cAcos(f)} computes the arccosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsin| (($ $) "\\spad{cAsin(f)} computes the arcsine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsc| (($ $) "\\spad{cCsc(f)} computes the cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSec| (($ $) "\\spad{cSec(f)} computes the secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCot| (($ $) "\\spad{cCot(f)} computes the cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTan| (($ $) "\\spad{cTan(f)} computes the tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCos| (($ $) "\\spad{cCos(f)} computes the cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSin| (($ $) "\\spad{cSin(f)} computes the sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cLog| (($ $) "\\spad{cLog(f)} computes the logarithm of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cExp| (($ $) "\\spad{cExp(f)} computes the exponential of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cRationalPower| (($ $ (|Fraction| (|Integer|))) "\\spad{cRationalPower(f,r)} computes \\spad{f^r}. For use when the coefficient ring is commutative.")) (|cPower| (($ $ |#1|) "\\spad{cPower(f,r)} computes \\spad{f^r},{} where \\spad{f} has constant coefficient 1. For use when the coefficient ring is commutative.")) (|integrate| (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. Warning: function does not check for a term of degree \\spad{-1}.")) (|seriesToOutputForm| (((|OutputForm|) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) (|Reference| (|OrderedCompletion| (|Integer|))) (|Symbol|) |#1| (|Fraction| (|Integer|))) "\\spad{seriesToOutputForm(st,refer,var,cen,r)} prints the series \\spad{f((var - cen)^r)}.")) (|iCompose| (($ $ $) "\\spad{iCompose(f,g)} returns \\spad{f(g(x))}. This is an internal function which should only be called for Taylor series \\spad{f(x)} and \\spad{g(x)} such that the constant coefficient of \\spad{g(x)} is zero.")) (|taylorQuoByVar| (($ $) "\\spad{taylorQuoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...}")) (|iExquo| (((|Union| $ "failed") $ $ (|Boolean|)) "\\spad{iExquo(f,g,taylor?)} is the quotient of the power series \\spad{f} and \\spad{g}. If \\spad{taylor?} is \\spad{true},{} then we must have \\spad{order(f) >= order(g)}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(fn,f)} returns the series \\spad{sum(fn(n) * an * x^n,n = n0..)},{} where \\spad{f} is the series \\spad{sum(an * x^n,n = n0..)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(f)} tests if \\spad{f} is a single monomial.")) (|series| (($ (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.")) (|getStream| (((|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) $) "\\spad{getStream(f)} returns the stream of terms representing the series \\spad{f}.")) (|getRef| (((|Reference| (|OrderedCompletion| (|Integer|))) $) "\\spad{getRef(f)} returns a reference containing the order to which the terms of \\spad{f} have been computed.")) (|makeSeries| (($ (|Reference| (|OrderedCompletion| (|Integer|))) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{makeSeries(refer,str)} creates a power series from the reference \\spad{refer} and the stream \\spad{str}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-575)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-575)) (|devaluate| |#1|)))) (|HasCategory| (-575) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -2883) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-575)))))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-575)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-575)) (|devaluate| |#1|)))) (|HasCategory| (-575) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -2882) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-575)))))) (-607 |Coef|) ((|constructor| (NIL "Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a \\spadtype{Stream} of \\spadtype{Ring} elements. For univariate series,{} the \\spad{Stream} elements are the Taylor coefficients. For multivariate series,{} the \\spad{n}th Stream element is a form of degree \\spad{n} in the power series variables.")) (* (($ $ (|Integer|)) "\\spad{x*i} returns the product of integer \\spad{i} and the series \\spad{x}.")) (|order| (((|NonNegativeInteger|) $ (|NonNegativeInteger|)) "\\spad{order(x,n)} returns the minimum of \\spad{n} and the order of \\spad{x}.") (((|NonNegativeInteger|) $) "\\spad{order(x)} returns the order of a power series \\spad{x},{} \\indented{1}{\\spadignore{i.e.} the degree of the first non-zero term of the series.}")) (|pole?| (((|Boolean|) $) "\\spad{pole?(x)} tests if the series \\spad{x} has a pole. \\indented{1}{Note: this is \\spad{false} when \\spad{x} is a Taylor series.}")) (|series| (($ (|Stream| |#1|)) "\\spad{series(s)} creates a power series from a stream of \\indented{1}{ring elements.} \\indented{1}{For univariate series types,{} the stream \\spad{s} should be a stream} \\indented{1}{of Taylor coefficients. For multivariate series types,{} the} \\indented{1}{stream \\spad{s} should be a stream of forms the \\spad{n}th element} \\indented{1}{of which is a} \\indented{1}{form of degree \\spad{n} in the power series variables.}")) (|coefficients| (((|Stream| |#1|) $) "\\spad{coefficients(x)} returns a stream of ring elements. \\indented{1}{When \\spad{x} is a univariate series,{} this is a stream of Taylor} \\indented{1}{coefficients. When \\spad{x} is a multivariate series,{} the} \\indented{1}{\\spad{n}th element of the stream is a form of} \\indented{1}{degree \\spad{n} in the power series variables.}"))) (((-4462 "*") |has| |#1| (-567)) (-4453 |has| |#1| (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) @@ -2372,7 +2372,7 @@ NIL ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|map| (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|InfiniteTuple| |#1|) (|Stream| |#2|)) "\\spad{map(f,a,b)} \\undocumented") (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|Stream| |#1|) (|InfiniteTuple| |#2|)) "\\spad{map(f,a,b)} \\undocumented") (((|InfiniteTuple| |#3|) (|Mapping| |#3| |#1| |#2|) (|InfiniteTuple| |#1|) (|InfiniteTuple| |#2|)) "\\spad{map(f,a,b)} \\undocumented"))) NIL NIL -(-611 R -3029 FG) +(-611 R -3027 FG) ((|constructor| (NIL "This package provides transformations from trigonometric functions to exponentials and logarithms,{} and back. \\spad{F} and \\spad{FG} should be the same type of function space.")) (|trigs2explogs| ((|#3| |#3| (|List| (|Kernel| |#3|)) (|List| (|Symbol|))) "\\spad{trigs2explogs(f, [k1,...,kn], [x1,...,xm])} rewrites all the trigonometric functions appearing in \\spad{f} and involving one of the \\spad{xi's} in terms of complex logarithms and exponentials. A kernel of the form \\spad{tan(u)} is expressed using \\spad{exp(u)**2} if it is one of the \\spad{ki's},{} in terms of \\spad{exp(2*u)} otherwise.")) (|explogs2trigs| (((|Complex| |#2|) |#3|) "\\spad{explogs2trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (F2FG ((|#3| |#2|) "\\spad{F2FG(a + sqrt(-1) b)} returns \\spad{a + i b}.")) (FG2F ((|#2| |#3|) "\\spad{FG2F(a + i b)} returns \\spad{a + sqrt(-1) b}.")) (GF2FG ((|#3| (|Complex| |#2|)) "\\spad{GF2FG(a + i b)} returns \\spad{a + i b} viewed as a function with the \\spad{i} pushed down into the coefficient domain."))) NIL NIL @@ -2383,7 +2383,7 @@ NIL (-613 R |mn|) ((|constructor| (NIL "\\indented{2}{This type represents vector like objects with varying lengths} and a user-specified initial index."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1019))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1019))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-614 S |Index| |Entry|) ((|constructor| (NIL "An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.")) (|swap!| (((|Void|) $ |#2| |#2|) "\\spad{swap!(u,i,j)} interchanges elements \\spad{i} and \\spad{j} of aggregate \\spad{u}. No meaningful value is returned.")) (|fill!| (($ $ |#3|) "\\spad{fill!(u,x)} replaces each entry in aggregate \\spad{u} by \\spad{x}. The modified \\spad{u} is returned as value.")) (|first| ((|#3| $) "\\spad{first(u)} returns the first element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{first([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = \\spad{x}}. Error: if \\spad{u} is empty.")) (|minIndex| ((|#2| $) "\\spad{minIndex(u)} returns the minimum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{minIndex(a) = reduce(min,{}[\\spad{i} for \\spad{i} in indices a])}; for lists,{} \\axiom{minIndex(a) = 1}.")) (|maxIndex| ((|#2| $) "\\spad{maxIndex(u)} returns the maximum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{maxIndex(\\spad{u}) = reduce(max,{}[\\spad{i} for \\spad{i} in indices \\spad{u}])}; if \\spad{u} is a list,{} \\axiom{maxIndex(\\spad{u}) = \\#u}.")) (|entry?| (((|Boolean|) |#3| $) "\\spad{entry?(x,u)} tests if \\spad{x} equals \\axiom{\\spad{u} . \\spad{i}} for some index \\spad{i}.")) (|indices| (((|List| |#2|) $) "\\spad{indices(u)} returns a list of indices of aggregate \\spad{u} in no particular order.")) (|index?| (((|Boolean|) |#2| $) "\\spad{index?(i,u)} tests if \\spad{i} is an index of aggregate \\spad{u}.")) (|entries| (((|List| |#3|) $) "\\spad{entries(u)} returns a list of all the entries of aggregate \\spad{u} in no assumed order."))) NIL @@ -2402,8 +2402,8 @@ NIL NIL (-618 R A) ((|constructor| (NIL "\\indented{1}{AssociatedJordanAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$A}} \\indented{1}{to define the new multiplications \\spad{a*b := (a *\\$A b + b *\\$A a)/2}} \\indented{1}{(anticommutator).} \\indented{1}{The usual notation \\spad{{a,b}_+} cannot be used due to} \\indented{1}{restrictions in the current language.} \\indented{1}{This domain only gives a Jordan algebra if the} \\indented{1}{Jordan-identity \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds} \\indented{1}{for all \\spad{a},{}\\spad{b},{}\\spad{c} in \\spad{A}.} \\indented{1}{This relation can be checked by} \\indented{1}{\\spadfun{jordanAdmissible?()\\$A}.} \\blankline If the underlying algebra is of type \\spadtype{FramedNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank,{} together with a fixed \\spad{R}-module basis),{} then the same is \\spad{true} for the associated Jordan algebra. Moreover,{} if the underlying algebra is of type \\spadtype{FiniteRankNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank),{} then the same \\spad{true} for the associated Jordan algebra.")) (|coerce| (($ |#2|) "\\spad{coerce(a)} coerces the element \\spad{a} of the algebra \\spad{A} to an element of the Jordan algebra \\spadtype{AssociatedJordanAlgebra}(\\spad{R},{}A)."))) -((-4457 -3765 (-3226 (|has| |#2| (-377 |#1|)) (|has| |#1| (-567))) (-12 (|has| |#2| (-428 |#1|)) (|has| |#1| (-567)))) (-4455 . T) (-4454 . T)) -((-3765 (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) +((-4457 -3763 (-3224 (|has| |#2| (-377 |#1|)) (|has| |#1| (-567))) (-12 (|has| |#2| (-428 |#1|)) (|has| |#1| (-567)))) (-4455 . T) (-4454 . T)) +((-3763 (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) (-619 |Entry|) ((|constructor| (NIL "This domain allows a random access file to be viewed both as a table and as a file object.")) (|pack!| (($ $) "\\spad{pack!(f)} reorganizes the file \\spad{f} on disk to recover unused space."))) ((-4460 . T) (-4461 . T)) @@ -2432,7 +2432,7 @@ NIL ((|constructor| (NIL "A is convertible to \\spad{B} means any element of A can be converted into an element of \\spad{B},{} but not automatically by the interpreter.")) (|convert| ((|#1| $) "\\spad{convert(a)} transforms a into an element of \\spad{S}."))) NIL NIL -(-626 -3029 UP) +(-626 -3027 UP) ((|constructor| (NIL "\\spadtype{Kovacic} provides a modified Kovacic\\spad{'s} algorithm for solving explicitely irreducible 2nd order linear ordinary differential equations.")) (|kovacic| (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{kovacic(a_0,a_1,a_2,ezfactor)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$e^{\\int(-a_1/2a_2)} e^{\\int u}\\$} is a solution of \\indented{5}{\\spad{\\$a_2 y'' + a_1 y' + a0 y = 0\\$}} whenever \\spad{u} is a solution of \\spad{P u = 0}. The equation must be already irreducible over the rational functions. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{kovacic(a_0,a_1,a_2)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$e^{\\int(-a_1/2a_2)} e^{\\int u}\\$} is a solution of \\indented{5}{\\spad{a_2 y'' + a_1 y' + a0 y = 0}} whenever \\spad{u} is a solution of \\spad{P u = 0}. The equation must be already irreducible over the rational functions."))) NIL NIL @@ -2460,7 +2460,7 @@ NIL ((|constructor| (NIL "LocalAlgebra produces the localization of an algebra,{} \\spadignore{i.e.} fractions whose numerators come from some \\spad{R} algebra.")) (|denom| ((|#3| $) "\\spad{denom x} returns the denominator of \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer x} returns the numerator of \\spad{x}.")) (/ (($ |#1| |#3|) "\\spad{a / d} divides the element \\spad{a} by \\spad{d}.") (($ $ |#3|) "\\spad{x / d} divides the element \\spad{x} by \\spad{d}."))) ((-4454 . T) (-4455 . T) (-4457 . T)) ((|HasCategory| |#1| (QUOTE (-859)))) -(-633 R -3029) +(-633 R -3027) ((|constructor| (NIL "This package computes the forward Laplace Transform.")) (|laplace| ((|#2| |#2| (|Symbol|) (|Symbol|)) "\\spad{laplace(f, t, s)} returns the Laplace transform of \\spad{f(t)} using \\spad{s} as the new variable. This is \\spad{integral(exp(-s*t)*f(t), t = 0..\\%plusInfinity)}. Returns the formal object \\spad{laplace(f, t, s)} if it cannot compute the transform."))) NIL NIL @@ -2492,18 +2492,18 @@ NIL ((|constructor| (NIL "Category for the transcendental Liouvillian functions.")) (|erf| (($ $) "\\spad{erf(x)} returns the error function of \\spad{x},{} \\spadignore{i.e.} \\spad{2 / sqrt(\\%pi)} times the integral of \\spad{exp(-x**2) dx}.")) (|dilog| (($ $) "\\spad{dilog(x)} returns the dilogarithm of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{log(x) / (1 - x) dx}.")) (|li| (($ $) "\\spad{li(x)} returns the logarithmic integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{dx / log(x)}.")) (|Ci| (($ $) "\\spad{Ci(x)} returns the cosine integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{cos(x) / x dx}.")) (|Si| (($ $) "\\spad{Si(x)} returns the sine integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{sin(x) / x dx}.")) (|Ei| (($ $) "\\spad{Ei(x)} returns the exponential integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{exp(x)/x dx}."))) NIL NIL -(-641 R -3029) +(-641 R -3027) ((|constructor| (NIL "This package provides liouvillian functions over an integral domain.")) (|integral| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{integral(f,x = a..b)} denotes the definite integral of \\spad{f} with respect to \\spad{x} from \\spad{a} to \\spad{b}.") ((|#2| |#2| (|Symbol|)) "\\spad{integral(f,x)} indefinite integral of \\spad{f} with respect to \\spad{x}.")) (|dilog| ((|#2| |#2|) "\\spad{dilog(f)} denotes the dilogarithm")) (|erf| ((|#2| |#2|) "\\spad{erf(f)} denotes the error function")) (|li| ((|#2| |#2|) "\\spad{li(f)} denotes the logarithmic integral")) (|Ci| ((|#2| |#2|) "\\spad{Ci(f)} denotes the cosine integral")) (|Si| ((|#2| |#2|) "\\spad{Si(f)} denotes the sine integral")) (|Ei| ((|#2| |#2|) "\\spad{Ei(f)} denotes the exponential integral")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns the Liouvillian operator based on \\spad{op}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} checks if \\spad{op} is Liouvillian"))) NIL NIL -(-642 |lv| -3029) +(-642 |lv| -3027) ((|constructor| (NIL "\\indented{1}{Given a Groebner basis \\spad{B} with respect to the total degree ordering for} a zero-dimensional ideal \\spad{I},{} compute a Groebner basis with respect to the lexicographical ordering by using linear algebra.")) (|transform| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{transform }\\undocumented")) (|choosemon| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{choosemon }\\undocumented")) (|intcompBasis| (((|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{intcompBasis }\\undocumented")) (|anticoord| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|List| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{anticoord }\\undocumented")) (|coord| (((|Vector| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{coord }\\undocumented")) (|computeBasis| (((|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{computeBasis }\\undocumented")) (|minPol| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|)) "\\spad{minPol }\\undocumented") (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|)) "\\spad{minPol }\\undocumented")) (|totolex| (((|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{totolex }\\undocumented")) (|groebgen| (((|Record| (|:| |glbase| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |glval| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{groebgen }\\undocumented")) (|linGenPos| (((|Record| (|:| |gblist| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |gvlist| (|List| (|Integer|)))) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{linGenPos }\\undocumented"))) NIL NIL (-643) ((|constructor| (NIL "This domain provides a simple way to save values in files.")) (|setelt| (((|Any|) $ (|Symbol|) (|Any|)) "\\spad{lib.k := v} saves the value \\spad{v} in the library \\spad{lib}. It can later be extracted using the key \\spad{k}.")) (|pack!| (($ $) "\\spad{pack!(f)} reorganizes the file \\spad{f} on disk to recover unused space.")) (|library| (($ (|FileName|)) "\\spad{library(ln)} creates a new library file."))) ((-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1176))) (LIST (QUOTE |:|) (QUOTE -3179) (QUOTE (-52))))))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-52) (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -318) (QUOTE (-52))))) (|HasCategory| (-1176) (QUOTE (-861))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117)))) +((-12 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1176))) (LIST (QUOTE |:|) (QUOTE -3179) (QUOTE (-52))))))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-52) (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -318) (QUOTE (-52))))) (|HasCategory| (-1176) (QUOTE (-861))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 (-52))) (QUOTE (-1117)))) (-644 S R) ((|constructor| (NIL "\\axiom{JacobiIdentity} means that \\axiom{[\\spad{x},{}[\\spad{y},{}\\spad{z}]]+[\\spad{y},{}[\\spad{z},{}\\spad{x}]]+[\\spad{z},{}[\\spad{x},{}\\spad{y}]] = 0} holds.")) (/ (($ $ |#2|) "\\axiom{\\spad{x/r}} returns the division of \\axiom{\\spad{x}} by \\axiom{\\spad{r}}.")) (|construct| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) NIL @@ -2514,8 +2514,8 @@ NIL NIL (-646 R A) ((|constructor| (NIL "AssociatedLieAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$A} to define the Lie bracket \\spad{a*b := (a *\\$A b - b *\\$A a)} (commutator). Note that the notation \\spad{[a,b]} cannot be used due to restrictions of the current compiler. This domain only gives a Lie algebra if the Jacobi-identity \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds for all \\spad{a},{}\\spad{b},{}\\spad{c} in \\spad{A}. This relation can be checked by \\spad{lieAdmissible?()\\$A}. \\blankline If the underlying algebra is of type \\spadtype{FramedNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank,{} together with a fixed \\spad{R}-module basis),{} then the same is \\spad{true} for the associated Lie algebra. Also,{} if the underlying algebra is of type \\spadtype{FiniteRankNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank),{} then the same is \\spad{true} for the associated Lie algebra.")) (|coerce| (($ |#2|) "\\spad{coerce(a)} coerces the element \\spad{a} of the algebra \\spad{A} to an element of the Lie algebra \\spadtype{AssociatedLieAlgebra}(\\spad{R},{}A)."))) -((-4457 -3765 (-3226 (|has| |#2| (-377 |#1|)) (|has| |#1| (-567))) (-12 (|has| |#2| (-428 |#1|)) (|has| |#1| (-567)))) (-4455 . T) (-4454 . T)) -((-3765 (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) +((-4457 -3763 (-3224 (|has| |#2| (-377 |#1|)) (|has| |#1| (-567))) (-12 (|has| |#2| (-428 |#1|)) (|has| |#1| (-567)))) (-4455 . T) (-4454 . T)) +((-3763 (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (LIST (QUOTE -428) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -377) (|devaluate| |#1|)))) (-647 R FE) ((|constructor| (NIL "PowerSeriesLimitPackage implements limits of expressions in one or more variables as one of the variables approaches a limiting value. Included are two-sided limits,{} left- and right- hand limits,{} and limits at plus or minus infinity.")) (|complexLimit| (((|Union| (|OnePointCompletion| |#2|) "failed") |#2| (|Equation| (|OnePointCompletion| |#2|))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit \\spad{lim(x -> a,f(x))}.")) (|limit| (((|Union| (|OrderedCompletion| |#2|) "failed") |#2| (|Equation| |#2|) (|String|)) "\\spad{limit(f(x),x=a,\"left\")} computes the left hand real limit \\spad{lim(x -> a-,f(x))}; \\spad{limit(f(x),x=a,\"right\")} computes the right hand real limit \\spad{lim(x -> a+,f(x))}.") (((|Union| (|OrderedCompletion| |#2|) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| |#2|) "failed")) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| |#2|) "failed"))) "failed") |#2| (|Equation| (|OrderedCompletion| |#2|))) "\\spad{limit(f(x),x = a)} computes the real limit \\spad{lim(x -> a,f(x))}."))) NIL @@ -2527,7 +2527,7 @@ NIL (-649 S R) ((|constructor| (NIL "Test for linear dependence.")) (|solveLinear| (((|Union| (|Vector| (|Fraction| |#1|)) "failed") (|Vector| |#2|) |#2|) "\\spad{solveLinear([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such \\spad{ci}\\spad{'s} exist in the quotient field of \\spad{S}.") (((|Union| (|Vector| |#1|) "failed") (|Vector| |#2|) |#2|) "\\spad{solveLinear([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such \\spad{ci}\\spad{'s} exist in \\spad{S}.")) (|linearDependence| (((|Union| (|Vector| |#1|) "failed") (|Vector| |#2|)) "\\spad{linearDependence([v1,...,vn])} returns \\spad{[c1,...,cn]} if \\spad{c1*v1 + ... + cn*vn = 0} and not all the \\spad{ci}\\spad{'s} are 0,{} \"failed\" if the \\spad{vi}\\spad{'s} are linearly independent over \\spad{S}.")) (|linearlyDependent?| (((|Boolean|) (|Vector| |#2|)) "\\spad{linearlyDependent?([v1,...,vn])} returns \\spad{true} if the \\spad{vi}\\spad{'s} are linearly dependent over \\spad{S},{} \\spad{false} otherwise."))) NIL -((-3215 (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-373)))) +((-3213 (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-373)))) (-650 R) ((|constructor| (NIL "An extension of left-module with an explicit linear dependence test.")) (|reducedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| |#1|))) (|Matrix| $) (|Vector| $)) "\\spad{reducedSystem(A, v)} returns a matrix \\spad{B} and a vector \\spad{w} such that \\spad{A x = v} and \\spad{B x = w} have the same solutions in \\spad{R}.") (((|Matrix| |#1|) (|Matrix| $)) "\\spad{reducedSystem(A)} returns a matrix \\spad{B} such that \\spad{A x = 0} and \\spad{B x = 0} have the same solutions in \\spad{R}.") (((|Matrix| |#1|) (|Vector| $)) "\\spad{reducedSystem [v1,...,vn]} returns a matrix \\spad{M} with coefficients in \\spad{R} such that the system of equations \\spad{c1*v1 + ... + cn*vn = 0\\$\\%} has the same solution as \\spad{c * M = 0} where \\spad{c} is the row vector \\spad{[c1,...cn]}."))) NIL @@ -2551,7 +2551,7 @@ NIL (-655 S) ((|constructor| (NIL "\\spadtype{List} implements singly-linked lists that are addressable by indices; the index of the first element is 1. In addition to the operations provided by \\spadtype{IndexedList},{} this constructor provides some LISP-like functions such as \\spadfun{null} and \\spadfun{cons}.")) (|setDifference| (($ $ $) "\\spad{setDifference(u1,u2)} returns a list of the elements of \\spad{u1} that are not also in \\spad{u2}. The order of elements in the resulting list is unspecified.")) (|setIntersection| (($ $ $) "\\spad{setIntersection(u1,u2)} returns a list of the elements that lists \\spad{u1} and \\spad{u2} have in common. The order of elements in the resulting list is unspecified.")) (|setUnion| (($ $ $) "\\spad{setUnion(u1,u2)} appends the two lists \\spad{u1} and \\spad{u2},{} then removes all duplicates. The order of elements in the resulting list is unspecified.")) (|append| (($ $ $) "\\spad{append(u1,u2)} appends the elements of list \\spad{u1} onto the front of list \\spad{u2}. This new list and \\spad{u2} will share some structure.")) (|cons| (($ |#1| $) "\\spad{cons(element,u)} appends \\spad{element} onto the front of list \\spad{u} and returns the new list. This new list and the old one will share some structure.")) (|null| (((|Boolean|) $) "\\spad{null(u)} tests if list \\spad{u} is the empty list.")) (|nil| (($) "\\spad{nil} is the empty list."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-839))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-839))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-656 T$) ((|constructor| (NIL "This domain represents AST for Spad literals."))) NIL @@ -2563,7 +2563,7 @@ NIL (-658 S) ((|substitute| (($ |#1| |#1| $) "\\spad{substitute(x,y,d)} replace \\spad{x}\\spad{'s} with \\spad{y}\\spad{'s} in dictionary \\spad{d}.")) (|duplicates?| (((|Boolean|) $) "\\spad{duplicates?(d)} tests if dictionary \\spad{d} has duplicate entries."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-659 R) ((|constructor| (NIL "The category of left modules over an \\spad{rng} (ring not necessarily with unit). This is an abelian group which supports left multiplation by elements of the \\spad{rng}. \\blankline"))) NIL @@ -2580,7 +2580,7 @@ NIL ((|constructor| (NIL "A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \\spadfun{concat} of two lists needs only to copy its first argument,{} whereas \\spadfun{concat} of two arrays needs to copy both arguments. Most of the operations exported here apply to infinite objects (\\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \\spadtype{FiniteLinearAggregate}.")) (|setelt| ((|#1| $ (|UniversalSegment| (|Integer|)) |#1|) "\\spad{setelt(u,i..j,x)} (also written: \\axiom{\\spad{u}(\\spad{i}..\\spad{j}) \\spad{:=} \\spad{x}}) destructively replaces each element in the segment \\axiom{\\spad{u}(\\spad{i}..\\spad{j})} by \\spad{x}. The value \\spad{x} is returned. Note: \\spad{u} is destructively change so that \\axiom{\\spad{u}.\\spad{k} \\spad{:=} \\spad{x} for \\spad{k} in \\spad{i}..\\spad{j}}; its length remains unchanged.")) (|insert| (($ $ $ (|Integer|)) "\\spad{insert(v,u,k)} returns a copy of \\spad{u} having \\spad{v} inserted beginning at the \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{v},{}\\spad{u},{}\\spad{k}) = concat( \\spad{u}(0..\\spad{k}-1),{} \\spad{v},{} \\spad{u}(\\spad{k}..) )}.") (($ |#1| $ (|Integer|)) "\\spad{insert(x,u,i)} returns a copy of \\spad{u} having \\spad{x} as its \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{x},{}a,{}\\spad{k}) = concat(concat(a(0..\\spad{k}-1),{}\\spad{x}),{}a(\\spad{k}..))}.")) (|delete| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete(u,i..j)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th through \\axiom{\\spad{j}}th element deleted. Note: \\axiom{delete(a,{}\\spad{i}..\\spad{j}) = concat(a(0..\\spad{i}-1),{}a(\\spad{j+1}..))}.") (($ $ (|Integer|)) "\\spad{delete(u,i)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th element deleted. Note: for lists,{} \\axiom{delete(a,{}\\spad{i}) \\spad{==} concat(a(0..\\spad{i} - 1),{}a(\\spad{i} + 1,{}..))}.")) (|map| (($ (|Mapping| |#1| |#1| |#1|) $ $) "\\spad{map(f,u,v)} returns a new collection \\spad{w} with elements \\axiom{\\spad{z} = \\spad{f}(\\spad{x},{}\\spad{y})} for corresponding elements \\spad{x} and \\spad{y} from \\spad{u} and \\spad{v}. Note: for linear aggregates,{} \\axiom{\\spad{w}.\\spad{i} = \\spad{f}(\\spad{u}.\\spad{i},{}\\spad{v}.\\spad{i})}.")) (|concat| (($ (|List| $)) "\\spad{concat(u)},{} where \\spad{u} is a lists of aggregates \\axiom{[a,{}\\spad{b},{}...,{}\\spad{c}]},{} returns a single aggregate consisting of the elements of \\axiom{a} followed by those of \\spad{b} followed ... by the elements of \\spad{c}. Note: \\axiom{concat(a,{}\\spad{b},{}...,{}\\spad{c}) = concat(a,{}concat(\\spad{b},{}...,{}\\spad{c}))}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} then \\axiom{\\spad{w}.\\spad{i} = \\spad{u}.\\spad{i} for \\spad{i} in indices \\spad{u}} and \\axiom{\\spad{w}.(\\spad{j} + maxIndex \\spad{u}) = \\spad{v}.\\spad{j} for \\spad{j} in indices \\spad{v}}.") (($ |#1| $) "\\spad{concat(x,u)} returns aggregate \\spad{u} with additional element at the front. Note: for lists: \\axiom{concat(\\spad{x},{}\\spad{u}) \\spad{==} concat([\\spad{x}],{}\\spad{u})}.") (($ $ |#1|) "\\spad{concat(u,x)} returns aggregate \\spad{u} with additional element \\spad{x} at the end. Note: for lists,{} \\axiom{concat(\\spad{u},{}\\spad{x}) \\spad{==} concat(\\spad{u},{}[\\spad{x}])}")) (|new| (($ (|NonNegativeInteger|) |#1|) "\\spad{new(n,x)} returns \\axiom{fill!(new \\spad{n},{}\\spad{x})}."))) NIL NIL -(-663 R -3029 L) +(-663 R -3027 L) ((|constructor| (NIL "\\spad{ElementaryFunctionLODESolver} provides the top-level functions for finding closed form solutions of linear ordinary differential equations and initial value problems.")) (|solve| (((|Union| |#2| "failed") |#3| |#2| (|Symbol|) |#2| (|List| |#2|)) "\\spad{solve(op, g, x, a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{op y = g, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; \\spad{x} is the dependent variable.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) "failed") |#3| |#2| (|Symbol|)) "\\spad{solve(op, g, x)} returns either a solution of the ordinary differential equation \\spad{op y = g} or \"failed\" if no non-trivial solution can be found; When found,{} the solution is returned in the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{op y = 0}. A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; \\spad{x} is the dependent variable."))) NIL NIL @@ -2600,11 +2600,11 @@ NIL ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|directSum| (($ $ $) "\\spad{directSum(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}.")) (|symmetricSquare| (($ $) "\\spad{symmetricSquare(a)} computes \\spad{symmetricProduct(a,a)} using a more efficient method.")) (|symmetricPower| (($ $ (|NonNegativeInteger|)) "\\spad{symmetricPower(a,n)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}.")) (|symmetricProduct| (($ $ $) "\\spad{symmetricProduct(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}.")) (|adjoint| (($ $) "\\spad{adjoint(a)} returns the adjoint operator of a.")) (D (($) "\\spad{D()} provides the operator corresponding to a derivation in the ring \\spad{A}."))) ((-4454 . T) (-4455 . T) (-4457 . T)) NIL -(-668 -3029 UP) +(-668 -3027 UP) ((|constructor| (NIL "\\spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a factorizer for linear ordinary differential operators whose coefficients are rational functions.")) (|factor1| (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{factor1(a)} returns the factorisation of a,{} assuming that a has no first-order right factor.")) (|factor| (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{factor(a)} returns the factorisation of a.") (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{factor(a, zeros)} returns the factorisation of a. \\spad{zeros} is a zero finder in \\spad{UP}."))) NIL ((|HasCategory| |#1| (QUOTE (-27)))) -(-669 A -3523) +(-669 A -3902) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator} defines a ring of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}"))) ((-4454 . T) (-4455 . T) (-4457 . T)) ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-373)))) @@ -2640,11 +2640,11 @@ NIL ((|constructor| (NIL "A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.")) (|list| (($ |#1|) "\\spad{list(x)} returns the list of one element \\spad{x}."))) ((-4461 . T) (-4460 . T)) NIL -(-678 -3029) +(-678 -3027) ((|constructor| (NIL "This package solves linear system in the matrix form \\spad{AX = B}. It is essentially a particular instantiation of the package \\spadtype{LinearSystemMatrixPackage} for Matrix and Vector. This package\\spad{'s} existence makes it easier to use \\spadfun{solve} in the AXIOM interpreter.")) (|rank| (((|NonNegativeInteger|) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{rank(A,B)} computes the rank of the complete matrix \\spad{(A|B)} of the linear system \\spad{AX = B}.")) (|hasSolution?| (((|Boolean|) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{hasSolution?(A,B)} tests if the linear system \\spad{AX = B} has a solution.")) (|particularSolution| (((|Union| (|Vector| |#1|) "failed") (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|))))) (|List| (|List| |#1|)) (|List| (|Vector| |#1|))) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|List| (|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|))))) (|Matrix| |#1|) (|List| (|Vector| |#1|))) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|)))) (|List| (|List| |#1|)) (|Vector| |#1|)) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}.") (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|)))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}."))) NIL NIL -(-679 -3029 |Row| |Col| M) +(-679 -3027 |Row| |Col| M) ((|constructor| (NIL "This package solves linear system in the matrix form \\spad{AX = B}.")) (|rank| (((|NonNegativeInteger|) |#4| |#3|) "\\spad{rank(A,B)} computes the rank of the complete matrix \\spad{(A|B)} of the linear system \\spad{AX = B}.")) (|hasSolution?| (((|Boolean|) |#4| |#3|) "\\spad{hasSolution?(A,B)} tests if the linear system \\spad{AX = B} has a solution.")) (|particularSolution| (((|Union| |#3| "failed") |#4| |#3|) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| |#3| "failed")) (|:| |basis| (|List| |#3|)))) |#4| (|List| |#3|)) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|Record| (|:| |particular| (|Union| |#3| "failed")) (|:| |basis| (|List| |#3|))) |#4| |#3|) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}."))) NIL NIL @@ -2655,7 +2655,7 @@ NIL (-681 |n| R) ((|constructor| (NIL "LieSquareMatrix(\\spad{n},{}\\spad{R}) implements the Lie algebra of the \\spad{n} by \\spad{n} matrices over the commutative ring \\spad{R}. The Lie bracket (commutator) of the algebra is given by \\spad{a*b := (a *\\$SQMATRIX(n,R) b - b *\\$SQMATRIX(n,R) a)},{} where \\spadfun{*\\$SQMATRIX(\\spad{n},{}\\spad{R})} is the usual matrix multiplication."))) ((-4457 . T) (-4460 . T) (-4454 . T) (-4455 . T)) -((|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (QUOTE (-237))) (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (-12 (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))))) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-567))) (-3765 (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) +((|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (QUOTE (-237))) (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (-12 (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))))) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-567))) (-3763 (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) (-682) ((|constructor| (NIL "This domain represents `literal sequence' syntax.")) (|elements| (((|List| (|SpadAst|)) $) "\\spad{elements(e)} returns the list of expressions in the `literal' list `e'."))) NIL @@ -2675,7 +2675,7 @@ NIL (-686 R) ((|constructor| (NIL "This domain represents three dimensional matrices over a general object type")) (|matrixDimensions| (((|Vector| (|NonNegativeInteger|)) $) "\\spad{matrixDimensions(x)} returns the dimensions of a matrix")) (|matrixConcat3D| (($ (|Symbol|) $ $) "\\spad{matrixConcat3D(s,x,y)} concatenates two 3-\\spad{D} matrices along a specified axis")) (|coerce| (((|PrimitiveArray| (|PrimitiveArray| (|PrimitiveArray| |#1|))) $) "\\spad{coerce(x)} moves from the domain to the representation type") (($ (|PrimitiveArray| (|PrimitiveArray| (|PrimitiveArray| |#1|)))) "\\spad{coerce(p)} moves from the representation type (PrimitiveArray PrimitiveArray PrimitiveArray \\spad{R}) to the domain")) (|setelt!| ((|#1| $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\spad{setelt!(x,i,j,k,s)} (or \\spad{x}.\\spad{i}.\\spad{j}.k:=s) sets a specific element of the array to some value of type \\spad{R}")) (|elt| ((|#1| $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{elt(x,i,j,k)} extract an element from the matrix \\spad{x}")) (|construct| (($ (|List| (|List| (|List| |#1|)))) "\\spad{construct(lll)} creates a 3-\\spad{D} matrix from a List List List \\spad{R} \\spad{lll}")) (|plus| (($ $ $) "\\spad{plus(x,y)} adds two matrices,{} term by term we note that they must be the same size")) (|identityMatrix| (($ (|NonNegativeInteger|)) "\\spad{identityMatrix(n)} create an identity matrix we note that this must be square")) (|zeroMatrix| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{zeroMatrix(i,j,k)} create a matrix with all zero terms"))) NIL -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-687) ((|constructor| (NIL "This domain represents the syntax of a macro definition.")) (|body| (((|SpadAst|) $) "\\spad{body(m)} returns the right hand side of the definition \\spad{`m'}.")) (|head| (((|HeadAst|) $) "\\spad{head(m)} returns the head of the macro definition \\spad{`m'}. This is a list of identifiers starting with the name of the macro followed by the name of the parameters,{} if any."))) NIL @@ -2731,7 +2731,7 @@ NIL (-700 R) ((|constructor| (NIL "\\spadtype{Matrix} is a matrix domain where 1-based indexing is used for both rows and columns.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|diagonalMatrix| (($ (|Vector| |#1|)) "\\spad{diagonalMatrix(v)} returns a diagonal matrix where the elements of \\spad{v} appear on the diagonal."))) ((-4460 . T) (-4461 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-567))) (|HasAttribute| |#1| (QUOTE (-4462 "*"))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-316))) (|HasCategory| |#1| (QUOTE (-567))) (|HasAttribute| |#1| (QUOTE (-4462 "*"))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-701 R) ((|constructor| (NIL "This package provides standard arithmetic operations on matrices. The functions in this package store the results of computations in existing matrices,{} rather than creating new matrices. This package works only for matrices of type Matrix and uses the internal representation of this type.")) (** (((|Matrix| |#1|) (|Matrix| |#1|) (|NonNegativeInteger|)) "\\spad{x ** n} computes the \\spad{n}-th power of a square matrix. The power \\spad{n} is assumed greater than 1.")) (|power!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|NonNegativeInteger|)) "\\spad{power!(a,b,c,m,n)} computes \\spad{m} \\spad{**} \\spad{n} and stores the result in \\spad{a}. The matrices \\spad{b} and \\spad{c} are used to store intermediate results. Error: if \\spad{a},{} \\spad{b},{} \\spad{c},{} and \\spad{m} are not square and of the same dimensions.")) (|times!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{times!(c,a,b)} computes the matrix product \\spad{a * b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have compatible dimensions.")) (|rightScalarTimes!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rightScalarTimes!(c,a,r)} computes the scalar product \\spad{a * r} and stores the result in the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions.")) (|leftScalarTimes!| (((|Matrix| |#1|) (|Matrix| |#1|) |#1| (|Matrix| |#1|)) "\\spad{leftScalarTimes!(c,r,a)} computes the scalar product \\spad{r * a} and stores the result in the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions.")) (|minus!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{!minus!(c,a,b)} computes the matrix difference \\spad{a - b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have the same dimensions.") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{minus!(c,a)} computes \\spad{-a} and stores the result in the matrix \\spad{c}. Error: if a and \\spad{c} do not have the same dimensions.")) (|plus!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{plus!(c,a,b)} computes the matrix sum \\spad{a + b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have the same dimensions.")) (|copy!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{copy!(c,a)} copies the matrix \\spad{a} into the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions."))) NIL @@ -2740,7 +2740,7 @@ NIL ((|constructor| (NIL "This domain implements the notion of optional value,{} where a computation may fail to produce expected value.")) (|nothing| (($) "\\spad{nothing} represents failure or absence of value.")) (|autoCoerce| ((|#1| $) "\\spad{autoCoerce} is a courtesy coercion function used by the compiler in case it knows that \\spad{`x'} really is a \\spadtype{T}.")) (|case| (((|Boolean|) $ (|[\|\|]| |nothing|)) "\\spad{x case nothing} holds if the value for \\spad{x} is missing.") (((|Boolean|) $ (|[\|\|]| |#1|)) "\\spad{x case T} returns \\spad{true} if \\spad{x} is actually a data of type \\spad{T}.")) (|just| (($ |#1|) "\\spad{just x} injects the value \\spad{`x'} into \\%."))) NIL NIL -(-703 S -3029 FLAF FLAS) +(-703 S -3027 FLAF FLAS) ((|constructor| (NIL "\\indented{1}{\\spadtype{MultiVariableCalculusFunctions} Package provides several} \\indented{1}{functions for multivariable calculus.} These include gradient,{} hessian and jacobian,{} divergence and laplacian. Various forms for banded and sparse storage of matrices are included.")) (|bandedJacobian| (((|Matrix| |#2|) |#3| |#4| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{bandedJacobian(vf,xlist,kl,ku)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \\spad{vf},{} \\spad{vf} a vector function of the variables listed in \\spad{xlist},{} \\spad{kl} is the number of nonzero subdiagonals,{} \\spad{ku} is the number of nonzero superdiagonals,{} kl+ku+1 being actual bandwidth. Stores the nonzero band in a matrix,{} dimensions kl+ku+1 by \\#xlist. The upper triangle is in the top \\spad{ku} rows,{} the diagonal is in row ku+1,{} the lower triangle in the last \\spad{kl} rows. Entries in a column in the band store correspond to entries in same column of full store. (The notation conforms to LAPACK/NAG-\\spad{F07} conventions.)")) (|jacobian| (((|Matrix| |#2|) |#3| |#4|) "\\spad{jacobian(vf,xlist)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \\spad{vf},{} \\spad{vf} a vector function of the variables listed in \\spad{xlist}.")) (|bandedHessian| (((|Matrix| |#2|) |#2| |#4| (|NonNegativeInteger|)) "\\spad{bandedHessian(v,xlist,k)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist},{} \\spad{k} is the semi-bandwidth,{} the number of nonzero subdiagonals,{} 2*k+1 being actual bandwidth. Stores the nonzero band in lower triangle in a matrix,{} dimensions \\spad{k+1} by \\#xlist,{} whose rows are the vectors formed by diagonal,{} subdiagonal,{} etc. of the real,{} full-matrix,{} hessian. (The notation conforms to LAPACK/NAG-\\spad{F07} conventions.)")) (|hessian| (((|Matrix| |#2|) |#2| |#4|) "\\spad{hessian(v,xlist)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist}.")) (|laplacian| ((|#2| |#2| |#4|) "\\spad{laplacian(v,xlist)} computes the laplacian of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist}.")) (|divergence| ((|#2| |#3| |#4|) "\\spad{divergence(vf,xlist)} computes the divergence of the vector field \\spad{vf},{} \\spad{vf} a vector function of the variables listed in \\spad{xlist}.")) 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T) (-4459 |has| (-710) (-6 -4459)) (-4456 |has| (-710) (-6 -4456)) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . 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As for any dictionary,{} its size is assumed large so that copying (non-destructive) operations are generally to be avoided.")) (|duplicates| (((|List| (|Record| (|:| |entry| |#1|) (|:| |count| (|NonNegativeInteger|)))) $) "\\spad{duplicates(d)} returns a list of values which have duplicates in \\spad{d}")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(d)} destructively removes any duplicate values in dictionary \\spad{d}.")) (|insert!| (($ |#1| $ (|NonNegativeInteger|)) "\\spad{insert!(x,d,n)} destructively inserts \\spad{n} copies of \\spad{x} into dictionary \\spad{d}."))) ((-4461 . T)) @@ -2764,13 +2764,13 @@ NIL ((|constructor| (NIL "\\indented{1}{<description of package>} Author: Jim Wen Date Created: \\spad{??} Date Last Updated: October 1991 by Jon Steinbach Keywords: Examples: References:")) (|ptFunc| (((|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{ptFunc(a,b,c,d)} is an internal function exported in order to compile packages.")) (|meshPar1Var| (((|ThreeSpace| (|DoubleFloat|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar1Var(s,t,u,f,s1,l)} \\undocumented")) (|meshFun2Var| (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Union| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "undefined") (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshFun2Var(f,g,s1,s2,l)} \\undocumented")) (|meshPar2Var| (((|ThreeSpace| (|DoubleFloat|)) (|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(sp,f,s1,s2,l)} \\undocumented") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,s1,s2,l)} \\undocumented") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Union| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "undefined") (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,g,h,j,s1,s2,l)} \\undocumented"))) NIL NIL -(-709 OV E -3029 PG) +(-709 OV E -3027 PG) ((|constructor| (NIL "Package for factorization of multivariate polynomials over finite fields.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factor(p)} produces the complete factorization of the multivariate polynomial \\spad{p} over a finite field. \\spad{p} is represented as a univariate polynomial with multivariate coefficients over a finite field.") (((|Factored| |#4|) |#4|) "\\spad{factor(p)} produces the complete factorization of the multivariate polynomial \\spad{p} over a finite field."))) NIL NIL (-710) ((|constructor| (NIL "A domain which models the floating point representation used by machines in the AXIOM-NAG link.")) (|changeBase| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{changeBase(exp,man,base)} \\undocumented{}")) (|exponent| (((|Integer|) $) "\\spad{exponent(u)} returns the exponent of \\spad{u}")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(u)} returns the mantissa of \\spad{u}")) (|coerce| (($ (|MachineInteger|)) "\\spad{coerce(u)} transforms a MachineInteger into a MachineFloat") (((|Float|) $) "\\spad{coerce(u)} transforms a MachineFloat to a standard Float")) (|minimumExponent| (((|Integer|)) "\\spad{minimumExponent()} returns the minimum exponent in the model") (((|Integer|) (|Integer|)) "\\spad{minimumExponent(e)} sets the minimum exponent in the model to \\spad{e}")) (|maximumExponent| (((|Integer|)) "\\spad{maximumExponent()} returns the maximum exponent in the model") (((|Integer|) (|Integer|)) "\\spad{maximumExponent(e)} sets the maximum exponent in the model to \\spad{e}")) (|base| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{base(b)} sets the base of the model to \\spad{b}")) (|precision| (((|PositiveInteger|)) "\\spad{precision()} returns the number of digits in the model") (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(p)} sets the number of digits in the model to \\spad{p}"))) -((-3494 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) +((-3493 . T) (-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-711 R) ((|constructor| (NIL "\\indented{1}{Modular hermitian row reduction.} Author: Manuel Bronstein Date Created: 22 February 1989 Date Last Updated: 24 November 1993 Keywords: matrix,{} reduction.")) (|normalizedDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{normalizedDivide(n,d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \\spadignore{e.g.} positive remainders")) (|rowEchelonLocal| (((|Matrix| |#1|) (|Matrix| |#1|) |#1| |#1|) "\\spad{rowEchelonLocal(m, d, p)} computes the row-echelon form of \\spad{m} concatenated with \\spad{d} times the identity matrix over a local ring where \\spad{p} is the only prime.")) (|rowEchLocal| (((|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rowEchLocal(m,p)} computes a modular row-echelon form of \\spad{m},{} finding an appropriate modulus over a local ring where \\spad{p} is the only prime.")) (|rowEchelon| (((|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rowEchelon(m, d)} computes a modular row-echelon form mod \\spad{d} of \\indented{3}{[\\spad{d}\\space{5}]} \\indented{3}{[\\space{2}\\spad{d}\\space{3}]} \\indented{3}{[\\space{4}. ]} \\indented{3}{[\\space{5}\\spad{d}]} \\indented{3}{[\\space{3}\\spad{M}\\space{2}]} where \\spad{M = m mod d}.")) (|rowEch| (((|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{rowEch(m)} computes a modular row-echelon form of \\spad{m},{} finding an appropriate modulus."))) @@ -2796,7 +2796,7 @@ NIL ((|constructor| (NIL "MakeRecord is used internally by the interpreter to create record types which are used for doing parallel iterations on streams.")) (|makeRecord| (((|Record| (|:| |part1| |#1|) (|:| |part2| |#2|)) |#1| |#2|) "\\spad{makeRecord(a,b)} creates a record object with type Record(part1:S,{} part2:R),{} where part1 is \\spad{a} and part2 is \\spad{b}."))) NIL NIL -(-717 S -3430 I) +(-717 S -3428 I) ((|constructor| (NIL "transforms top-level objects into compiled functions.")) (|compiledFunction| (((|Mapping| |#3| |#2|) |#1| (|Symbol|)) "\\spad{compiledFunction(expr, x)} returns a function \\spad{f: D -> I} defined by \\spad{f(x) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{D}.")) (|unaryFunction| (((|Mapping| |#3| |#2|) (|Symbol|)) "\\spad{unaryFunction(a)} is a local function"))) NIL NIL @@ -2816,14 +2816,14 @@ NIL ((|constructor| (NIL "\\spadtype{MathMLFormat} provides a coercion from \\spadtype{OutputForm} to MathML format.")) (|display| (((|Void|) (|String|)) "prints the string returned by coerce,{} adding <math ...> tags.")) (|exprex| (((|String|) (|OutputForm|)) "coverts \\spadtype{OutputForm} to \\spadtype{String} with the structure preserved with braces. Actually this is not quite accurate. The function \\spadfun{precondition} is first applied to the \\spadtype{OutputForm} expression before \\spadfun{exprex}. The raw \\spadtype{OutputForm} and the nature of the \\spadfun{precondition} function is still obscure to me at the time of this writing (2007-02-14).")) (|coerceL| (((|String|) (|OutputForm|)) "coerceS(\\spad{o}) changes \\spad{o} in the standard output format to MathML format and displays result as one long string.")) (|coerceS| (((|String|) (|OutputForm|)) "\\spad{coerceS(o)} changes \\spad{o} in the standard output format to MathML format and displays formatted result.")) (|coerce| (((|String|) (|OutputForm|)) "coerceS(\\spad{o}) changes \\spad{o} in the standard output format to MathML format."))) 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T)) NIL (-723 R |Rep|) ((|constructor| (NIL "This package \\undocumented")) (|frobenius| (($ $) "\\spad{frobenius(x)} \\undocumented")) (|computePowers| (((|PrimitiveArray| $)) "\\spad{computePowers()} \\undocumented")) (|pow| (((|PrimitiveArray| $)) "\\spad{pow()} \\undocumented")) (|An| (((|Vector| |#1|) $) "\\spad{An(x)} \\undocumented")) (|UnVectorise| (($ (|Vector| |#1|)) "\\spad{UnVectorise(v)} \\undocumented")) (|Vectorise| (((|Vector| |#1|) $) "\\spad{Vectorise(x)} \\undocumented")) (|lift| ((|#2| $) "\\spad{lift(x)} \\undocumented")) (|reduce| (($ |#2|) "\\spad{reduce(x)} \\undocumented")) (|modulus| ((|#2|) "\\spad{modulus()} \\undocumented")) (|setPoly| ((|#2| |#2|) "\\spad{setPoly(x)} \\undocumented"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4456 |has| |#1| (-373)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . 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(-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-1169))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-238))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-724 IS E |ff|) ((|constructor| (NIL "This package \\undocumented")) (|construct| (($ |#1| |#2|) "\\spad{construct(i,e)} \\undocumented")) (|index| ((|#1| $) "\\spad{index(x)} \\undocumented")) (|exponent| ((|#2| $) "\\spad{exponent(x)} \\undocumented"))) NIL @@ -2832,7 +2832,7 @@ NIL ((|constructor| (NIL "Algebra of ADDITIVE operators on a module.")) (|makeop| (($ |#1| (|FreeGroup| (|BasicOperator|))) "\\spad{makeop should} be local but conditional")) (|opeval| ((|#2| (|BasicOperator|) |#2|) "\\spad{opeval should} be local but conditional")) (** (($ $ (|Integer|)) "\\spad{op**n} \\undocumented") (($ (|BasicOperator|) (|Integer|)) "\\spad{op**n} \\undocumented")) (|evaluateInverse| (($ $ (|Mapping| |#2| |#2|)) "\\spad{evaluateInverse(x,f)} \\undocumented")) (|evaluate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{evaluate(f, u +-> g u)} attaches the map \\spad{g} to \\spad{f}. \\spad{f} must be a basic operator \\spad{g} MUST be additive,{} \\spadignore{i.e.} \\spad{g(a + b) = g(a) + g(b)} for any \\spad{a},{} \\spad{b} in \\spad{M}. This implies that \\spad{g(n a) = n g(a)} for any \\spad{a} in \\spad{M} and integer \\spad{n > 0}.")) (|conjug| ((|#1| |#1|) "\\spad{conjug(x)}should be local but conditional")) (|adjoint| (($ $ $) "\\spad{adjoint(op1, op2)} sets the adjoint of \\spad{op1} to be op2. \\spad{op1} must be a basic operator") (($ $) "\\spad{adjoint(op)} returns the adjoint of the operator \\spad{op}."))) ((-4455 |has| |#1| (-174)) (-4454 |has| |#1| (-174)) (-4457 . T)) ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148)))) -(-726 R |Mod| -4256 -3794 |exactQuo|) +(-726 R |Mod| -1745 -4167 |exactQuo|) ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{EuclideanModularRing} ,{}\\spadtype{ModularField}")) (|inv| (($ $) "\\spad{inv(x)} \\undocumented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} \\undocumented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#1| |#2|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#1| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#2| $) "\\spad{modulus(x)} \\undocumented"))) ((-4457 . T)) NIL @@ -2844,7 +2844,7 @@ NIL ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) ((-4455 . T) (-4454 . T)) NIL -(-729 -3029) +(-729 -3027) ((|constructor| (NIL "\\indented{1}{MoebiusTransform(\\spad{F}) is the domain of fractional linear (Moebius)} transformations over \\spad{F}.")) (|eval| (((|OnePointCompletion| |#1|) $ (|OnePointCompletion| |#1|)) "\\spad{eval(m,x)} returns \\spad{(a*x + b)/(c*x + d)} where \\spad{m = moebius(a,b,c,d)} (see \\spadfunFrom{moebius}{MoebiusTransform}).") ((|#1| $ |#1|) "\\spad{eval(m,x)} returns \\spad{(a*x + b)/(c*x + d)} where \\spad{m = moebius(a,b,c,d)} (see \\spadfunFrom{moebius}{MoebiusTransform}).")) (|recip| (($ $) "\\spad{recip(m)} = recip() * \\spad{m}") (($) "\\spad{recip()} returns \\spad{matrix [[0,1],[1,0]]} representing the map \\spad{x -> 1 / x}.")) (|scale| (($ $ |#1|) "\\spad{scale(m,h)} returns \\spad{scale(h) * m} (see \\spadfunFrom{shift}{MoebiusTransform}).") (($ |#1|) "\\spad{scale(k)} returns \\spad{matrix [[k,0],[0,1]]} representing the map \\spad{x -> k * x}.")) (|shift| (($ $ |#1|) "\\spad{shift(m,h)} returns \\spad{shift(h) * m} (see \\spadfunFrom{shift}{MoebiusTransform}).") (($ |#1|) "\\spad{shift(k)} returns \\spad{matrix [[1,k],[0,1]]} representing the map \\spad{x -> x + k}.")) (|moebius| (($ |#1| |#1| |#1| |#1|) "\\spad{moebius(a,b,c,d)} returns \\spad{matrix [[a,b],[c,d]]}."))) ((-4457 . T)) NIL @@ -2880,7 +2880,7 @@ NIL ((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with a multiplicative identity element. \\blankline")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} tries to compute the multiplicative inverse for \\spad{x} or \"failed\" if it cannot find the inverse (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (|one?| (((|Boolean|) $) "\\spad{one?(x)} tests if \\spad{x} is equal to 1.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) ((|One|) (($) "1 is the multiplicative identity."))) NIL NIL -(-738 -3029 UP) +(-738 -3027 UP) ((|constructor| (NIL "Tools for handling monomial extensions.")) (|decompose| (((|Record| (|:| |poly| |#2|) (|:| |normal| (|Fraction| |#2|)) (|:| |special| (|Fraction| |#2|))) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{decompose(f, D)} returns \\spad{[p,n,s]} such that \\spad{f = p+n+s},{} all the squarefree factors of \\spad{denom(n)} are normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{denom(s)} is special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and \\spad{n} and \\spad{s} are proper fractions (no pole at infinity). \\spad{D} is the derivation to use.")) (|normalDenom| ((|#2| (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{normalDenom(f, D)} returns the product of all the normal factors of \\spad{denom(f)}. \\spad{D} is the derivation to use.")) (|splitSquarefree| (((|Record| (|:| |normal| (|Factored| |#2|)) (|:| |special| (|Factored| |#2|))) |#2| (|Mapping| |#2| |#2|)) "\\spad{splitSquarefree(p, D)} returns \\spad{[n_1 n_2\\^2 ... n_m\\^m, s_1 s_2\\^2 ... s_q\\^q]} such that \\spad{p = n_1 n_2\\^2 ... n_m\\^m s_1 s_2\\^2 ... s_q\\^q},{} each \\spad{n_i} is normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D} and each \\spad{s_i} is special \\spad{w}.\\spad{r}.\\spad{t} \\spad{D}. \\spad{D} is the derivation to use.")) (|split| (((|Record| (|:| |normal| |#2|) (|:| |special| |#2|)) |#2| (|Mapping| |#2| |#2|)) "\\spad{split(p, D)} returns \\spad{[n,s]} such that \\spad{p = n s},{} all the squarefree factors of \\spad{n} are normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and \\spad{s} is special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D}. \\spad{D} is the derivation to use."))) NIL NIL @@ -2899,7 +2899,7 @@ NIL (-742 |vl| R) ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are from a user specified list of symbols. The ordering is specified by the position of the variable in the list. The coefficient ring may be non commutative,{} but the variables are assumed to commute."))) (((-4462 "*") |has| |#2| (-174)) (-4453 |has| |#2| (-567)) (-4458 |has| |#2| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#2| (QUOTE (-924))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) +((|HasCategory| |#2| (QUOTE (-924))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-875 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) (-743 E OV R PRF) ((|constructor| (NIL "\\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are rational functions over some ring \\spad{R} over which we can factor. It is used internally by packages such as primary decomposition which need to work with polynomials with rational function coefficients,{} \\spadignore{i.e.} themselves fractions of polynomials.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(prf)} factors a polynomial with rational function coefficients.")) (|pushuconst| ((|#4| (|Fraction| (|Polynomial| |#3|)) |#2|) "\\spad{pushuconst(r,var)} takes a rational function and raises all occurances of the variable \\spad{var} to the polynomial level.")) (|pushucoef| ((|#4| (|SparseUnivariatePolynomial| (|Polynomial| |#3|)) |#2|) "\\spad{pushucoef(upoly,var)} converts the anonymous univariate polynomial \\spad{upoly} to a polynomial in \\spad{var} over rational functions.")) (|pushup| ((|#4| |#4| |#2|) "\\spad{pushup(prf,var)} raises all occurences of the variable \\spad{var} in the coefficients of the polynomial \\spad{prf} back to the polynomial level.")) (|pushdterm| ((|#4| (|SparseUnivariatePolynomial| |#4|) |#2|) "\\spad{pushdterm(monom,var)} pushes all top level occurences of the variable \\spad{var} into the coefficient domain for the monomial \\spad{monom}.")) (|pushdown| ((|#4| |#4| |#2|) "\\spad{pushdown(prf,var)} pushes all top level occurences of the variable \\spad{var} into the coefficient domain for the polynomial \\spad{prf}.")) (|totalfract| (((|Record| (|:| |sup| (|Polynomial| |#3|)) (|:| |inf| (|Polynomial| |#3|))) |#4|) "\\spad{totalfract(prf)} takes a polynomial whose coefficients are themselves fractions of polynomials and returns a record containing the numerator and denominator resulting from putting \\spad{prf} over a common denominator.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) NIL @@ -3032,11 +3032,11 @@ NIL ((|constructor| (NIL "This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the complex rational numbers. The results are expressed either as complex floating numbers or as complex rational numbers depending on the type of the precision parameter.")) (|complexEigenvectors| (((|List| (|Record| (|:| |outval| (|Complex| |#1|)) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| (|Complex| |#1|)))))) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) |#1|) "\\spad{complexEigenvectors(m,eps)} returns a list of records each one containing a complex eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \\spad{eps} and are expressed as complex floats or complex rational numbers depending on the type of \\spad{eps} (float or rational).")) (|complexEigenvalues| (((|List| (|Complex| |#1|)) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) |#1|) "\\spad{complexEigenvalues(m,eps)} computes the eigenvalues of the matrix \\spad{m} to precision \\spad{eps}. The eigenvalues are expressed as complex floats or complex rational numbers depending on the type of \\spad{eps} (float or rational).")) (|characteristicPolynomial| (((|Polynomial| (|Complex| (|Fraction| (|Integer|)))) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) (|Symbol|)) "\\spad{characteristicPolynomial(m,x)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over Complex Rationals with variable \\spad{x}.") (((|Polynomial| (|Complex| (|Fraction| (|Integer|)))) (|Matrix| (|Complex| (|Fraction| (|Integer|))))) "\\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over complex rationals with a new symbol as variable."))) NIL NIL -(-776 -3029) +(-776 -3027) ((|constructor| (NIL "\\spadtype{NumericContinuedFraction} provides functions \\indented{2}{for converting floating point numbers to continued fractions.}")) (|continuedFraction| (((|ContinuedFraction| (|Integer|)) |#1|) "\\spad{continuedFraction(f)} converts the floating point number \\spad{f} to a reduced continued fraction."))) NIL NIL -(-777 P -3029) +(-777 P -3027) ((|constructor| (NIL "This package provides a division and related operations for \\spadtype{MonogenicLinearOperator}\\spad{s} over a \\spadtype{Field}. Since the multiplication is in general non-commutative,{} these operations all have left- and right-hand versions. This package provides the operations based on left-division.")) (|leftLcm| ((|#1| |#1| |#1|) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftGcd| ((|#1| |#1| |#1|) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| |#1| "failed") |#1| |#1|) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| ((|#1| |#1| |#1|) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| ((|#1| |#1| |#1|) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division\\spad{''}."))) NIL NIL @@ -3044,7 +3044,7 @@ NIL NIL NIL NIL -(-779 UP -3029) +(-779 UP -3027) ((|constructor| (NIL "In this package \\spad{F} is a framed algebra over the integers (typically \\spad{F = Z[a]} for some algebraic integer a). The package provides functions to compute the integral closure of \\spad{Z} in the quotient quotient field of \\spad{F}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| (|Integer|))) (|:| |basisDen| (|Integer|)) (|:| |basisInv| (|Matrix| (|Integer|)))) (|Integer|)) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{Z} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{Z}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| (|Integer|))) (|:| |basisDen| (|Integer|)) (|:| |basisInv| (|Matrix| (|Integer|))))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{Z} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{Z}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|discriminant| (((|Integer|)) "\\spad{discriminant()} returns the discriminant of the integral closure of \\spad{Z} in the quotient field of the framed algebra \\spad{F}."))) NIL NIL @@ -3060,7 +3060,7 @@ NIL ((|constructor| (NIL "\\spadtype{NonNegativeInteger} provides functions for non \\indented{2}{negative integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : \\spad{x*y = y*x}.")) (|random| (($ $) "\\spad{random(n)} returns a random integer from 0 to \\spad{n-1}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} bits.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} returns the quotient of \\spad{a} and \\spad{b},{} or \"failed\" if \\spad{b} is zero or \\spad{a} rem \\spad{b} is zero.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(a,b)} returns a record containing both remainder and quotient.")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two non negative integers \\spad{a} and \\spad{b}.")) (|rem| (($ $ $) "\\spad{a rem b} returns the remainder of \\spad{a} and \\spad{b}.")) (|quo| (($ $ $) "\\spad{a quo b} returns the quotient of \\spad{a} and \\spad{b},{} forgetting the remainder."))) (((-4462 "*") . T)) NIL -(-783 R -3029) +(-783 R -3027) ((|constructor| (NIL "NonLinearFirstOrderODESolver provides a function for finding closed form first integrals of nonlinear ordinary differential equations of order 1.")) (|solve| (((|Union| |#2| "failed") |#2| |#2| (|BasicOperator|) (|Symbol|)) "\\spad{solve(M(x,y), N(x,y), y, x)} returns \\spad{F(x,y)} such that \\spad{F(x,y) = c} for a constant \\spad{c} is a first integral of the equation \\spad{M(x,y) dx + N(x,y) dy = 0},{} or \"failed\" if no first-integral can be found."))) NIL NIL @@ -3080,7 +3080,7 @@ NIL ((|constructor| (NIL "A package for computing normalized assocites of univariate polynomials with coefficients in a tower of simple extensions of a field.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of \\spad{gcd} over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of AAECC11} \\indented{5}{Paris,{} 1995.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.}")) (|normInvertible?| (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{normInvertible?(\\spad{p},{}\\spad{ts})} is an internal subroutine,{} exported only for developement.")) (|outputArgs| (((|Void|) (|String|) (|String|) |#4| |#5|) "\\axiom{outputArgs(\\spad{s1},{}\\spad{s2},{}\\spad{p},{}\\spad{ts})} is an internal subroutine,{} exported only for developement.")) (|normalize| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{normalize(\\spad{p},{}\\spad{ts})} normalizes \\axiom{\\spad{p}} \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}.")) (|normalizedAssociate| ((|#4| |#4| |#5|) "\\axiom{normalizedAssociate(\\spad{p},{}\\spad{ts})} returns a normalized polynomial \\axiom{\\spad{n}} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts} such that \\axiom{\\spad{n}} and \\axiom{\\spad{p}} are associates \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts} and assuming that \\axiom{\\spad{p}} is invertible \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}.")) (|recip| (((|Record| (|:| |num| |#4|) (|:| |den| |#4|)) |#4| |#5|) "\\axiom{recip(\\spad{p},{}\\spad{ts})} returns the inverse of \\axiom{\\spad{p}} \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts} assuming that \\axiom{\\spad{p}} is invertible \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}."))) NIL NIL -(-788 -3029 |ExtF| |SUEx| |ExtP| |n|) +(-788 -3027 |ExtF| |SUEx| |ExtP| |n|) ((|constructor| (NIL "This package \\undocumented")) (|Frobenius| ((|#4| |#4|) "\\spad{Frobenius(x)} \\undocumented")) (|retractIfCan| (((|Union| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|)) "failed") |#4|) "\\spad{retractIfCan(x)} \\undocumented")) (|normFactors| (((|List| |#4|) |#4|) "\\spad{normFactors(x)} \\undocumented"))) NIL NIL @@ -3095,7 +3095,7 @@ NIL (-791 R |VarSet|) ((|constructor| (NIL "A post-facto extension for \\axiomType{\\spad{SMP}} in order to speed up operations related to pseudo-division and \\spad{gcd}. This domain is based on the \\axiomType{NSUP} constructor which is itself a post-facto extension of the \\axiomType{SUP} constructor."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) 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(-575))))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-1194)))) (-3213 (|HasCategory| |#1| (LIST (QUOTE -38) (QUOTE (-575))))) (-3213 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-1194)))) (-3213 (|HasCategory| |#1| (LIST (QUOTE -1009) (QUOTE (-575))))))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-792 R S) ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from sparse univariate polynomial over \\spad{R} to a sparse univariate polynomial over \\spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|NewSparseUnivariatePolynomial| |#2|) (|Mapping| |#2| |#1|) (|NewSparseUnivariatePolynomial| |#1|)) "\\axiom{map(func,{} poly)} creates a new polynomial by applying func to every non-zero coefficient of the polynomial poly."))) NIL @@ -3103,7 +3103,7 @@ NIL (-793 R) ((|constructor| (NIL "A post-facto extension for \\axiomType{SUP} in order to speed up operations related to pseudo-division and \\spad{gcd} for both \\axiomType{SUP} and,{} consequently,{} \\axiomType{NSMP}.")) (|halfExtendedResultant2| (((|Record| (|:| |resultant| |#1|) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedResultant2(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca]} such that \\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{} \\spad{cb}]}")) (|halfExtendedResultant1| (((|Record| (|:| |resultant| |#1|) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedResultant1(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca]} such that \\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{} \\spad{cb}]}")) (|extendedResultant| (((|Record| (|:| |resultant| |#1|) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{}\\spad{cb}]} such that \\axiom{\\spad{r}} is the resultant of \\axiom{a} and \\axiom{\\spad{b}} and \\axiom{\\spad{r} = ca * a + \\spad{cb} * \\spad{b}}")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedSubResultantGcd2(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}\\spad{cb}]} such that \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} \\spad{cb}]}")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedSubResultantGcd1(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} such that \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} \\spad{cb}]}")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} \\spad{cb}]} such that \\axiom{\\spad{g}} is a \\spad{gcd} of \\axiom{a} and \\axiom{\\spad{b}} in \\axiom{\\spad{R^}(\\spad{-1}) \\spad{P}} and \\axiom{\\spad{g} = ca * a + \\spad{cb} * \\spad{b}}")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns \\axiom{resultant(a,{}\\spad{b})} if \\axiom{a} and \\axiom{\\spad{b}} has no non-trivial \\spad{gcd} in \\axiom{\\spad{R^}(\\spad{-1}) \\spad{P}} otherwise the non-zero sub-resultant with smallest index.")) (|subResultantsChain| (((|List| $) $ $) "\\axiom{subResultantsChain(a,{}\\spad{b})} returns the list of the non-zero sub-resultants of \\axiom{a} and \\axiom{\\spad{b}} sorted by increasing degree.")) (|lazyPseudoQuotient| (($ $ $) "\\axiom{lazyPseudoQuotient(a,{}\\spad{b})} returns \\axiom{\\spad{q}} if \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}")) (|lazyPseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{c^n} * a = \\spad{q*b} \\spad{+r}} and \\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]} where \\axiom{\\spad{n} + \\spad{g} = max(0,{} degree(\\spad{b}) - degree(a) + 1)}.")) (|lazyPseudoRemainder| (($ $ $) "\\axiom{lazyPseudoRemainder(a,{}\\spad{b})} returns \\axiom{\\spad{r}} if \\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]}. This lazy pseudo-remainder is computed by means of the \\axiomOpFrom{fmecg}{NewSparseUnivariatePolynomial} operation.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| |#1|) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]} such that \\axiom{\\spad{r}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{b}} divides \\axiom{\\spad{c^n} * a - \\spad{r}} where \\axiom{\\spad{c}} is \\axiom{leadingCoefficient(\\spad{b})} and \\axiom{\\spad{n}} is as small as possible with the previous properties.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} returns \\axiom{\\spad{r}} such that \\axiom{\\spad{r}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{b}} divides \\axiom{a \\spad{-r}} where \\axiom{\\spad{b}} is monic.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#1| $) "\\axiom{fmecg(\\spad{p1},{}\\spad{e},{}\\spad{r},{}\\spad{p2})} returns \\axiom{\\spad{p1} - \\spad{r} * X**e * \\spad{p2}} where \\axiom{\\spad{X}} is \\axiom{monomial(1,{}1)}"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4456 |has| |#1| (-373)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-1169))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-238))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) +((|HasCategory| |#1| (QUOTE (-924))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-1169))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-238))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-794 R) ((|constructor| (NIL "This package provides polynomials as functions on a ring.")) (|eulerE| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{eulerE(n,r)} \\undocumented")) (|bernoulliB| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{bernoulliB(n,r)} \\undocumented")) (|cyclotomic| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{cyclotomic(n,r)} \\undocumented"))) NIL @@ -3164,23 +3164,23 @@ NIL ((|constructor| (NIL "OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.")) (|inv| (($ $) "\\spad{inv(o)} returns the inverse of \\spad{o} if it exists.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \\spad{o} is not rational.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(o)} tests if \\spad{o} is rational,{} \\spadignore{i.e.} that all seven imaginary parts are 0.")) (|abs| ((|#1| $) "\\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \\spadfunFrom{norm}{Octonion}.")) (|octon| (($ |#1| |#1| |#1| |#1| |#1| |#1| |#1| |#1|) "\\spad{octon(re,ri,rj,rk,rE,rI,rJ,rK)} constructs an octonion from scalars.")) (|norm| ((|#1| $) "\\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.")) (|imagK| ((|#1| $) "\\spad{imagK(o)} extracts the imaginary \\spad{K} part of octonion \\spad{o}.")) (|imagJ| ((|#1| $) "\\spad{imagJ(o)} extracts the imaginary \\spad{J} part of octonion \\spad{o}.")) (|imagI| ((|#1| $) "\\spad{imagI(o)} extracts the imaginary \\spad{I} part of octonion \\spad{o}.")) (|imagE| ((|#1| $) "\\spad{imagE(o)} extracts the imaginary \\spad{E} part of octonion \\spad{o}.")) (|imagk| ((|#1| $) "\\spad{imagk(o)} extracts the \\spad{k} part of octonion \\spad{o}.")) (|imagj| ((|#1| $) "\\spad{imagj(o)} extracts the \\spad{j} part of octonion \\spad{o}.")) (|imagi| ((|#1| $) "\\spad{imagi(o)} extracts the \\spad{i} part of octonion \\spad{o}.")) (|real| ((|#1| $) "\\spad{real(o)} extracts real part of octonion \\spad{o}.")) (|conjugate| (($ $) "\\spad{conjugate(o)} negates the imaginary parts \\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{E},{}\\spad{I},{}\\spad{J},{}\\spad{K} of octonian \\spad{o}."))) ((-4454 . T) (-4455 . T) (-4457 . T)) NIL -(-809 -3765 R OS S) +(-809 -3763 R OS S) ((|constructor| (NIL "OctonionCategoryFunctions2 implements functions between two octonion domains defined over different rings. The function map is used to coerce between octonion types.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the component parts of the octonion \\spad{u}."))) NIL NIL (-810 R) ((|constructor| (NIL "Octonion implements octonions (Cayley-Dixon algebra) over a commutative ring,{} an eight-dimensional non-associative algebra,{} doubling the quaternions in the same way as doubling the complex numbers to get the quaternions the main constructor function is {\\em octon} which takes 8 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j} imaginary part,{} the \\spad{k} imaginary part,{} (as with quaternions) and in addition the imaginary parts \\spad{E},{} \\spad{I},{} \\spad{J},{} \\spad{K}.")) (|octon| (($ (|Quaternion| |#1|) (|Quaternion| |#1|)) "\\spad{octon(qe,qE)} constructs an octonion from two quaternions using the relation {\\em O = Q + QE}."))) ((-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (-3765 (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3765 (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-1077))) (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (-3763 (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3763 (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-1077))) (|HasCategory| |#1| (QUOTE (-556))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1016 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (-811) ((|ODESolve| (((|Result|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{ODESolve(args)} performs the integration of the function given the strategy or method returned by \\axiomFun{measure}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|))) (|RoutinesTable|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve a problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far."))) NIL NIL -(-812 R -3029 L) +(-812 R -3027 L) ((|constructor| (NIL "Solution of linear ordinary differential equations,{} constant coefficient case.")) (|constDsolve| (((|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Symbol|)) "\\spad{constDsolve(op, g, x)} returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular solution of the equation \\spad{op y = g},{} and the \\spad{yi}\\spad{'s} form a basis for the solutions of \\spad{op y = 0}."))) NIL NIL -(-813 R -3029) +(-813 R -3027) ((|constructor| (NIL "\\spad{ElementaryFunctionODESolver} provides the top-level functions for finding closed form solutions of ordinary differential equations and initial value problems.")) (|solve| (((|Union| |#2| "failed") |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{solve(eq, y, x = a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{eq, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \\spad{dy/dx = f(x,y)}.") (((|Union| |#2| "failed") (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{solve(eq, y, x = a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{eq, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \\spad{dy/dx = f(x,y)}.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#2| "failed") |#2| (|BasicOperator|) (|Symbol|)) "\\spad{solve(eq, y, x)} returns either a solution of the ordinary differential equation \\spad{eq} or \"failed\" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{f(x,y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \\spad{f}(\\spad{x},{}\\spad{y})},{} a solution is of the form \\spad{h(x,y)} where \\spad{h(x,y) = c} is a first integral of the equation for any constant \\spad{c}.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#2| "failed") (|Equation| |#2|) (|BasicOperator|) (|Symbol|)) "\\spad{solve(eq, y, x)} returns either a solution of the ordinary differential equation \\spad{eq} or \"failed\" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{f(x,y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \\spad{f}(\\spad{x},{}\\spad{y})},{} a solution is of the form \\spad{h(x,y)} where \\spad{h(x,y) = c} is a first integral of the equation for any constant \\spad{c}; error if the equation is not one of those 2 forms.") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|List| |#2|) (|List| (|BasicOperator|)) (|Symbol|)) "\\spad{solve([eq_1,...,eq_n], [y_1,...,y_n], x)} returns either \"failed\" or,{} if the equations form a fist order linear system,{} a solution of the form \\spad{[y_p, [b_1,...,b_n]]} where \\spad{h_p} is a particular solution and \\spad{[b_1,...b_m]} are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Symbol|)) "\\spad{solve([eq_1,...,eq_n], [y_1,...,y_n], x)} returns either \"failed\" or,{} if the equations form a fist order linear system,{} a solution of the form \\spad{[y_p, [b_1,...,b_n]]} where \\spad{h_p} is a particular solution and \\spad{[b_1,...b_m]} are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system") (((|Union| (|List| (|Vector| |#2|)) "failed") (|Matrix| |#2|) (|Symbol|)) "\\spad{solve(m, x)} returns a basis for the solutions of \\spad{D y = m y}. \\spad{x} is the dependent variable.") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|Matrix| |#2|) (|Vector| |#2|) (|Symbol|)) "\\spad{solve(m, v, x)} returns \\spad{[v_p, [v_1,...,v_m]]} such that the solutions of the system \\spad{D y = m y + v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{D y = m y}. \\spad{x} is the dependent variable."))) NIL NIL @@ -3188,7 +3188,7 @@ NIL ((|constructor| (NIL "\\axiom{ODEIntensityFunctionsTable()} provides a dynamic table and a set of functions to store details found out about sets of ODE\\spad{'s}.")) (|showIntensityFunctions| (((|Union| (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|))) "failed") (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{showIntensityFunctions(k)} returns the entries in the table of intensity functions \\spad{k}.")) (|insert!| (($ (|Record| (|:| |key| (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|)))))) "\\spad{insert!(r)} inserts an entry \\spad{r} into theIFTable")) (|iFTable| (($ (|List| (|Record| (|:| |key| (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|))))))) "\\spad{iFTable(l)} creates an intensity-functions table from the elements of \\spad{l}.")) (|keys| (((|List| (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) $) "\\spad{keys(tab)} returns the list of keys of \\spad{f}")) (|clearTheIFTable| (((|Void|)) "\\spad{clearTheIFTable()} clears the current table of intensity functions.")) (|showTheIFTable| (($) "\\spad{showTheIFTable()} returns the current table of intensity functions."))) NIL NIL -(-815 R -3029) +(-815 R -3027) ((|constructor| (NIL "\\spadtype{ODEIntegration} provides an interface to the integrator. This package is intended for use by the differential equations solver but not at top-level.")) (|diff| (((|Mapping| |#2| |#2|) (|Symbol|)) "\\spad{diff(x)} returns the derivation with respect to \\spad{x}.")) (|expint| ((|#2| |#2| (|Symbol|)) "\\spad{expint(f, x)} returns e^{the integral of \\spad{f} with respect to \\spad{x}}.")) (|int| ((|#2| |#2| (|Symbol|)) "\\spad{int(f, x)} returns the integral of \\spad{f} with respect to \\spad{x}."))) NIL NIL @@ -3196,11 +3196,11 @@ NIL ((|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalODEProblem|) (|RoutinesTable|)) "\\spad{measure(prob,R)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical ODE problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} listed in \\axiom{\\spad{R}} of \\axiom{category} \\axiomType{OrdinaryDifferentialEquationsSolverCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information. It predicts the likely most effective NAG numerical Library routine to solve the input set of ODEs by checking various attributes of the system of ODEs and calculating a measure of compatibility of each routine to these attributes.") (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalODEProblem|)) "\\spad{measure(prob)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical ODE problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} of \\axiom{category} \\axiomType{OrdinaryDifferentialEquationsSolverCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information. It predicts the likely most effective NAG numerical Library routine to solve the input set of ODEs by checking various attributes of the system of ODEs and calculating a measure of compatibility of each routine to these attributes.")) (|solve| (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Expression| (|Float|)) (|List| (|Float|)) (|Float|) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,G,intVals,epsabs,epsrel)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to an absolute error requirement \\axiom{\\spad{epsabs}} and relative error \\axiom{\\spad{epsrel}}. The values of \\spad{Y}[1]..\\spad{Y}[\\spad{n}] will be output for the values of \\spad{X} in \\axiom{\\spad{intVals}}. The calculation will stop if the function \\spad{G}(\\spad{X},{}\\spad{Y}[1],{}..,{}\\spad{Y}[\\spad{n}]) evaluates to zero before \\spad{X} = \\spad{xEnd}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Expression| (|Float|)) (|List| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,G,intVals,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. The values of \\spad{Y}[1]..\\spad{Y}[\\spad{n}] will be output for the values of \\spad{X} in \\axiom{\\spad{intVals}}. The calculation will stop if the function \\spad{G}(\\spad{X},{}\\spad{Y}[1],{}..,{}\\spad{Y}[\\spad{n}]) evaluates to zero before \\spad{X} = \\spad{xEnd}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|List| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,intVals,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. The values of \\spad{Y}[1]..\\spad{Y}[\\spad{n}] will be output for the values of \\spad{X} in \\axiom{\\spad{intVals}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Expression| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,G,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. The calculation will stop if the function \\spad{G}(\\spad{X},{}\\spad{Y}[1],{}..,{}\\spad{Y}[\\spad{n}]) evaluates to zero before \\spad{X} = \\spad{xEnd}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|))) "\\spad{solve(f,xStart,xEnd,yInitial)} is a top level ANNA function to solve numerically a system of ordinary differential equations \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}],{} together with a starting value for \\spad{X} and \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (called the initial conditions) and a final value of \\spad{X}. A default value is used for the accuracy requirement. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|NumericalODEProblem|) (|RoutinesTable|)) "\\spad{solve(odeProblem,R)} is a top level ANNA function to solve numerically a system of ordinary differential equations \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}],{} together with starting values for \\spad{X} and \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (called the initial conditions),{} a final value of \\spad{X},{} an accuracy requirement and any intermediate points at which the result is required. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|NumericalODEProblem|)) "\\spad{solve(odeProblem)} is a top level ANNA function to solve numerically a system of ordinary differential equations \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}],{} together with starting values for \\spad{X} and \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (called the initial conditions),{} a final value of \\spad{X},{} an accuracy requirement and any intermediate points at which the result is required. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine."))) NIL NIL -(-817 -3029 UP UPUP R) +(-817 -3027 UP UPUP R) ((|constructor| (NIL "In-field solution of an linear ordinary differential equation,{} pure algebraic case.")) (|algDsolve| (((|Record| (|:| |particular| (|Union| |#4| "failed")) (|:| |basis| (|List| |#4|))) (|LinearOrdinaryDifferentialOperator1| |#4|) |#4|) "\\spad{algDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no solution in \\spad{R}. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{y_i's} form a basis for the solutions in \\spad{R} of the homogeneous equation."))) NIL NIL -(-818 -3029 UP L LQ) +(-818 -3027 UP L LQ) ((|constructor| (NIL "\\spad{PrimitiveRatDE} provides functions for in-field solutions of linear \\indented{1}{ordinary differential equations,{} in the transcendental case.} \\indented{1}{The derivation to use is given by the parameter \\spad{L}.}")) (|splitDenominator| (((|Record| (|:| |eq| |#3|) (|:| |rh| (|List| (|Fraction| |#2|)))) |#4| (|List| (|Fraction| |#2|))) "\\spad{splitDenominator(op, [g1,...,gm])} returns \\spad{op0, [h1,...,hm]} such that the equations \\spad{op y = c1 g1 + ... + cm gm} and \\spad{op0 y = c1 h1 + ... + cm hm} have the same solutions.")) (|indicialEquation| ((|#2| |#4| |#1|) "\\spad{indicialEquation(op, a)} returns the indicial equation of \\spad{op} at \\spad{a}.") ((|#2| |#3| |#1|) "\\spad{indicialEquation(op, a)} returns the indicial equation of \\spad{op} at \\spad{a}.")) (|indicialEquations| (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#4| |#2|) "\\spad{indicialEquations(op, p)} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op} above the roots of \\spad{p},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#4|) "\\spad{indicialEquations op} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#3| |#2|) "\\spad{indicialEquations(op, p)} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op} above the roots of \\spad{p},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#3|) "\\spad{indicialEquations op} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.")) (|denomLODE| ((|#2| |#3| (|List| (|Fraction| |#2|))) "\\spad{denomLODE(op, [g1,...,gm])} returns a polynomial \\spad{d} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{p/d} for some polynomial \\spad{p}.") (((|Union| |#2| "failed") |#3| (|Fraction| |#2|)) "\\spad{denomLODE(op, g)} returns a polynomial \\spad{d} such that any rational solution of \\spad{op y = g} is of the form \\spad{p/d} for some polynomial \\spad{p},{} and \"failed\",{} if the equation has no rational solution."))) NIL NIL @@ -3208,38 +3208,38 @@ NIL ((|retract| (((|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))) $) "\\spad{retract(x)} \\undocumented{}")) (|coerce| (($ (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{coerce(x)} \\undocumented{}"))) NIL NIL -(-820 -3029 UP L LQ) +(-820 -3027 UP L LQ) ((|constructor| (NIL "In-field solution of Riccati equations,{} primitive case.")) (|changeVar| ((|#3| |#3| (|Fraction| |#2|)) "\\spad{changeVar(+/[ai D^i], a)} returns the operator \\spad{+/[ai (D+a)^i]}.") ((|#3| |#3| |#2|) "\\spad{changeVar(+/[ai D^i], a)} returns the operator \\spad{+/[ai (D+a)^i]}.")) (|singRicDE| (((|List| (|Record| (|:| |frac| (|Fraction| |#2|)) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{singRicDE(op, zeros, ezfactor)} returns \\spad{[[f1, L1], [f2, L2], ... , [fk, Lk]]} such that the singular part of any rational solution of the associated Riccati equation of \\spad{op y=0} must be one of the \\spad{fi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z=y e^{-int p}} is \\spad{Li z=0}. \\spad{zeros(C(x),H(x,y))} returns all the \\spad{P_i(x)}\\spad{'s} such that \\spad{H(x,P_i(x)) = 0 modulo C(x)}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.")) (|polyRicDE| (((|List| (|Record| (|:| |poly| |#2|) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#1|) |#2|)) "\\spad{polyRicDE(op, zeros)} returns \\spad{[[p1, L1], [p2, L2], ... , [pk, Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y=0} must be one of the \\spad{pi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z=y e^{-int p}} is \\spad{Li z =0}. \\spad{zeros} is a zero finder in \\spad{UP}.")) (|constantCoefficientRicDE| (((|List| (|Record| (|:| |constant| |#1|) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#1|) |#2|)) "\\spad{constantCoefficientRicDE(op, ric)} returns \\spad{[[a1, L1], [a2, L2], ... , [ak, Lk]]} such that any rational solution with no polynomial part of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{ai}\\spad{'s} in which case the equation for \\spad{z = y e^{-int ai}} is \\spad{Li z = 0}. \\spad{ric} is a Riccati equation solver over \\spad{F},{} whose input is the associated linear equation.")) (|leadingCoefficientRicDE| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |eq| |#2|))) |#3|) "\\spad{leadingCoefficientRicDE(op)} returns \\spad{[[m1, p1], [m2, p2], ... , [mk, pk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must have degree \\spad{mj} for some \\spad{j},{} and its leading coefficient is then a zero of \\spad{pj}. In addition,{}\\spad{m1>m2> ... >mk}.")) (|denomRicDE| ((|#2| |#3|) "\\spad{denomRicDE(op)} returns a polynomial \\spad{d} such that any rational solution of the associated Riccati equation of \\spad{op y = 0} is of the form \\spad{p/d + q'/q + r} for some polynomials \\spad{p} and \\spad{q} and a reduced \\spad{r}. Also,{} \\spad{deg(p) < deg(d)} and {\\spad{gcd}(\\spad{d},{}\\spad{q}) = 1}."))) NIL NIL -(-821 -3029 UP) +(-821 -3027 UP) ((|constructor| (NIL "\\spad{RationalLODE} provides functions for in-field solutions of linear \\indented{1}{ordinary differential equations,{} in the rational case.}")) (|indicialEquationAtInfinity| ((|#2| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))) "\\spad{indicialEquationAtInfinity op} returns the indicial equation of \\spad{op} at infinity.") ((|#2| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{indicialEquationAtInfinity op} returns the indicial equation of \\spad{op} at infinity.")) (|ratDsolve| (((|Record| (|:| |basis| (|List| (|Fraction| |#2|))) (|:| |mat| (|Matrix| |#1|))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|List| (|Fraction| |#2|))) "\\spad{ratDsolve(op, [g1,...,gm])} returns \\spad{[[h1,...,hq], M]} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{d1 h1 + ... + dq hq} where \\spad{M [d1,...,dq,c1,...,cm] = 0}.") (((|Record| (|:| |particular| (|Union| (|Fraction| |#2|) "failed")) (|:| |basis| (|List| (|Fraction| |#2|)))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{ratDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no rational solution. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{yi}\\spad{'s} form a basis for the rational solutions of the homogeneous equation.") (((|Record| (|:| |basis| (|List| (|Fraction| |#2|))) (|:| |mat| (|Matrix| |#1|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|List| (|Fraction| |#2|))) "\\spad{ratDsolve(op, [g1,...,gm])} returns \\spad{[[h1,...,hq], M]} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{d1 h1 + ... + dq hq} where \\spad{M [d1,...,dq,c1,...,cm] = 0}.") (((|Record| (|:| |particular| (|Union| (|Fraction| |#2|) "failed")) (|:| |basis| (|List| (|Fraction| |#2|)))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{ratDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no rational solution. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{yi}\\spad{'s} form a basis for the rational solutions of the homogeneous equation."))) NIL NIL -(-822 -3029 L UP A LO) +(-822 -3027 L UP A LO) ((|constructor| (NIL "Elimination of an algebraic from the coefficentss of a linear ordinary differential equation.")) (|reduceLODE| (((|Record| (|:| |mat| (|Matrix| |#2|)) (|:| |vec| (|Vector| |#1|))) |#5| |#4|) "\\spad{reduceLODE(op, g)} returns \\spad{[m, v]} such that any solution in \\spad{A} of \\spad{op z = g} is of the form \\spad{z = (z_1,...,z_m) . (b_1,...,b_m)} where the \\spad{b_i's} are the basis of \\spad{A} over \\spad{F} returned by \\spadfun{basis}() from \\spad{A},{} and the \\spad{z_i's} satisfy the differential system \\spad{M.z = v}."))) NIL NIL -(-823 -3029 UP) +(-823 -3027 UP) ((|constructor| (NIL "In-field solution of Riccati equations,{} rational case.")) (|polyRicDE| (((|List| (|Record| (|:| |poly| |#2|) (|:| |eq| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{polyRicDE(op, zeros)} returns \\spad{[[p1, L1], [p2, L2], ... , [pk,Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{pi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z = y e^{-int p}} is \\spad{Li z = 0}. \\spad{zeros} is a zero finder in \\spad{UP}.")) (|singRicDE| (((|List| (|Record| (|:| |frac| (|Fraction| |#2|)) (|:| |eq| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{singRicDE(op, ezfactor)} returns \\spad{[[f1,L1], [f2,L2],..., [fk,Lk]]} such that the singular \\spad{++} part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{fi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z = y e^{-int ai}} is \\spad{Li z = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.")) (|ricDsolve| (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))) "\\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, zeros, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{ricDsolve(op, zeros)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, zeros, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{ricDsolve(op, zeros)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}."))) NIL ((|HasCategory| |#1| (QUOTE (-27)))) -(-824 -3029 LO) +(-824 -3027 LO) ((|constructor| (NIL "SystemODESolver provides tools for triangulating and solving some systems of linear ordinary differential equations.")) (|solveInField| (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|)))) (|Matrix| |#2|) (|Vector| |#1|) (|Mapping| (|Record| (|:| |particular| (|Union| |#1| "failed")) (|:| |basis| (|List| |#1|))) |#2| |#1|)) "\\spad{solveInField(m, v, solve)} returns \\spad{[[v_1,...,v_m], v_p]} such that the solutions in \\spad{F} of the system \\spad{m x = v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{m x = 0}. Argument \\spad{solve} is a function for solving a single linear ordinary differential equation in \\spad{F}.")) (|solve| (((|Union| (|Record| (|:| |particular| (|Vector| |#1|)) (|:| |basis| (|Matrix| |#1|))) "failed") (|Matrix| |#1|) (|Vector| |#1|) (|Mapping| (|Union| (|Record| (|:| |particular| |#1|) (|:| |basis| (|List| |#1|))) "failed") |#2| |#1|)) "\\spad{solve(m, v, solve)} returns \\spad{[[v_1,...,v_m], v_p]} such that the solutions in \\spad{F} of the system \\spad{D x = m x + v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{D x = m x}. Argument \\spad{solve} is a function for solving a single linear ordinary differential equation in \\spad{F}.")) (|triangulate| (((|Record| (|:| |mat| (|Matrix| |#2|)) (|:| |vec| (|Vector| |#1|))) (|Matrix| |#2|) (|Vector| |#1|)) "\\spad{triangulate(m, v)} returns \\spad{[m_0, v_0]} such that \\spad{m_0} is upper triangular and the system \\spad{m_0 x = v_0} is equivalent to \\spad{m x = v}.") (((|Record| (|:| A (|Matrix| |#1|)) (|:| |eqs| (|List| (|Record| (|:| C (|Matrix| |#1|)) (|:| |g| (|Vector| |#1|)) (|:| |eq| |#2|) (|:| |rh| |#1|))))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{triangulate(M,v)} returns \\spad{A,[[C_1,g_1,L_1,h_1],...,[C_k,g_k,L_k,h_k]]} such that under the change of variable \\spad{y = A z},{} the first order linear system \\spad{D y = M y + v} is uncoupled as \\spad{D z_i = C_i z_i + g_i} and each \\spad{C_i} is a companion matrix corresponding to the scalar equation \\spad{L_i z_j = h_i}."))) NIL NIL -(-825 -3029 LODO) +(-825 -3027 LODO) ((|constructor| (NIL "\\spad{ODETools} provides tools for the linear ODE solver.")) (|particularSolution| (((|Union| |#1| "failed") |#2| |#1| (|List| |#1|) (|Mapping| |#1| |#1|)) "\\spad{particularSolution(op, g, [f1,...,fm], I)} returns a particular solution \\spad{h} of the equation \\spad{op y = g} where \\spad{[f1,...,fm]} are linearly independent and \\spad{op(fi)=0}. The value \"failed\" is returned if no particular solution is found. Note: the method of variations of parameters is used.")) (|variationOfParameters| (((|Union| (|Vector| |#1|) "failed") |#2| |#1| (|List| |#1|)) "\\spad{variationOfParameters(op, g, [f1,...,fm])} returns \\spad{[u1,...,um]} such that a particular solution of the equation \\spad{op y = g} is \\spad{f1 int(u1) + ... + fm int(um)} where \\spad{[f1,...,fm]} are linearly independent and \\spad{op(fi)=0}. The value \"failed\" is returned if \\spad{m < n} and no particular solution is found.")) (|wronskianMatrix| (((|Matrix| |#1|) (|List| |#1|) (|NonNegativeInteger|)) "\\spad{wronskianMatrix([f1,...,fn], q, D)} returns the \\spad{q x n} matrix \\spad{m} whose i^th row is \\spad{[f1^(i-1),...,fn^(i-1)]}.") (((|Matrix| |#1|) (|List| |#1|)) "\\spad{wronskianMatrix([f1,...,fn])} returns the \\spad{n x n} matrix \\spad{m} whose i^th row is \\spad{[f1^(i-1),...,fn^(i-1)]}."))) NIL NIL -(-826 -2833 S |f|) +(-826 -2831 S |f|) ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The ordering on the type is determined by its third argument which represents the less than function on vectors. This type is a suitable third argument for \\spadtype{GeneralDistributedMultivariatePolynomial}."))) ((-4454 |has| |#2| (-1066)) (-4455 |has| |#2| (-1066)) (-4457 |has| |#2| (-6 -4457)) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-132))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-378))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-737))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-804))) (|HasCategory| |#2| (LIST 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(|HasCategory| |#2| (QUOTE (-1066)))) (-12 (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194))))) (|HasCategory| |#2| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))))) (-827 R) ((|constructor| (NIL "\\spadtype{OrderlyDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is orderly. This is analogous to the domain \\spadtype{Polynomial}. \\blankline"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) +((|HasCategory| |#1| (QUOTE (-924))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-829 (-1194)) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-828 |Kernels| R |var|) ((|constructor| (NIL "This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable."))) (((-4462 "*") |has| |#2| (-373)) (-4453 |has| |#2| (-373)) (-4458 |has| |#2| (-373)) (-4452 |has| |#2| (-373)) (-4457 . T) (-4455 . T) (-4454 . T)) @@ -3307,7 +3307,7 @@ NIL (-844 R) ((|constructor| (NIL "Adjunction of a complex infinity to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a finite rational number if it is one,{} \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a finite rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a finite rational number.")) (|infinite?| (((|Boolean|) $) "\\spad{infinite?(x)} tests if \\spad{x} is infinite.")) (|finite?| (((|Boolean|) $) "\\spad{finite?(x)} tests if \\spad{x} is finite.")) (|infinity| (($) "\\spad{infinity()} returns infinity."))) ((-4457 |has| |#1| (-859))) -((|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (QUOTE (-21))) (-3765 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-859)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3765 (|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-556)))) +((|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (QUOTE (-21))) (-3763 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-859)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3763 (|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-556)))) (-845 A S) ((|constructor| (NIL "This category specifies the interface for operators used to build terms,{} in the sense of Universal Algebra. The domain parameter \\spad{S} provides representation for the `external name' of an operator.")) (|is?| (((|Boolean|) $ |#2|) "\\spad{is?(op,n)} holds if the name of the operator \\spad{op} is \\spad{n}.")) (|arity| (((|Arity|) $) "\\spad{arity(op)} returns the arity of the operator \\spad{op}.")) (|name| ((|#2| $) "\\spad{name(op)} returns the externam name of \\spad{op}."))) NIL @@ -3347,12 +3347,12 @@ NIL (-854 R) ((|constructor| (NIL "Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a finite rational number if it is one and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a finite rational number. Error: if \\spad{x} cannot be so converted.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a finite rational number.")) (|whatInfinity| (((|SingleInteger|) $) "\\spad{whatInfinity(x)} returns 0 if \\spad{x} is finite,{} 1 if \\spad{x} is +infinity,{} and \\spad{-1} if \\spad{x} is -infinity.")) (|infinite?| (((|Boolean|) $) "\\spad{infinite?(x)} tests if \\spad{x} is +infinity or -infinity,{}")) (|finite?| (((|Boolean|) $) "\\spad{finite?(x)} tests if \\spad{x} is finite.")) (|minusInfinity| (($) "\\spad{minusInfinity()} returns -infinity.")) (|plusInfinity| (($) "\\spad{plusInfinity()} returns +infinity."))) ((-4457 |has| |#1| (-859))) -((|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (QUOTE (-21))) (-3765 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-859)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3765 (|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-556)))) +((|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (QUOTE (-21))) (-3763 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-859)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3763 (|HasCategory| |#1| (QUOTE (-859))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-556)))) (-855) ((|constructor| (NIL "Ordered finite sets.")) (|max| (($) "\\spad{max} is the maximum value of \\%.")) (|min| (($) "\\spad{min} is the minimum value of \\%."))) NIL NIL -(-856 -2833 S) +(-856 -2831 S) ((|constructor| (NIL "\\indented{3}{This package provides ordering functions on vectors which} are suitable parameters for OrderedDirectProduct.")) (|reverseLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{reverseLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by the reverse lexicographic ordering.")) (|totalLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{totalLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by lexicographic ordering.")) (|pureLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{pureLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the lexicographic ordering."))) NIL NIL @@ -3388,11 +3388,11 @@ NIL ((|constructor| (NIL "\\spad{UnivariateSkewPolynomialCategoryOps} provides products and \\indented{1}{divisions of univariate skew polynomials.}")) (|rightDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{rightDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|leftDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{leftDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|monicRightDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{monicRightDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|monicLeftDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{monicLeftDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|apply| ((|#1| |#2| |#1| |#1| (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{apply(p, c, m, sigma, delta)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|times| ((|#2| |#2| |#2| (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{times(p, q, sigma, delta)} returns \\spad{p * q}. \\spad{\\sigma} and \\spad{\\delta} are the maps to use."))) NIL ((|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) -(-865 R |sigma| -3594) +(-865 R |sigma| -3592) ((|constructor| (NIL "This is the domain of sparse univariate skew polynomials over an Ore coefficient field. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}.")) (|outputForm| (((|OutputForm|) $ (|OutputForm|)) "\\spad{outputForm(p, x)} returns the output form of \\spad{p} using \\spad{x} for the otherwise anonymous variable."))) ((-4454 . T) (-4455 . T) (-4457 . T)) ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-373)))) -(-866 |x| R |sigma| -3594) +(-866 |x| R |sigma| -3592) ((|constructor| (NIL "This is the domain of univariate skew polynomials over an Ore coefficient field in a named variable. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}."))) ((-4454 . T) (-4455 . T) (-4457 . T)) ((|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-373)))) @@ -3459,15 +3459,15 @@ NIL (-882 |p|) ((|constructor| (NIL "Stream-based implementation of \\spad{Qp:} numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i) where the a[\\spad{i}] lie in 0,{}1,{}...,{}(\\spad{p} - 1)."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-881 |#1|) (QUOTE (-924))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-881 |#1|) (QUOTE (-146))) (|HasCategory| (-881 |#1|) (QUOTE (-148))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-881 |#1|) (QUOTE (-1039))) (|HasCategory| (-881 |#1|) (QUOTE (-831))) (-3765 (|HasCategory| (-881 |#1|) (QUOTE (-831))) (|HasCategory| (-881 |#1|) (QUOTE (-861)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-881 |#1|) (QUOTE (-1169))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| (-881 |#1|) (QUOTE (-237))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-881 |#1|) (QUOTE (-238))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -525) (QUOTE (-1194)) (LIST (QUOTE -881) (|devaluate| |#1|)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -318) (LIST (QUOTE -881) (|devaluate| |#1|)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -295) (LIST (QUOTE -881) (|devaluate| |#1|)) (LIST (QUOTE -881) (|devaluate| |#1|)))) (|HasCategory| (-881 |#1|) (QUOTE (-316))) (|HasCategory| (-881 |#1|) (QUOTE (-556))) (|HasCategory| (-881 |#1|) (QUOTE (-861))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-881 |#1|) (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-881 |#1|) (QUOTE (-924)))) (|HasCategory| (-881 |#1|) (QUOTE (-146))))) +((|HasCategory| (-881 |#1|) (QUOTE (-924))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-881 |#1|) (QUOTE (-146))) (|HasCategory| (-881 |#1|) (QUOTE (-148))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-881 |#1|) (QUOTE (-1039))) (|HasCategory| (-881 |#1|) (QUOTE (-831))) (-3763 (|HasCategory| (-881 |#1|) (QUOTE (-831))) (|HasCategory| (-881 |#1|) (QUOTE (-861)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-881 |#1|) (QUOTE (-1169))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| (-881 |#1|) (QUOTE (-237))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-881 |#1|) (QUOTE (-238))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -525) (QUOTE (-1194)) (LIST (QUOTE -881) (|devaluate| |#1|)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -318) (LIST (QUOTE -881) (|devaluate| |#1|)))) (|HasCategory| (-881 |#1|) (LIST (QUOTE -295) (LIST (QUOTE -881) (|devaluate| |#1|)) (LIST (QUOTE -881) (|devaluate| |#1|)))) (|HasCategory| (-881 |#1|) (QUOTE (-316))) (|HasCategory| (-881 |#1|) (QUOTE (-556))) (|HasCategory| (-881 |#1|) (QUOTE (-861))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-881 |#1|) (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-881 |#1|) (QUOTE (-924)))) (|HasCategory| (-881 |#1|) (QUOTE (-146))))) (-883 |p| PADIC) ((|constructor| (NIL "This is the category of stream-based representations of \\spad{Qp}.")) (|removeZeroes| (($ (|Integer|) $) "\\spad{removeZeroes(n,x)} removes up to \\spad{n} leading zeroes from the \\spad{p}-adic rational \\spad{x}.") (($ $) "\\spad{removeZeroes(x)} removes leading zeroes from the representation of the \\spad{p}-adic rational \\spad{x}. A \\spad{p}-adic rational is represented by (1) an exponent and (2) a \\spad{p}-adic integer which may have leading zero digits. When the \\spad{p}-adic integer has a leading zero digit,{} a 'leading zero' is removed from the \\spad{p}-adic rational as follows: the number is rewritten by increasing the exponent by 1 and dividing the \\spad{p}-adic integer by \\spad{p}. Note: \\spad{removeZeroes(f)} removes all leading zeroes from \\spad{f}.")) (|continuedFraction| (((|ContinuedFraction| (|Fraction| (|Integer|))) $) "\\spad{continuedFraction(x)} converts the \\spad{p}-adic rational number \\spad{x} to a continued fraction.")) (|approximate| (((|Fraction| (|Integer|)) $ (|Integer|)) "\\spad{approximate(x,n)} returns a rational number \\spad{y} such that \\spad{y = x (mod p^n)}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#2| (QUOTE (-924))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (QUOTE (-1039))) (|HasCategory| |#2| (QUOTE (-831))) (-3765 (|HasCategory| |#2| (QUOTE (-831))) (|HasCategory| |#2| (QUOTE (-861)))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-1169))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-237))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -295) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-556))) (|HasCategory| |#2| (QUOTE (-861))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) +((|HasCategory| |#2| (QUOTE (-924))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (QUOTE (-1039))) (|HasCategory| |#2| (QUOTE (-831))) (-3763 (|HasCategory| |#2| (QUOTE (-831))) (|HasCategory| |#2| (QUOTE (-861)))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-1169))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-237))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -295) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-556))) (|HasCategory| |#2| (QUOTE (-861))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) (-884 S T$) ((|constructor| (NIL "\\indented{1}{This domain provides a very simple representation} of the notion of `pair of objects'. It does not try to achieve all possible imaginable things.")) (|second| ((|#2| $) "\\spad{second(p)} extracts the second components of \\spad{`p'}.")) (|first| ((|#1| $) "\\spad{first(p)} extracts the first component of \\spad{`p'}.")) (|construct| (($ |#1| |#2|) "\\spad{construct(s,t)} is same as pair(\\spad{s},{}\\spad{t}),{} with syntactic sugar.")) (|pair| (($ |#1| |#2|) "\\spad{pair(s,t)} returns a pair object composed of \\spad{`s'} and \\spad{`t'}."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))))) (-885) ((|constructor| (NIL "This domain describes four groups of color shades (palettes).")) (|coerce| (($ (|Color|)) "\\spad{coerce(c)} sets the average shade for the palette to that of the indicated color \\spad{c}.")) (|shade| (((|Integer|) $) "\\spad{shade(p)} returns the shade index of the indicated palette \\spad{p}.")) (|hue| (((|Color|) $) "\\spad{hue(p)} returns the hue field of the indicated palette \\spad{p}.")) (|light| (($ (|Color|)) "\\spad{light(c)} sets the shade of a hue,{} \\spad{c},{} to it\\spad{'s} highest value.")) (|pastel| (($ (|Color|)) "\\spad{pastel(c)} sets the shade of a hue,{} \\spad{c},{} above bright,{} but below light.")) (|bright| (($ (|Color|)) "\\spad{bright(c)} sets the shade of a hue,{} \\spad{c},{} above dim,{} but below pastel.")) (|dim| (($ (|Color|)) "\\spad{dim(c)} sets the shade of a hue,{} \\spad{c},{} above dark,{} but below bright.")) (|dark| (($ (|Color|)) "\\spad{dark(c)} sets the shade of the indicated hue of \\spad{c} to it\\spad{'s} lowest value."))) NIL @@ -3527,7 +3527,7 @@ NIL (-899 |Base| |Subject| |Pat|) ((|constructor| (NIL "This package provides the top-level pattern macthing functions.")) (|Is| (((|PatternMatchResult| |#1| |#2|) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a match of the form \\spad{[v1 = e1,...,vn = en]}; returns an empty match if \\spad{expr} is exactly equal to pat. returns a \\spadfun{failed} match if pat does not match \\spad{expr}.") (((|List| (|Equation| (|Polynomial| |#2|))) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a list of matches \\spad{[v1 = e1,...,vn = en]}; returns an empty list if either \\spad{expr} is exactly equal to pat or if pat does not match \\spad{expr}.") (((|List| (|Equation| |#2|)) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a list of matches \\spad{[v1 = e1,...,vn = en]}; returns an empty list if either \\spad{expr} is exactly equal to pat or if pat does not match \\spad{expr}.") (((|PatternMatchListResult| |#1| |#2| (|List| |#2|)) (|List| |#2|) |#3|) "\\spad{Is([e1,...,en], pat)} matches the pattern pat on the list of expressions \\spad{[e1,...,en]} and returns the result.")) (|is?| (((|Boolean|) (|List| |#2|) |#3|) "\\spad{is?([e1,...,en], pat)} tests if the list of expressions \\spad{[e1,...,en]} matches the pattern pat.") (((|Boolean|) |#2| |#3|) "\\spad{is?(expr, pat)} tests if the expression \\spad{expr} matches the pattern pat."))) NIL -((-12 (-3215 (|HasCategory| |#2| (QUOTE (-1066)))) (-3215 (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194)))))) (-12 (|HasCategory| |#2| (QUOTE (-1066))) (-3215 (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194))))) +((-12 (-3213 (|HasCategory| |#2| (QUOTE (-1066)))) (-3213 (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194)))))) (-12 (|HasCategory| |#2| (QUOTE (-1066))) (-3213 (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-1194))))) (-900 R A B) ((|constructor| (NIL "Lifts maps to pattern matching results.")) (|map| (((|PatternMatchResult| |#1| |#3|) (|Mapping| |#3| |#2|) (|PatternMatchResult| |#1| |#2|)) "\\spad{map(f, [(v1,a1),...,(vn,an)])} returns the matching result [(\\spad{v1},{}\\spad{f}(a1)),{}...,{}(\\spad{vn},{}\\spad{f}(an))]."))) NIL @@ -3536,7 +3536,7 @@ NIL ((|constructor| (NIL "A PatternMatchResult is an object internally returned by the pattern matcher; It is either a failed match,{} or a list of matches of the form (var,{} expr) meaning that the variable var matches the expression expr.")) (|satisfy?| (((|Union| (|Boolean|) "failed") $ (|Pattern| |#1|)) "\\spad{satisfy?(r, p)} returns \\spad{true} if the matches satisfy the top-level predicate of \\spad{p},{} \\spad{false} if they don\\spad{'t},{} and \"failed\" if not enough variables of \\spad{p} are matched in \\spad{r} to decide.")) (|construct| (($ (|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| |#2|)))) "\\spad{construct([v1,e1],...,[vn,en])} returns the match result containing the matches (\\spad{v1},{}e1),{}...,{}(\\spad{vn},{}en).")) (|destruct| (((|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| |#2|))) $) "\\spad{destruct(r)} returns the list of matches (var,{} expr) in \\spad{r}. Error: if \\spad{r} is a failed match.")) (|addMatchRestricted| (($ (|Pattern| |#1|) |#2| $ |#2|) "\\spad{addMatchRestricted(var, expr, r, val)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} provided that \\spad{expr} satisfies the predicates attached to \\spad{var},{} that \\spad{var} is not matched to another expression already,{} and that either \\spad{var} is an optional pattern variable or that \\spad{expr} is not equal to val (usually an identity).")) (|insertMatch| (($ (|Pattern| |#1|) |#2| $) "\\spad{insertMatch(var, expr, r)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} without checking predicates or previous matches for \\spad{var}.")) (|addMatch| (($ (|Pattern| |#1|) |#2| $) "\\spad{addMatch(var, expr, r)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} provided that \\spad{expr} satisfies the predicates attached to \\spad{var},{} and that \\spad{var} is not matched to another expression already.")) (|getMatch| (((|Union| |#2| "failed") (|Pattern| |#1|) $) "\\spad{getMatch(var, r)} returns the expression that \\spad{var} matches in the result \\spad{r},{} and \"failed\" if \\spad{var} is not matched in \\spad{r}.")) (|union| (($ $ $) "\\spad{union(a, b)} makes the set-union of two match results.")) (|new| (($) "\\spad{new()} returns a new empty match result.")) (|failed| (($) "\\spad{failed()} returns a failed match.")) (|failed?| (((|Boolean|) $) "\\spad{failed?(r)} tests if \\spad{r} is a failed match."))) NIL NIL -(-902 R -3430) +(-902 R -3428) ((|constructor| (NIL "Tools for patterns.")) (|badValues| (((|List| |#2|) (|Pattern| |#1|)) "\\spad{badValues(p)} returns the list of \"bad values\" for \\spad{p}; \\spad{p} is not allowed to match any of its \"bad values\".")) (|addBadValue| (((|Pattern| |#1|) (|Pattern| |#1|) |#2|) "\\spad{addBadValue(p, v)} adds \\spad{v} to the list of \"bad values\" for \\spad{p}; \\spad{p} is not allowed to match any of its \"bad values\".")) (|satisfy?| (((|Boolean|) (|List| |#2|) (|Pattern| |#1|)) "\\spad{satisfy?([v1,...,vn], p)} returns \\spad{f(v1,...,vn)} where \\spad{f} is the top-level predicate attached to \\spad{p}.") (((|Boolean|) |#2| (|Pattern| |#1|)) "\\spad{satisfy?(v, p)} returns \\spad{f}(\\spad{v}) where \\spad{f} is the predicate attached to \\spad{p}.")) (|predicate| (((|Mapping| (|Boolean|) |#2|) (|Pattern| |#1|)) "\\spad{predicate(p)} returns the predicate attached to \\spad{p},{} the constant function \\spad{true} if \\spad{p} has no predicates attached to it.")) (|suchThat| (((|Pattern| |#1|) (|Pattern| |#1|) (|List| (|Symbol|)) (|Mapping| (|Boolean|) (|List| |#2|))) "\\spad{suchThat(p, [a1,...,an], f)} returns a copy of \\spad{p} with the top-level predicate set to \\spad{f(a1,...,an)}.") (((|Pattern| |#1|) (|Pattern| |#1|) (|List| (|Mapping| (|Boolean|) |#2|))) "\\spad{suchThat(p, [f1,...,fn])} makes a copy of \\spad{p} and adds the predicate \\spad{f1} and ... and \\spad{fn} to the copy,{} which is returned.") (((|Pattern| |#1|) (|Pattern| |#1|) (|Mapping| (|Boolean|) |#2|)) "\\spad{suchThat(p, f)} makes a copy of \\spad{p} and adds the predicate \\spad{f} to the copy,{} which is returned."))) NIL NIL @@ -3568,7 +3568,7 @@ NIL ((|PDESolve| (((|Result|) (|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|)))) "\\spad{PDESolve(args)} performs the integration of the function given the strategy or method returned by \\axiomFun{measure}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|))) (|RoutinesTable|) (|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|)))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve a problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far."))) NIL NIL -(-910 UP -3029) +(-910 UP -3027) ((|constructor| (NIL "This package \\undocumented")) (|rightFactorCandidate| ((|#1| |#1| (|NonNegativeInteger|)) "\\spad{rightFactorCandidate(p,n)} \\undocumented")) (|leftFactor| (((|Union| |#1| "failed") |#1| |#1|) "\\spad{leftFactor(p,q)} \\undocumented")) (|decompose| (((|Union| (|Record| (|:| |left| |#1|) (|:| |right| |#1|)) "failed") |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{decompose(up,m,n)} \\undocumented") (((|List| |#1|) |#1|) "\\spad{decompose(up)} \\undocumented"))) NIL NIL @@ -3595,7 +3595,7 @@ NIL (-916 S) ((|constructor| (NIL "\\indented{1}{A PendantTree(\\spad{S})is either a leaf? and is an \\spad{S} or has} a left and a right both PendantTree(\\spad{S})\\spad{'s}")) (|ptree| (($ $ $) "\\spad{ptree(x,y)} \\undocumented") (($ |#1|) "\\spad{ptree(s)} is a leaf? pendant tree"))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-917 |n| R) ((|constructor| (NIL "Permanent implements the functions {\\em permanent},{} the permanent for square matrices.")) (|permanent| ((|#2| (|SquareMatrix| |#1| |#2|)) "\\spad{permanent(x)} computes the permanent of a square matrix \\spad{x}. The {\\em permanent} is equivalent to the \\spadfun{determinant} except that coefficients have no change of sign. This function is much more difficult to compute than the {\\em determinant}. The formula used is by \\spad{H}.\\spad{J}. Ryser,{} improved by [Nijenhuis and Wilf,{} \\spad{Ch}. 19]. Note: permanent(\\spad{x}) choose one of three algorithms,{} depending on the underlying ring \\spad{R} and on \\spad{n},{} the number of rows (and columns) of \\spad{x:}\\begin{items} \\item 1. if 2 has an inverse in \\spad{R} we can use the algorithm of \\indented{3}{[Nijenhuis and Wilf,{} \\spad{ch}.19,{}\\spad{p}.158]; if 2 has no inverse,{}} \\indented{3}{some modifications are necessary:} \\item 2. if {\\em n > 6} and \\spad{R} is an integral domain with characteristic \\indented{3}{different from 2 (the algorithm works if and only 2 is not a} \\indented{3}{zero-divisor of \\spad{R} and {\\em characteristic()\\$R ~= 2},{}} \\indented{3}{but how to check that for any given \\spad{R} ?),{}} \\indented{3}{the local function {\\em permanent2} is called;} \\item 3. else,{} the local function {\\em permanent3} is called \\indented{3}{(works for all commutative rings \\spad{R}).} \\end{items}"))) NIL @@ -3611,7 +3611,7 @@ NIL (-920 S) ((|constructor| (NIL "Permutation(\\spad{S}) implements the group of all bijections \\indented{2}{on a set \\spad{S},{} which move only a finite number of points.} \\indented{2}{A permutation is considered as a map from \\spad{S} into \\spad{S}. In particular} \\indented{2}{multiplication is defined as composition of maps:} \\indented{2}{{\\em pi1 * pi2 = pi1 o pi2}.} \\indented{2}{The internal representation of permuatations are two lists} \\indented{2}{of equal length representing preimages and images.}")) (|coerceImages| (($ (|List| |#1|)) "\\spad{coerceImages(ls)} coerces the list {\\em ls} to a permutation whose image is given by {\\em ls} and the preimage is fixed to be {\\em [1,...,n]}. Note: {coerceImages(\\spad{ls})=coercePreimagesImages([1,{}...,{}\\spad{n}],{}\\spad{ls})}. We assume that both preimage and image do not contain repetitions.")) (|fixedPoints| (((|Set| |#1|) $) "\\spad{fixedPoints(p)} returns the points fixed by the permutation \\spad{p}.")) (|sort| (((|List| $) (|List| $)) "\\spad{sort(lp)} sorts a list of permutations {\\em lp} according to cycle structure first according to length of cycles,{} second,{} if \\spad{S} has \\spadtype{Finite} or \\spad{S} has \\spadtype{OrderedSet} according to lexicographical order of entries in cycles of equal length.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(p)} returns \\spad{true} if and only if \\spad{p} is an odd permutation \\spadignore{i.e.} {\\em sign(p)} is {\\em -1}.")) (|even?| (((|Boolean|) $) "\\spad{even?(p)} returns \\spad{true} if and only if \\spad{p} is an even permutation,{} \\spadignore{i.e.} {\\em sign(p)} is 1.")) (|sign| (((|Integer|) $) "\\spad{sign(p)} returns the signum of the permutation \\spad{p},{} \\spad{+1} or \\spad{-1}.")) (|numberOfCycles| (((|NonNegativeInteger|) $) "\\spad{numberOfCycles(p)} returns the number of non-trivial cycles of the permutation \\spad{p}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of a permutation \\spad{p} as a group element.")) (|cyclePartition| (((|Partition|) $) "\\spad{cyclePartition(p)} returns the cycle structure of a permutation \\spad{p} including cycles of length 1 only if \\spad{S} is finite.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} retuns the number of points moved by the permutation \\spad{p}.")) (|coerceListOfPairs| (($ (|List| (|List| |#1|))) "\\spad{coerceListOfPairs(lls)} coerces a list of pairs {\\em lls} to a permutation. Error: if not consistent,{} \\spadignore{i.e.} the set of the first elements coincides with the set of second elements. coerce(\\spad{p}) generates output of the permutation \\spad{p} with domain OutputForm.")) (|coerce| (($ (|List| |#1|)) "\\spad{coerce(ls)} coerces a cycle {\\em ls},{} \\spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.") (($ (|List| (|List| |#1|))) "\\spad{coerce(lls)} coerces a list of cycles {\\em lls} to a permutation,{} each cycle being a list with no repetitions,{} is coerced to the permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.")) (|coercePreimagesImages| (($ (|List| (|List| |#1|))) "\\spad{coercePreimagesImages(lls)} coerces the representation {\\em lls} of a permutation as a list of preimages and images to a permutation. We assume that both preimage and image do not contain repetitions.")) (|listRepresentation| (((|Record| (|:| |preimage| (|List| |#1|)) (|:| |image| (|List| |#1|))) $) "\\spad{listRepresentation(p)} produces a representation {\\em rep} of the permutation \\spad{p} as a list of preimages and images,{} \\spad{i}.\\spad{e} \\spad{p} maps {\\em (rep.preimage).k} to {\\em (rep.image).k} for all indices \\spad{k}. Elements of \\spad{S} not in {\\em (rep.preimage).k} are fixed points,{} and these are the only fixed points of the permutation."))) ((-4457 . T)) -((-3765 (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-861)))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-861)))) +((-3763 (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-861)))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-861)))) (-921 R E |VarSet| S) ((|constructor| (NIL "PolynomialFactorizationByRecursion(\\spad{R},{}\\spad{E},{}\\spad{VarSet},{}\\spad{S}) is used for factorization of sparse univariate polynomials over a domain \\spad{S} of multivariate polynomials over \\spad{R}.")) (|factorSFBRlcUnit| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|List| |#3|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \\spad{p} (see \\spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \\spad{p} is a unit.")) (|bivariateSLPEBR| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|List| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|) |#3|) "\\spad{bivariateSLPEBR(lp,p,v)} implements the bivariate case of \\spadfunFrom{solveLinearPolynomialEquationByRecursion}{PolynomialFactorizationByRecursionUnivariate}; its implementation depends on \\spad{R}")) (|randomR| ((|#1|) "\\spad{randomR produces} a random element of \\spad{R}")) (|factorSquareFreeByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \\spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorSquareFreePolynomial}).")) (|factorByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorByRecursion(p)} factors polynomial \\spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorPolynomial})")) (|solveLinearPolynomialEquationByRecursion| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|List| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{solveLinearPolynomialEquationByRecursion([p1,...,pn],p)} returns the list of polynomials \\spad{[q1,...,qn]} such that \\spad{sum qi/pi = p / prod pi},{} a recursion step for solveLinearPolynomialEquation as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{solveLinearPolynomialEquation}). If no such list of \\spad{qi} exists,{} then \"failed\" is returned."))) NIL @@ -3632,7 +3632,7 @@ NIL ((|constructor| (NIL "PrimeField(\\spad{p}) implements the field with \\spad{p} elements if \\spad{p} is a prime number. Error: if \\spad{p} is not prime. Note: this domain does not check that argument is a prime."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) ((|HasCategory| $ (QUOTE (-148))) (|HasCategory| $ (QUOTE (-146))) (|HasCategory| $ (QUOTE (-378)))) -(-926 R0 -3029 UP UPUP R) +(-926 R0 -3027 UP UPUP R) ((|constructor| (NIL "This package provides function for testing whether a divisor on a curve is a torsion divisor.")) (|torsionIfCan| (((|Union| (|Record| (|:| |order| (|NonNegativeInteger|)) (|:| |function| |#5|)) "failed") (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{torsionIfCan(f)}\\\\ undocumented")) (|torsion?| (((|Boolean|) (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{torsion?(f)} \\undocumented")) (|order| (((|Union| (|NonNegativeInteger|) "failed") (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{order(f)} \\undocumented"))) NIL NIL @@ -3660,7 +3660,7 @@ NIL ((|constructor| (NIL "PermutationGroupExamples provides permutation groups for some classes of groups: symmetric,{} alternating,{} dihedral,{} cyclic,{} direct products of cyclic,{} which are in fact the finite abelian groups of symmetric groups called Young subgroups. Furthermore,{} Rubik\\spad{'s} group as permutation group of 48 integers and a list of sporadic simple groups derived from the atlas of finite groups.")) (|youngGroup| (((|PermutationGroup| (|Integer|)) (|Partition|)) "\\spad{youngGroup(lambda)} constructs the direct product of the symmetric groups given by the parts of the partition {\\em lambda}.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{youngGroup([n1,...,nk])} constructs the direct product of the symmetric groups {\\em Sn1},{}...,{}{\\em Snk}.")) (|rubiksGroup| (((|PermutationGroup| (|Integer|))) "\\spad{rubiksGroup constructs} the permutation group representing Rubic\\spad{'s} Cube acting on integers {\\em 10*i+j} for {\\em 1 <= i <= 6},{} {\\em 1 <= j <= 8}. The faces of Rubik\\spad{'s} Cube are labelled in the obvious way Front,{} Right,{} Up,{} Down,{} Left,{} Back and numbered from 1 to 6 in this given ordering,{} the pieces on each face (except the unmoveable center piece) are clockwise numbered from 1 to 8 starting with the piece in the upper left corner. The moves of the cube are represented as permutations on these pieces,{} represented as a two digit integer {\\em ij} where \\spad{i} is the numer of theface (1 to 6) and \\spad{j} is the number of the piece on this face. The remaining ambiguities are resolved by looking at the 6 generators,{} which represent a 90 degree turns of the faces,{} or from the following pictorial description. Permutation group representing Rubic\\spad{'s} Cube acting on integers 10*i+j for 1 \\spad{<=} \\spad{i} \\spad{<=} 6,{} 1 \\spad{<=} \\spad{j} \\spad{<=8}. \\blankline\\begin{verbatim}Rubik's Cube: +-----+ +-- B where: marks Side # : / U /|/ / / | F(ront) <-> 1 L --> +-----+ R| R(ight) <-> 2 | | + U(p) <-> 3 | F | / D(own) <-> 4 | |/ L(eft) <-> 5 +-----+ B(ack) <-> 6 ^ | DThe Cube's surface: The pieces on each side +---+ (except the unmoveable center |567| piece) are clockwise numbered |4U8| from 1 to 8 starting with the |321| piece in the upper left +---+---+---+ corner (see figure on the |781|123|345| left). The moves of the cube |6L2|8F4|2R6| are represented as |543|765|187| permutations on these pieces. +---+---+---+ Each of the pieces is |123| represented as a two digit |8D4| integer ij where i is the |765| # of the side ( 1 to 6 for +---+ F to B (see table above )) |567| and j is the # of the piece. |4B8| |321| +---+\\end{verbatim}")) (|janko2| (((|PermutationGroup| (|Integer|))) "\\spad{janko2 constructs} the janko group acting on the integers 1,{}...,{}100.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{janko2(li)} constructs the janko group acting on the 100 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 100 different entries")) (|mathieu24| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu24 constructs} the mathieu group acting on the integers 1,{}...,{}24.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu24(li)} constructs the mathieu group acting on the 24 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 24 different entries.")) (|mathieu23| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu23 constructs} the mathieu group acting on the integers 1,{}...,{}23.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu23(li)} constructs the mathieu group acting on the 23 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 23 different entries.")) (|mathieu22| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu22 constructs} the mathieu group acting on the integers 1,{}...,{}22.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu22(li)} constructs the mathieu group acting on the 22 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 22 different entries.")) (|mathieu12| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu12 constructs} the mathieu group acting on the integers 1,{}...,{}12.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu12(li)} constructs the mathieu group acting on the 12 integers given in the list {\\em li}. Note: duplicates in the list will be removed Error: if {\\em li} has less or more than 12 different entries.")) (|mathieu11| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu11 constructs} the mathieu group acting on the integers 1,{}...,{}11.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu11(li)} constructs the mathieu group acting on the 11 integers given in the list {\\em li}. Note: duplicates in the list will be removed. error,{} if {\\em li} has less or more than 11 different entries.")) (|dihedralGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{dihedralGroup([i1,...,ik])} constructs the dihedral group of order 2k acting on the integers out of {\\em i1},{}...,{}{\\em ik}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{dihedralGroup(n)} constructs the dihedral group of order 2n acting on integers 1,{}...,{}\\spad{N}.")) (|cyclicGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{cyclicGroup([i1,...,ik])} constructs the cyclic group of order \\spad{k} acting on the integers {\\em i1},{}...,{}{\\em ik}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{cyclicGroup(n)} constructs the cyclic group of order \\spad{n} acting on the integers 1,{}...,{}\\spad{n}.")) (|abelianGroup| (((|PermutationGroup| (|Integer|)) (|List| (|PositiveInteger|))) "\\spad{abelianGroup([n1,...,nk])} constructs the abelian group that is the direct product of cyclic groups with order {\\em ni}.")) (|alternatingGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{alternatingGroup(li)} constructs the alternating group acting on the integers in the list {\\em li},{} generators are in general the {\\em n-2}-cycle {\\em (li.3,...,li.n)} and the 3-cycle {\\em (li.1,li.2,li.3)},{} if \\spad{n} is odd and product of the 2-cycle {\\em (li.1,li.2)} with {\\em n-2}-cycle {\\em (li.3,...,li.n)} and the 3-cycle {\\em (li.1,li.2,li.3)},{} if \\spad{n} is even. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{alternatingGroup(n)} constructs the alternating group {\\em An} acting on the integers 1,{}...,{}\\spad{n},{} generators are in general the {\\em n-2}-cycle {\\em (3,...,n)} and the 3-cycle {\\em (1,2,3)} if \\spad{n} is odd and the product of the 2-cycle {\\em (1,2)} with {\\em n-2}-cycle {\\em (3,...,n)} and the 3-cycle {\\em (1,2,3)} if \\spad{n} is even.")) (|symmetricGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{symmetricGroup(li)} constructs the symmetric group acting on the integers in the list {\\em li},{} generators are the cycle given by {\\em li} and the 2-cycle {\\em (li.1,li.2)}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{symmetricGroup(n)} constructs the symmetric group {\\em Sn} acting on the integers 1,{}...,{}\\spad{n},{} generators are the {\\em n}-cycle {\\em (1,...,n)} and the 2-cycle {\\em (1,2)}."))) NIL NIL -(-933 -3029) +(-933 -3027) ((|constructor| (NIL "Groebner functions for \\spad{P} \\spad{F} \\indented{2}{This package is an interface package to the groebner basis} package which allows you to compute groebner bases for polynomials in either lexicographic ordering or total degree ordering refined by reverse lex. The input is the ordinary polynomial type which is internally converted to a type with the required ordering. The resulting grobner basis is converted back to ordinary polynomials. The ordering among the variables is controlled by an explicit list of variables which is passed as a second argument. The coefficient domain is allowed to be any \\spad{gcd} domain,{} but the groebner basis is computed as if the polynomials were over a field.")) (|totalGroebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{totalGroebner(lp,lv)} computes Groebner basis for the list of polynomials \\spad{lp} with the terms ordered first by total degree and then refined by reverse lexicographic ordering. The variables are ordered by their position in the list \\spad{lv}.")) (|lexGroebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{lexGroebner(lp,lv)} computes Groebner basis for the list of polynomials \\spad{lp} in lexicographic order. The variables are ordered by their position in the list \\spad{lv}."))) NIL NIL @@ -3676,11 +3676,11 @@ NIL ((|constructor| (NIL "\\spadtype{PositiveInteger} provides functions for \\indented{2}{positive integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : x*y = \\spad{y*x}")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two positive integers \\spad{a} and \\spad{b}."))) (((-4462 "*") . T)) NIL -(-937 -3029 P) +(-937 -3027 P) ((|constructor| (NIL "This package exports interpolation algorithms")) (|LagrangeInterpolation| ((|#2| (|List| |#1|) (|List| |#1|)) "\\spad{LagrangeInterpolation(l1,l2)} \\undocumented"))) NIL NIL -(-938 |xx| -3029) +(-938 |xx| -3027) ((|constructor| (NIL "This package exports interpolation algorithms")) (|interpolate| (((|SparseUnivariatePolynomial| |#2|) (|List| |#2|) (|List| |#2|)) "\\spad{interpolate(lf,lg)} \\undocumented") (((|UnivariatePolynomial| |#1| |#2|) (|UnivariatePolynomial| |#1| |#2|) (|List| |#2|) (|List| |#2|)) "\\spad{interpolate(u,lf,lg)} \\undocumented"))) NIL NIL @@ -3704,7 +3704,7 @@ NIL ((|constructor| (NIL "This package exports plotting tools")) (|calcRanges| (((|List| (|Segment| (|DoubleFloat|))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{calcRanges(l)} \\undocumented"))) NIL NIL -(-944 R -3029) +(-944 R -3027) ((|constructor| (NIL "Attaching assertions to symbols for pattern matching; Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|multiple| ((|#2| |#2|) "\\spad{multiple(x)} tells the pattern matcher that \\spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\\spad{x}) tells the pattern matcher that \\spad{x} should match a list instead of an element of a list. Error: if \\spad{x} is not a symbol.")) (|optional| ((|#2| |#2|) "\\spad{optional(x)} tells the pattern matcher that \\spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation). Error: if \\spad{x} is not a symbol.")) (|constant| ((|#2| |#2|) "\\spad{constant(x)} tells the pattern matcher that \\spad{x} should match only the symbol \\spad{'x} and no other quantity. Error: if \\spad{x} is not a symbol.")) (|assert| ((|#2| |#2| (|Identifier|)) "\\spad{assert(x, s)} makes the assertion \\spad{s} about \\spad{x}. Error: if \\spad{x} is not a symbol."))) NIL NIL @@ -3716,7 +3716,7 @@ NIL ((|constructor| (NIL "This packages provides tools for matching recursively in type towers.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#2| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches. Note: this function handles type towers by changing the predicates and calling the matching function provided by \\spad{A}.")) (|fixPredicate| (((|Mapping| (|Boolean|) |#2|) (|Mapping| (|Boolean|) |#3|)) "\\spad{fixPredicate(f)} returns \\spad{g} defined by \\spad{g}(a) = \\spad{f}(a::B)."))) NIL NIL -(-947 S R -3029) +(-947 S R -3027) ((|constructor| (NIL "This package provides pattern matching functions on function spaces.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL NIL @@ -3736,11 +3736,11 @@ NIL ((|constructor| (NIL "This package provides pattern matching functions on polynomials.")) (|patternMatch| (((|PatternMatchResult| |#1| |#5|) |#5| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|)) "\\spad{patternMatch(p, pat, res)} matches the pattern \\spad{pat} to the polynomial \\spad{p}; res contains the variables of \\spad{pat} which are already matched and their matches.") (((|PatternMatchResult| |#1| |#5|) |#5| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|) (|Mapping| (|PatternMatchResult| |#1| |#5|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|))) "\\spad{patternMatch(p, pat, res, vmatch)} matches the pattern \\spad{pat} to the polynomial \\spad{p}. \\spad{res} contains the variables of \\spad{pat} which are already matched and their matches; vmatch is the matching function to use on the variables."))) NIL ((|HasCategory| |#3| (LIST (QUOTE -898) (|devaluate| |#1|)))) -(-952 R -3029 -3430) +(-952 R -3027 -3428) ((|constructor| (NIL "Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|suchThat| ((|#2| |#2| (|List| (|Mapping| (|Boolean|) |#3|))) "\\spad{suchThat(x, [f1, f2, ..., fn])} attaches the predicate \\spad{f1} and \\spad{f2} and ... and \\spad{fn} to \\spad{x}. Error: if \\spad{x} is not a symbol.") ((|#2| |#2| (|Mapping| (|Boolean|) |#3|)) "\\spad{suchThat(x, foo)} attaches the predicate foo to \\spad{x}; error if \\spad{x} is not a symbol."))) NIL NIL -(-953 -3430) +(-953 -3428) ((|constructor| (NIL "Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|suchThat| (((|Expression| (|Integer|)) (|Symbol|) (|List| (|Mapping| (|Boolean|) |#1|))) "\\spad{suchThat(x, [f1, f2, ..., fn])} attaches the predicate \\spad{f1} and \\spad{f2} and ... and \\spad{fn} to \\spad{x}.") (((|Expression| (|Integer|)) (|Symbol|) (|Mapping| (|Boolean|) |#1|)) "\\spad{suchThat(x, foo)} attaches the predicate foo to \\spad{x}."))) NIL NIL @@ -3763,7 +3763,7 @@ NIL (-958 R) ((|constructor| (NIL "This domain implements points in coordinate space"))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1019))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1019))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-959 |lv| R) ((|constructor| (NIL "Package with the conversion functions among different kind of polynomials")) (|pToDmp| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|Polynomial| |#2|)) "\\spad{pToDmp(p)} converts \\spad{p} from a \\spadtype{POLY} to a \\spadtype{DMP}.")) (|dmpToP| (((|Polynomial| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{dmpToP(p)} converts \\spad{p} from a \\spadtype{DMP} to a \\spadtype{POLY}.")) (|hdmpToP| (((|Polynomial| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{hdmpToP(p)} converts \\spad{p} from a \\spadtype{HDMP} to a \\spadtype{POLY}.")) (|pToHdmp| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|Polynomial| |#2|)) "\\spad{pToHdmp(p)} converts \\spad{p} from a \\spadtype{POLY} to a \\spadtype{HDMP}.")) (|hdmpToDmp| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{hdmpToDmp(p)} converts \\spad{p} from a \\spadtype{HDMP} to a \\spadtype{DMP}.")) (|dmpToHdmp| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{dmpToHdmp(p)} converts \\spad{p} from a \\spadtype{DMP} to a \\spadtype{HDMP}."))) NIL @@ -3788,7 +3788,7 @@ NIL ((|constructor| (NIL "The category for general multi-variate polynomials over a ring \\spad{R},{} in variables from VarSet,{} with exponents from the \\spadtype{OrderedAbelianMonoidSup}.")) (|canonicalUnitNormal| ((|attribute|) "we can choose a unique representative for each associate class. This normalization is chosen to be normalization of leading coefficient (by default).")) (|squareFreePart| (($ $) "\\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \\spad{p} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(p)} returns the square free factorization of the polynomial \\spad{p}.")) (|primitivePart| (($ $ |#3|) "\\spad{primitivePart(p,v)} returns the unitCanonical associate of the polynomial \\spad{p} with its content with respect to the variable \\spad{v} divided out.") (($ $) "\\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \\spad{p} with its content divided out.")) (|content| (($ $ |#3|) "\\spad{content(p,v)} is the \\spad{gcd} of the coefficients of the polynomial \\spad{p} when \\spad{p} is viewed as a univariate polynomial with respect to the variable \\spad{v}. Thus,{} for polynomial 7*x**2*y + 14*x*y**2,{} the \\spad{gcd} of the coefficients with respect to \\spad{x} is 7*y.")) (|discriminant| (($ $ |#3|) "\\spad{discriminant(p,v)} returns the disriminant of the polynomial \\spad{p} with respect to the variable \\spad{v}.")) (|resultant| (($ $ $ |#3|) "\\spad{resultant(p,q,v)} returns the resultant of the polynomials \\spad{p} and \\spad{q} with respect to the variable \\spad{v}.")) (|primitiveMonomials| (((|List| $) $) "\\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \\spad{p} with their coefficients removed. Note: \\spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}.")) (|variables| (((|List| |#3|) $) "\\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \\spad{p}.")) (|totalDegree| (((|NonNegativeInteger|) $ (|List| |#3|)) "\\spad{totalDegree(p, lv)} returns the maximum sum (over all monomials of polynomial \\spad{p}) of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $) "\\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#3|) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if polynomial \\spad{p} has the form \\spad{x**n} and \\spad{n > 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if polynomial \\spad{p = a1 ... an} and \\spad{n >= 2},{} and,{} for each \\spad{i},{} \\spad{ai} is either a nontrivial constant in \\spad{R} or else of the form \\spad{x**e},{} where \\spad{e > 0} is an integer and \\spad{x} in a member of VarSet.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if polynomial \\spad{p = m1 + ... + mn} and \\spad{n >= 2} and each \\spad{mi} is a nonzero monomial.")) (|multivariate| (($ (|SparseUnivariatePolynomial| $) |#3|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.") (($ (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.")) (|monomial| (($ $ (|List| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[v1..vn],[e1..en])} returns \\spad{a*prod(vi**ei)}.") (($ $ |#3| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} creates the monomial \\spad{a*x**n} where \\spad{a} is a polynomial,{} \\spad{x} is a variable and \\spad{n} is a nonnegative integer.")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $ |#3|) "\\spad{monicDivide(a,b,v)} divides the polynomial a by the polynomial \\spad{b},{} with each viewed as a univariate polynomial in \\spad{v} returning both the quotient and remainder. Error: if \\spad{b} is not monic with respect to \\spad{v}.")) (|minimumDegree| (((|List| (|NonNegativeInteger|)) $ (|List| |#3|)) "\\spad{minimumDegree(p, lv)} gives the list of minimum degrees of the polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}") (((|NonNegativeInteger|) $ |#3|) "\\spad{minimumDegree(p,v)} gives the minimum degree of polynomial \\spad{p} with respect to \\spad{v},{} \\spadignore{i.e.} viewed a univariate polynomial in \\spad{v}")) (|mainVariable| (((|Union| |#3| "failed") $) "\\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \\spad{p},{} or \"failed\" if no variables are present. fails precisely if polynomial satisfies ground?")) (|univariate| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{univariate(p)} converts the multivariate polynomial \\spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate") (((|SparseUnivariatePolynomial| $) $ |#3|) "\\spad{univariate(p,v)} converts the multivariate polynomial \\spad{p} into a univariate polynomial in \\spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).")) (|monomials| (((|List| $) $) "\\spad{monomials(p)} returns the list of non-zero monomials of polynomial \\spad{p},{} \\spadignore{i.e.} \\spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.")) (|coefficient| (($ $ (|List| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(p, lv, ln)} views the polynomial \\spad{p} as a polynomial in the variables of \\spad{lv} and returns the coefficient of the term \\spad{lv**ln},{} \\spadignore{i.e.} \\spad{prod(lv_i ** ln_i)}.") (($ $ |#3| (|NonNegativeInteger|)) "\\spad{coefficient(p,v,n)} views the polynomial \\spad{p} as a univariate polynomial in \\spad{v} and returns the coefficient of the \\spad{v**n} term.")) (|degree| (((|List| (|NonNegativeInteger|)) $ (|List| |#3|)) "\\spad{degree(p,lv)} gives the list of degrees of polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p,v)} gives the degree of polynomial \\spad{p} with respect to the variable \\spad{v}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) NIL -(-965 E V R P -3029) +(-965 E V R P -3027) ((|constructor| (NIL "This package transforms multivariate polynomials or fractions into univariate polynomials or fractions,{} and back.")) (|isPower| (((|Union| (|Record| (|:| |val| |#5|) (|:| |exponent| (|Integer|))) "failed") |#5|) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0},{} \"failed\" otherwise.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#2|) (|:| |exponent| (|Integer|))) "failed") |#5|) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0},{} \"failed\" otherwise.")) (|isTimes| (((|Union| (|List| |#5|) "failed") |#5|) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1 ... an} and \\spad{n > 1},{} \"failed\" otherwise.")) (|isPlus| (((|Union| (|List| |#5|) "failed") |#5|) "\\spad{isPlus(p)} returns [\\spad{m1},{}...,{}\\spad{mn}] if \\spad{p = m1 + ... + mn} and \\spad{n > 1},{} \"failed\" otherwise.")) (|multivariate| ((|#5| (|Fraction| (|SparseUnivariatePolynomial| |#5|)) |#2|) "\\spad{multivariate(f, v)} applies both the numerator and denominator of \\spad{f} to \\spad{v}.")) (|univariate| (((|SparseUnivariatePolynomial| |#5|) |#5| |#2| (|SparseUnivariatePolynomial| |#5|)) "\\spad{univariate(f, x, p)} returns \\spad{f} viewed as a univariate polynomial in \\spad{x},{} using the side-condition \\spad{p(x) = 0}.") (((|Fraction| (|SparseUnivariatePolynomial| |#5|)) |#5| |#2|) "\\spad{univariate(f, v)} returns \\spad{f} viewed as a univariate rational function in \\spad{v}.")) (|mainVariable| (((|Union| |#2| "failed") |#5|) "\\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \\spad{f},{} \"failed\" if \\spad{f} has no variables.")) (|variables| (((|List| |#2|) |#5|) "\\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \\spad{f}."))) NIL NIL @@ -3799,8 +3799,8 @@ NIL (-967 R) ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are arbitrary symbols. The ordering is alphabetic determined by the Symbol type. The coefficient ring may be non commutative,{} but the variables are assumed to commute.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(p,x)} computes the integral of \\spad{p*dx},{} \\spadignore{i.e.} integrates the polynomial \\spad{p} with respect to the variable \\spad{x}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) -(-968 E V R P -3029) +((|HasCategory| |#1| (QUOTE (-924))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1194) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) +(-968 E V R P -3027) ((|constructor| (NIL "computes \\spad{n}-th roots of quotients of multivariate polynomials")) (|nthr| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#4|) (|:| |radicand| (|List| |#4|))) |#4| (|NonNegativeInteger|)) "\\spad{nthr(p,n)} should be local but conditional")) (|froot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) |#5| (|NonNegativeInteger|)) "\\spad{froot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|qroot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) (|Fraction| (|Integer|)) (|NonNegativeInteger|)) "\\spad{qroot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|rroot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) |#3| (|NonNegativeInteger|)) "\\spad{rroot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|denom| ((|#4| $) "\\spad{denom(x)} \\undocumented")) (|numer| ((|#4| $) "\\spad{numer(x)} \\undocumented"))) NIL ((|HasCategory| |#3| (QUOTE (-463)))) @@ -3823,12 +3823,12 @@ NIL (-973 S) ((|constructor| (NIL "\\indented{1}{This provides a fast array type with no bound checking on elt\\spad{'s}.} Minimum index is 0 in this type,{} cannot be changed"))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-974) ((|constructor| (NIL "Category for the functions defined by integrals.")) (|integral| (($ $ (|SegmentBinding| $)) "\\spad{integral(f, x = a..b)} returns the formal definite integral of \\spad{f} \\spad{dx} for \\spad{x} between \\spad{a} and \\spad{b}.") (($ $ (|Symbol|)) "\\spad{integral(f, x)} returns the formal integral of \\spad{f} \\spad{dx}."))) NIL NIL -(-975 -3029) +(-975 -3027) ((|constructor| (NIL "PrimitiveElement provides functions to compute primitive elements in algebraic extensions.")) (|primitiveElement| (((|Record| (|:| |coef| (|List| (|Integer|))) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#1|))) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|)) (|Symbol|)) "\\spad{primitiveElement([p1,...,pn], [a1,...,an], a)} returns \\spad{[[c1,...,cn], [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} where \\spad{a = a1 c1 + ... + an cn},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The \\spad{pi}\\spad{'s} are the defining polynomials for the \\spad{ai}\\spad{'s}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}.") (((|Record| (|:| |coef| (|List| (|Integer|))) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#1|))) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{primitiveElement([p1,...,pn], [a1,...,an])} returns \\spad{[[c1,...,cn], [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} where \\spad{a = a1 c1 + ... + an cn},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The \\spad{pi}\\spad{'s} are the defining polynomials for the \\spad{ai}\\spad{'s}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}.") (((|Record| (|:| |coef1| (|Integer|)) (|:| |coef2| (|Integer|)) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|Polynomial| |#1|) (|Symbol|) (|Polynomial| |#1|) (|Symbol|)) "\\spad{primitiveElement(p1, a1, p2, a2)} returns \\spad{[c1, c2, q]} such that \\spad{k(a1, a2) = k(a)} where \\spad{a = c1 a1 + c2 a2, and q(a) = 0}. The \\spad{pi}\\spad{'s} are the defining polynomials for the \\spad{ai}\\spad{'s}. The \\spad{p2} may involve \\spad{a1},{} but \\spad{p1} must not involve a2. This operation uses \\spadfun{resultant}."))) NIL NIL @@ -3843,11 +3843,11 @@ NIL (-978 R E) ((|constructor| (NIL "This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and terms indexed by their exponents (from an arbitrary ordered abelian monoid). This type is used,{} for example,{} by the \\spadtype{DistributedMultivariatePolynomial} domain where the exponent domain is a direct product of non negative integers.")) (|canonicalUnitNormal| ((|attribute|) "canonicalUnitNormal guarantees that the function unitCanonical returns the same representative for all associates of any particular element.")) (|fmecg| (($ $ |#2| |#1| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-132)))) (|HasAttribute| |#1| (QUOTE -4458))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-132)))) (|HasAttribute| |#1| (QUOTE -4458))) (-979 A B) ((|constructor| (NIL "This domain implements cartesian product")) (|selectsecond| ((|#2| $) "\\spad{selectsecond(x)} \\undocumented")) (|selectfirst| ((|#1| $) "\\spad{selectfirst(x)} \\undocumented")) (|makeprod| (($ |#1| |#2|) "\\spad{makeprod(a,b)} \\undocumented"))) ((-4457 -12 (|has| |#2| (-484)) (|has| |#1| (-484)))) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804)))) (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-861))))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804))))) (-12 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-484)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-484)))) (-12 (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#2| (QUOTE (-737))))) (-12 (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#2| (QUOTE (-378)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-484)))) (-12 (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#2| (QUOTE (-737)))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804))))) (-12 (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#2| (QUOTE (-737)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-861))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804)))) (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-861))))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804))))) (-12 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-484)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-484)))) (-12 (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#2| (QUOTE (-737))))) (-12 (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#2| (QUOTE (-378)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-484))) (|HasCategory| |#2| (QUOTE (-484)))) (-12 (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#2| (QUOTE (-737)))) (-12 (|HasCategory| |#1| (QUOTE (-804))) (|HasCategory| |#2| (QUOTE (-804))))) (-12 (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#2| (QUOTE (-737)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-861))))) (-980) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. An `Property' is a pair of name and value.")) (|property| (($ (|Identifier|) (|SExpression|)) "\\spad{property(n,val)} constructs a property with name \\spad{`n'} and value `val'.")) (|value| (((|SExpression|) $) "\\spad{value(p)} returns value of property \\spad{p}")) (|name| (((|Identifier|) $) "\\spad{name(p)} returns the name of property \\spad{p}"))) NIL @@ -3936,7 +3936,7 @@ NIL ((|constructor| (NIL "This package \\undocumented{}")) (|map| ((|#4| (|Mapping| |#4| (|Polynomial| |#1|)) |#4|) "\\spad{map(f,p)} \\undocumented{}")) (|pushup| ((|#4| |#4| (|List| |#3|)) "\\spad{pushup(p,lv)} \\undocumented{}") ((|#4| |#4| |#3|) "\\spad{pushup(p,v)} \\undocumented{}")) (|pushdown| ((|#4| |#4| (|List| |#3|)) "\\spad{pushdown(p,lv)} \\undocumented{}") ((|#4| |#4| |#3|) "\\spad{pushdown(p,v)} \\undocumented{}")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) NIL NIL -(-1002 K R UP -3029) +(-1002 K R UP -3027) ((|constructor| (NIL "In this package \\spad{K} is a finite field,{} \\spad{R} is a ring of univariate polynomials over \\spad{K},{} and \\spad{F} is a monogenic algebra over \\spad{R}. We require that \\spad{F} is monogenic,{} \\spadignore{i.e.} that \\spad{F = K[x,y]/(f(x,y))},{} because the integral basis algorithm used will factor the polynomial \\spad{f(x,y)}. The package provides a function to compute the integral closure of \\spad{R} in the quotient field of \\spad{F} as well as a function to compute a \"local integral basis\" at a specific prime.")) (|reducedDiscriminant| ((|#2| |#3|) "\\spad{reducedDiscriminant(up)} \\undocumented")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) |#2|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv] } containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of the framed algebra \\spad{F}. \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If 'basis' is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix 'basisInv' contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if 'basisInv' is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv] } containing information regarding the integral closure of \\spad{R} in the quotient field of the framed algebra \\spad{F}. \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If 'basis' is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix 'basisInv' contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if 'basisInv' is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}."))) NIL NIL @@ -3995,11 +3995,11 @@ NIL (-1016 R) ((|constructor| (NIL "\\spadtype{Quaternion} implements quaternions over a \\indented{2}{commutative ring. The main constructor function is \\spadfun{quatern}} \\indented{2}{which takes 4 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j}} \\indented{2}{imaginary part and the \\spad{k} imaginary part.}"))) ((-4453 |has| |#1| (-299)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-373))) (-3765 (|HasCategory| |#1| (QUOTE (-299))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-299))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-1077))) (|HasCategory| |#1| (QUOTE (-556)))) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-373))) (-3763 (|HasCategory| |#1| (QUOTE (-299))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-299))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -525) (QUOTE (-1194)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -295) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-1077))) (|HasCategory| |#1| (QUOTE (-556)))) (-1017 S) ((|constructor| (NIL "Linked List implementation of a Queue")) (|queue| (($ (|List| |#1|)) "\\spad{queue([x,y,...,z])} creates a queue with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom) element \\spad{z}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-1018 S) ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|Fraction| (|Integer|))) "\\spad{x ** y} is the rational exponentiation of \\spad{x} by the power \\spad{y}.")) (|nthRoot| (($ $ (|Integer|)) "\\spad{nthRoot(x,n)} returns the \\spad{n}th root of \\spad{x}.")) (|sqrt| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) NIL @@ -4008,14 +4008,14 @@ NIL ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|Fraction| (|Integer|))) "\\spad{x ** y} is the rational exponentiation of \\spad{x} by the power \\spad{y}.")) (|nthRoot| (($ $ (|Integer|)) "\\spad{nthRoot(x,n)} returns the \\spad{n}th root of \\spad{x}.")) (|sqrt| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) NIL NIL -(-1020 -3029 UP UPUP |radicnd| |n|) +(-1020 -3027 UP UPUP |radicnd| |n|) ((|constructor| (NIL "Function field defined by y**n = \\spad{f}(\\spad{x})."))) ((-4453 |has| (-418 |#2|) (-373)) (-4458 |has| (-418 |#2|) (-373)) (-4452 |has| (-418 |#2|) (-373)) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-418 |#2|) (QUOTE (-146))) (|HasCategory| (-418 |#2|) (QUOTE (-148))) (|HasCategory| (-418 |#2|) (QUOTE (-359))) (-3765 (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-378))) (-3765 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3765 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3765 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-359))))) (-3765 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -650) (QUOTE (-575)))) (-3765 (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-378))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) +((|HasCategory| (-418 |#2|) (QUOTE (-146))) (|HasCategory| (-418 |#2|) (QUOTE (-148))) (|HasCategory| (-418 |#2|) (QUOTE (-359))) (-3763 (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))) (|HasCategory| (-418 |#2|) (QUOTE (-378))) (-3763 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3763 (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (QUOTE (-359)))) (-3763 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-359))))) (-3763 (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -650) (QUOTE (-575)))) (-3763 (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 |#2|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-378))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-237))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (QUOTE (-238))) (|HasCategory| (-418 |#2|) (QUOTE (-373)))) (-12 (|HasCategory| (-418 |#2|) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-418 |#2|) (QUOTE (-373))))) (-1021 |bb|) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions or more generally as repeating expansions in any base.")) (|fractRadix| (($ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{fractRadix(pre,cyc)} creates a fractional radix expansion from a list of prefix ragits and a list of cyclic ragits. For example,{} \\spad{fractRadix([1],[6])} will return \\spad{0.16666666...}.")) (|wholeRadix| (($ (|List| (|Integer|))) "\\spad{wholeRadix(l)} creates an integral radix expansion from a list of ragits. For example,{} \\spad{wholeRadix([1,3,4])} will return \\spad{134}.")) (|cycleRagits| (((|List| (|Integer|)) $) "\\spad{cycleRagits(rx)} returns the cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \\spad{x = 3/28 = 0.10 714285 714285 ...},{} then \\spad{cycleRagits(x) = [7,1,4,2,8,5]}.")) (|prefixRagits| (((|List| (|Integer|)) $) "\\spad{prefixRagits(rx)} returns the non-cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \\spad{x = 3/28 = 0.10 714285 714285 ...},{} then \\spad{prefixRagits(x)=[1,0]}.")) (|fractRagits| (((|Stream| (|Integer|)) $) "\\spad{fractRagits(rx)} returns the ragits of the fractional part of a radix expansion.")) (|wholeRagits| (((|List| (|Integer|)) $) "\\spad{wholeRagits(rx)} returns the ragits of the integer part of a radix expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(rx)} returns the fractional part of a radix expansion."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3765 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) +((|HasCategory| (-575) (QUOTE (-924))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-148))) (|HasCategory| (-575) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-1039))) (|HasCategory| (-575) (QUOTE (-831))) (-3763 (|HasCategory| (-575) (QUOTE (-831))) (|HasCategory| (-575) (QUOTE (-861)))) (|HasCategory| (-575) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-1169))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| (-575) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| (-575) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| (-575) (QUOTE (-237))) (|HasCategory| (-575) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| (-575) (QUOTE (-238))) (|HasCategory| (-575) (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| (-575) (LIST (QUOTE -525) (QUOTE (-1194)) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -318) (QUOTE (-575)))) (|HasCategory| (-575) (LIST (QUOTE -295) (QUOTE (-575)) (QUOTE (-575)))) (|HasCategory| (-575) (QUOTE (-316))) (|HasCategory| (-575) (QUOTE (-556))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-575) (LIST (QUOTE -650) (QUOTE (-575)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-575) (QUOTE (-924)))) (|HasCategory| (-575) (QUOTE (-146))))) (-1022) ((|constructor| (NIL "This package provides tools for creating radix expansions.")) (|radix| (((|Any|) (|Fraction| (|Integer|)) (|Integer|)) "\\spad{radix(x,b)} converts \\spad{x} to a radix expansion in base \\spad{b}."))) NIL @@ -4048,19 +4048,19 @@ NIL ((|constructor| (NIL "\\axiomType{RealClosedField} provides common acces functions for all real closed fields.")) (|approximate| (((|Fraction| (|Integer|)) $ $) "\\axiom{approximate(\\spad{n},{}\\spad{p})} gives an approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|rename| (($ $ (|OutputForm|)) "\\axiom{rename(\\spad{x},{}name)} gives a new number that prints as name")) (|rename!| (($ $ (|OutputForm|)) "\\axiom{rename!(\\spad{x},{}name)} changes the way \\axiom{\\spad{x}} is printed")) (|sqrt| (($ (|Integer|)) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ (|Fraction| (|Integer|))) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ $) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ $ (|PositiveInteger|)) "\\axiom{sqrt(\\spad{x},{}\\spad{n})} is \\axiom{\\spad{x} \\spad{**} (1/n)}")) (|allRootsOf| (((|List| $) (|Polynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely")) (|rootOf| (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} creates the \\spad{n}th root for the order of \\axiom{pol} and gives it unique name") (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|) (|OutputForm|)) "\\axiom{rootOf(pol,{}\\spad{n},{}name)} creates the \\spad{n}th root for the order of \\axiom{pol} and names it \\axiom{name}")) (|mainValue| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainValue(\\spad{x})} is the expression of \\axiom{\\spad{x}} in terms of \\axiom{SparseUnivariatePolynomial(\\$)}")) (|mainDefiningPolynomial| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainDefiningPolynomial(\\spad{x})} is the defining polynomial for the main algebraic quantity of \\axiom{\\spad{x}}")) (|mainForm| (((|Union| (|OutputForm|) "failed") $) "\\axiom{mainForm(\\spad{x})} is the main algebraic quantity name of \\axiom{\\spad{x}}"))) ((-4453 . T) (-4458 . T) (-4452 . T) (-4455 . T) (-4454 . T) ((-4462 "*") . T) (-4457 . T)) NIL -(-1030 R -3029) +(-1030 R -3027) ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 1 February 1988 Date Last Updated: 2 November 1995 Keywords: elementary,{} function,{} integration.")) (|rischDE| (((|Record| (|:| |ans| |#2|) (|:| |right| |#2|) (|:| |sol?| (|Boolean|))) (|Integer|) |#2| |#2| (|Symbol|) (|Mapping| (|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|List| |#2|)) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| |#2|)) "\\spad{rischDE(n, f, g, x, lim, ext)} returns \\spad{[y, h, b]} such that \\spad{dy/dx + n df/dx y = h} and \\spad{b := h = g}. The equation \\spad{dy/dx + n df/dx y = g} has no solution if \\spad{h \\~~= g} (\\spad{y} is a partial solution in that case). Notes: \\spad{lim} is a limited integration function,{} and ext is an extended integration function."))) NIL NIL -(-1031 R -3029) +(-1031 R -3027) ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 12 August 1992 Date Last Updated: 17 August 1992 Keywords: elementary,{} function,{} integration.")) (|rischDEsys| (((|Union| (|List| |#2|) "failed") (|Integer|) |#2| |#2| |#2| (|Symbol|) (|Mapping| (|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|List| |#2|)) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| |#2|)) "\\spad{rischDEsys(n, f, g_1, g_2, x,lim,ext)} returns \\spad{y_1.y_2} such that \\spad{(dy1/dx,dy2/dx) + ((0, - n df/dx),(n df/dx,0)) (y1,y2) = (g1,g2)} if \\spad{y_1,y_2} exist,{} \"failed\" otherwise. \\spad{lim} is a limited integration function,{} \\spad{ext} is an extended integration function."))) NIL NIL -(-1032 -3029 UP) +(-1032 -3027 UP) ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} transcendental case.} Author: Manuel Bronstein Date Created: Jan 1988 Date Last Updated: 2 November 1995")) (|polyRDE| (((|Union| (|:| |ans| (|Record| (|:| |ans| |#2|) (|:| |nosol| (|Boolean|)))) (|:| |eq| (|Record| (|:| |b| |#2|) (|:| |c| |#2|) (|:| |m| (|Integer|)) (|:| |alpha| |#2|) (|:| |beta| |#2|)))) |#2| |#2| |#2| (|Integer|) (|Mapping| |#2| |#2|)) "\\spad{polyRDE(a, B, C, n, D)} returns either: 1. \\spad{[Q, b]} such that \\spad{degree(Q) <= n} and \\indented{3}{\\spad{a Q'+ B Q = C} if \\spad{b = true},{} \\spad{Q} is a partial solution} \\indented{3}{otherwise.} 2. \\spad{[B1, C1, m, \\alpha, \\beta]} such that any polynomial solution \\indented{3}{of degree at most \\spad{n} of \\spad{A Q' + BQ = C} must be of the form} \\indented{3}{\\spad{Q = \\alpha H + \\beta} where \\spad{degree(H) <= m} and} \\indented{3}{\\spad{H} satisfies \\spad{H' + B1 H = C1}.} \\spad{D} is the derivation to use.")) (|baseRDE| (((|Record| (|:| |ans| (|Fraction| |#2|)) (|:| |nosol| (|Boolean|))) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{baseRDE(f, g)} returns a \\spad{[y, b]} such that \\spad{y' + fy = g} if \\spad{b = true},{} \\spad{y} is a partial solution otherwise (no solution in that case). \\spad{D} is the derivation to use.")) (|monomRDE| (((|Union| (|Record| (|:| |a| |#2|) (|:| |b| (|Fraction| |#2|)) (|:| |c| (|Fraction| |#2|)) (|:| |t| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomRDE(f,g,D)} returns \\spad{[A, B, C, T]} such that \\spad{y' + f y = g} has a solution if and only if \\spad{y = Q / T},{} where \\spad{Q} satisfies \\spad{A Q' + B Q = C} and has no normal pole. A and \\spad{T} are polynomials and \\spad{B} and \\spad{C} have no normal poles. \\spad{D} is the derivation to use."))) NIL NIL -(-1033 -3029 UP) +(-1033 -3027 UP) ((|constructor| (NIL "\\indented{1}{Risch differential equation system,{} transcendental case.} Author: Manuel Bronstein Date Created: 17 August 1992 Date Last Updated: 3 February 1994")) (|baseRDEsys| (((|Union| (|List| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{baseRDEsys(f, g1, g2)} returns fractions \\spad{y_1.y_2} such that \\spad{(y1', y2') + ((0, -f), (f, 0)) (y1,y2) = (g1,g2)} if \\spad{y_1,y_2} exist,{} \"failed\" otherwise.")) (|monomRDEsys| (((|Union| (|Record| (|:| |a| |#2|) (|:| |b| (|Fraction| |#2|)) (|:| |h| |#2|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |t| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomRDEsys(f,g1,g2,D)} returns \\spad{[A, B, H, C1, C2, T]} such that \\spad{(y1', y2') + ((0, -f), (f, 0)) (y1,y2) = (g1,g2)} has a solution if and only if \\spad{y1 = Q1 / T, y2 = Q2 / T},{} where \\spad{B,C1,C2,Q1,Q2} have no normal poles and satisfy A \\spad{(Q1', Q2') + ((H, -B), (B, H)) (Q1,Q2) = (C1,C2)} \\spad{D} is the derivation to use."))) NIL NIL @@ -4095,8 +4095,8 @@ NIL (-1041 |TheField|) ((|constructor| (NIL "This domain implements the real closure of an ordered field.")) (|relativeApprox| (((|Fraction| (|Integer|)) $ $) "\\axiom{relativeApprox(\\spad{n},{}\\spad{p})} gives a relative approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|mainCharacterization| (((|Union| (|RightOpenIntervalRootCharacterization| $ (|SparseUnivariatePolynomial| $)) "failed") $) "\\axiom{mainCharacterization(\\spad{x})} is the main algebraic quantity of \\axiom{\\spad{x}} (\\axiom{SEG})")) (|algebraicOf| (($ (|RightOpenIntervalRootCharacterization| $ (|SparseUnivariatePolynomial| $)) (|OutputForm|)) "\\axiom{algebraicOf(char)} is the external number"))) ((-4453 . T) (-4458 . T) (-4452 . T) (-4455 . T) (-4454 . T) ((-4462 "*") . T) (-4457 . T)) -((-3765 (|HasCategory| (-418 (-575)) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-418 (-575)) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 (-575)) (LIST (QUOTE -1055) (QUOTE (-575))))) -(-1042 -3029 L) +((-3763 (|HasCategory| (-418 (-575)) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-418 (-575)) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-418 (-575)) (LIST (QUOTE -1055) (QUOTE (-575))))) +(-1042 -3027 L) ((|constructor| (NIL "\\spadtype{ReductionOfOrder} provides functions for reducing the order of linear ordinary differential equations once some solutions are known.")) (|ReduceOrder| (((|Record| (|:| |eq| |#2|) (|:| |op| (|List| |#1|))) |#2| (|List| |#1|)) "\\spad{ReduceOrder(op, [f1,...,fk])} returns \\spad{[op1,[g1,...,gk]]} such that for any solution \\spad{z} of \\spad{op1 z = 0},{} \\spad{y = gk \\int(g_{k-1} \\int(... \\int(g1 \\int z)...)} is a solution of \\spad{op y = 0}. Each \\spad{fi} must satisfy \\spad{op fi = 0}.") ((|#2| |#2| |#1|) "\\spad{ReduceOrder(op, s)} returns \\spad{op1} such that for any solution \\spad{z} of \\spad{op1 z = 0},{} \\spad{y = s \\int z} is a solution of \\spad{op y = 0}. \\spad{s} must satisfy \\spad{op s = 0}."))) NIL NIL @@ -4132,14 +4132,14 @@ NIL ((|constructor| (NIL "This package provides coercions for the special types \\spadtype{Exit} and \\spadtype{Void}.")) (|coerce| ((|#1| (|Exit|)) "\\spad{coerce(e)} is never really evaluated. This coercion is used for formal type correctness when a function will not return directly to its caller.") (((|Void|) |#1|) "\\spad{coerce(s)} throws all information about \\spad{s} away. This coercion allows values of any type to appear in contexts where they will not be used. For example,{} it allows the resolution of different types in the \\spad{then} and \\spad{else} branches when an \\spad{if} is in a context where the resulting value is not used."))) NIL NIL -(-1051 -3029 |Expon| |VarSet| |FPol| |LFPol|) +(-1051 -3027 |Expon| |VarSet| |FPol| |LFPol|) ((|constructor| (NIL "ResidueRing is the quotient of a polynomial ring by an ideal. The ideal is given as a list of generators. The elements of the domain are equivalence classes expressed in terms of reduced elements")) (|lift| ((|#4| $) "\\spad{lift(x)} return the canonical representative of the equivalence class \\spad{x}")) (|coerce| (($ |#4|) "\\spad{coerce(f)} produces the equivalence class of \\spad{f} in the residue ring")) (|reduce| (($ |#4|) "\\spad{reduce(f)} produces the equivalence class of \\spad{f} in the residue ring"))) (((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-1052) ((|constructor| (NIL "A domain used to return the results from a call to the NAG Library. It prints as a list of names and types,{} though the user may choose to display values automatically if he or she wishes.")) (|showArrayValues| (((|Boolean|) (|Boolean|)) "\\spad{showArrayValues(true)} forces the values of array components to be \\indented{1}{displayed rather than just their types.}")) (|showScalarValues| (((|Boolean|) (|Boolean|)) "\\spad{showScalarValues(true)} forces the values of scalar components to be \\indented{1}{displayed rather than just their types.}"))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1194))) (LIST (QUOTE |:|) (QUOTE -3179) (QUOTE (-52))))))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-52) (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -318) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-1194) (QUOTE (-861))) (|HasCategory| (-52) (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1194))) (LIST (QUOTE |:|) (QUOTE -3179) (QUOTE (-52))))))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-52) (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -318) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-1194) (QUOTE (-861))) (|HasCategory| (-52) (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873))))) (-1053) ((|constructor| (NIL "This domain represents `return' expressions.")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression returned by `e'."))) NIL @@ -4196,7 +4196,7 @@ NIL ((|constructor| (NIL "The category of rings with unity,{} always associative,{} but not necessarily commutative.")) (|unitsKnown| ((|attribute|) "recip truly yields reciprocal or \"failed\" if not a unit. Note: \\spad{recip(0) = \"failed\"}.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \\spad{n} such that \\spad{n*x=0} for all \\spad{x} in the ring,{} or zero if no such \\spad{n} exists."))) ((-4457 . T)) NIL -(-1067 |xx| -3029) +(-1067 |xx| -3027) ((|constructor| (NIL "This package exports rational interpolation algorithms"))) NIL NIL @@ -4215,7 +4215,7 @@ NIL (-1071 |m| |n| R) ((|constructor| (NIL "\\spadtype{RectangularMatrix} is a matrix domain where the number of rows and the number of columns are parameters of the domain.")) (|rectangularMatrix| (($ (|Matrix| |#3|)) "\\spad{rectangularMatrix(m)} converts a matrix of type \\spadtype{Matrix} to a matrix of type \\spad{RectangularMatrix}."))) ((-4460 . T) (-4455 . T) (-4454 . T)) -((|HasCategory| |#3| (QUOTE (-174))) (-3765 (-12 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|))))) (|HasCategory| |#3| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-373)))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (QUOTE (-316))) (|HasCategory| |#3| (QUOTE (-567))) (-12 (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|)))) (|HasCategory| |#3| (LIST (QUOTE -624) (QUOTE (-873))))) +((|HasCategory| |#3| (QUOTE (-174))) (-3763 (-12 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|))))) (|HasCategory| |#3| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-373)))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (QUOTE (-316))) (|HasCategory| |#3| (QUOTE (-567))) (-12 (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|)))) (|HasCategory| |#3| (LIST (QUOTE -624) (QUOTE (-873))))) (-1072 |m| |n| R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) ((|constructor| (NIL "\\spadtype{RectangularMatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \\spadfun{map} and \\spadfun{reduce}.")) (|reduce| ((|#7| (|Mapping| |#7| |#3| |#7|) |#6| |#7|) "\\spad{reduce(f,m,r)} returns a matrix \\spad{n} where \\spad{n[i,j] = f(m[i,j],r)} for all indices spad{\\spad{i}} and \\spad{j}.")) (|map| ((|#10| (|Mapping| |#7| |#3|) |#6|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}."))) NIL @@ -4251,7 +4251,7 @@ NIL (-1080) ((|constructor| (NIL "\\axiomType{RoutinesTable} implements a database and associated tuning mechanisms for a set of known NAG routines")) (|recoverAfterFail| (((|Union| (|String|) "failed") $ (|String|) (|Integer|)) "\\spad{recoverAfterFail(routs,routineName,ifailValue)} acts on the instructions given by the ifail list")) (|showTheRoutinesTable| (($) "\\spad{showTheRoutinesTable()} returns the current table of NAG routines.")) (|deleteRoutine!| (($ $ (|Symbol|)) "\\spad{deleteRoutine!(R,s)} destructively deletes the given routine from the current database of NAG routines")) (|getExplanations| (((|List| (|String|)) $ (|String|)) "\\spad{getExplanations(R,s)} gets the explanations of the output parameters for the given NAG routine.")) (|getMeasure| (((|Float|) $ (|Symbol|)) "\\spad{getMeasure(R,s)} gets the current value of the maximum measure for the given NAG routine.")) (|changeMeasure| (($ $ (|Symbol|) (|Float|)) "\\spad{changeMeasure(R,s,newValue)} changes the maximum value for a measure of the given NAG routine.")) (|changeThreshhold| (($ $ (|Symbol|) (|Float|)) "\\spad{changeThreshhold(R,s,newValue)} changes the value below which,{} given a NAG routine generating a higher measure,{} the routines will make no attempt to generate a measure.")) (|selectMultiDimensionalRoutines| (($ $) "\\spad{selectMultiDimensionalRoutines(R)} chooses only those routines from the database which are designed for use with multi-dimensional expressions")) (|selectNonFiniteRoutines| (($ $) "\\spad{selectNonFiniteRoutines(R)} chooses only those routines from the database which are designed for use with non-finite expressions.")) (|selectSumOfSquaresRoutines| (($ $) "\\spad{selectSumOfSquaresRoutines(R)} chooses only those routines from the database which are designed for use with sums of squares")) (|selectFiniteRoutines| (($ $) "\\spad{selectFiniteRoutines(R)} chooses only those routines from the database which are designed for use with finite expressions")) (|selectODEIVPRoutines| (($ $) "\\spad{selectODEIVPRoutines(R)} chooses only those routines from the database which are for the solution of ODE\\spad{'s}")) (|selectPDERoutines| (($ $) "\\spad{selectPDERoutines(R)} chooses only those routines from the database which are for the solution of PDE\\spad{'s}")) (|selectOptimizationRoutines| (($ $) "\\spad{selectOptimizationRoutines(R)} chooses only those routines from the database which are for integration")) (|selectIntegrationRoutines| (($ $) "\\spad{selectIntegrationRoutines(R)} chooses only those routines from the database which are for integration")) (|routines| (($) "\\spad{routines()} initialises a database of known NAG routines")) (|concat| (($ $ $) "\\spad{concat(x,y)} merges two tables \\spad{x} and \\spad{y}"))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1194))) (LIST (QUOTE |:|) (QUOTE -3179) (QUOTE (-52))))))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-52) (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -318) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-1194) (QUOTE (-861))) (|HasCategory| (-52) (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1194))) (LIST (QUOTE |:|) (QUOTE -3179) (QUOTE (-52))))))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-52) (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| (-52) (QUOTE (-1117))) (|HasCategory| (-52) (LIST (QUOTE -318) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (QUOTE (-1117))) (|HasCategory| (-1194) (QUOTE (-861))) (|HasCategory| (-52) (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-52) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1194)) (|:| -3179 (-52))) (LIST (QUOTE -624) (QUOTE (-873))))) (-1081 S R E V) ((|constructor| (NIL "A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \\axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.")) (|mainSquareFreePart| (($ $) "\\axiom{mainSquareFreePart(\\spad{p})} returns the square free part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainPrimitivePart| (($ $) "\\axiom{mainPrimitivePart(\\spad{p})} returns the primitive part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainContent| (($ $) "\\axiom{mainContent(\\spad{p})} returns the content of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|primitivePart!| (($ $) "\\axiom{primitivePart!(\\spad{p})} replaces \\axiom{\\spad{p}} by its primitive part.")) (|gcd| ((|#2| |#2| $) "\\axiom{\\spad{gcd}(\\spad{r},{}\\spad{p})} returns the \\spad{gcd} of \\axiom{\\spad{r}} and the content of \\axiom{\\spad{p}}.")) (|nextsubResultant2| (($ $ $ $ $) "\\axiom{nextsubResultant2(\\spad{p},{}\\spad{q},{}\\spad{z},{}\\spad{s})} is the multivariate version of the operation \\axiomOpFrom{next_sousResultant2}{PseudoRemainderSequence} from the \\axiomType{PseudoRemainderSequence} constructor.")) (|LazardQuotient2| (($ $ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient2(\\spad{p},{}a,{}\\spad{b},{}\\spad{n})} returns \\axiom{(a**(\\spad{n}-1) * \\spad{p}) exquo \\spad{b**}(\\spad{n}-1)} assuming that this quotient does not fail.")) (|LazardQuotient| (($ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a**n exquo \\spad{b**}(\\spad{n}-1)} assuming that this quotient does not fail.")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns the last non-zero subresultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|subResultantChain| (((|List| $) $ $) "\\axiom{subResultantChain(a,{}\\spad{b})},{} where \\axiom{a} and \\axiom{\\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \\axiom{a} and \\axiom{\\spad{b}}.")) (|resultant| (($ $ $) "\\axiom{resultant(a,{}\\spad{b})} computes the resultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedSubResultantGcd2(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}\\spad{cb}]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}\\spad{cb}]} otherwise produces an error.")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedSubResultantGcd1(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}\\spad{cb}]} otherwise produces an error.")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[ca,{}\\spad{cb},{}\\spad{r}]} such that \\axiom{\\spad{r}} is \\axiom{subResultantGcd(a,{}\\spad{b})} and we have \\axiom{ca * a + \\spad{cb} * \\spad{cb} = \\spad{r}} .")) (|subResultantGcd| (($ $ $) "\\axiom{subResultantGcd(a,{}\\spad{b})} computes a \\spad{gcd} of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \\axiom{\\spad{R}}.")) (|exactQuotient!| (($ $ $) "\\axiom{exactQuotient!(a,{}\\spad{b})} replaces \\axiom{a} by \\axiom{exactQuotient(a,{}\\spad{b})}") (($ $ |#2|) "\\axiom{exactQuotient!(\\spad{p},{}\\spad{r})} replaces \\axiom{\\spad{p}} by \\axiom{exactQuotient(\\spad{p},{}\\spad{r})}.")) (|exactQuotient| (($ $ $) "\\axiom{exactQuotient(a,{}\\spad{b})} computes the exact quotient of \\axiom{a} by \\axiom{\\spad{b}},{} which is assumed to be a divisor of \\axiom{a}. No error is returned if this exact quotient fails!") (($ $ |#2|) "\\axiom{exactQuotient(\\spad{p},{}\\spad{r})} computes the exact quotient of \\axiom{\\spad{p}} by \\axiom{\\spad{r}},{} which is assumed to be a divisor of \\axiom{\\spad{p}}. No error is returned if this exact quotient fails!")) (|primPartElseUnitCanonical!| (($ $) "\\axiom{primPartElseUnitCanonical!(\\spad{p})} replaces \\axiom{\\spad{p}} by \\axiom{primPartElseUnitCanonical(\\spad{p})}.")) (|primPartElseUnitCanonical| (($ $) "\\axiom{primPartElseUnitCanonical(\\spad{p})} returns \\axiom{primitivePart(\\spad{p})} if \\axiom{\\spad{R}} is a \\spad{gcd}-domain,{} otherwise \\axiom{unitCanonical(\\spad{p})}.")) (|convert| (($ (|Polynomial| |#2|)) "\\axiom{convert(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}},{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.")) (|retract| (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.")) (|initiallyReduce| (($ $ $) "\\axiom{initiallyReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|headReduce| (($ $ $) "\\axiom{headReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| $) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{p},{}\\spad{q},{}\\spad{n}]} where \\axiom{\\spad{p} / q**n} represents the residue class of \\axiom{a} modulo \\axiom{\\spad{b}} and \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{q}} is \\axiom{init(\\spad{b})}.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} computes \\axiom{a mod \\spad{b}},{} if \\axiom{\\spad{b}} is monic as univariate polynomial in its main variable.")) (|pseudoDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{pseudoDivide(a,{}\\spad{b})} computes \\axiom{[pquo(a,{}\\spad{b}),{}prem(a,{}\\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}},{} if \\axiom{\\spad{b}} is not a constant polynomial.")) (|lazyPseudoDivide| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $ |#4|) "\\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})},{} \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}] = lazyPremWithDefault(a,{}\\spad{b})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.")) (|lazyPremWithDefault| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $ |#4|) "\\axiom{lazyPremWithDefault(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b},{}\\spad{v})}.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $) "\\axiom{lazyPremWithDefault(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b})} and \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b})}.")) (|lazyPquo| (($ $ $ |#4|) "\\axiom{lazyPquo(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.") (($ $ $) "\\axiom{lazyPquo(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.")) (|lazyPrem| (($ $ $ |#4|) "\\axiom{lazyPrem(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} viewed as univariate polynomials in the variable \\axiom{\\spad{v}} such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.") (($ $ $) "\\axiom{lazyPrem(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.")) (|pquo| (($ $ $ |#4|) "\\axiom{pquo(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{pquo(a,{}\\spad{b})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|prem| (($ $ $ |#4|) "\\axiom{prem(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{prem(a,{}\\spad{b})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|normalized?| (((|Boolean|) $ (|List| $)) "\\axiom{normalized?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{normalized?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{normalized?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{a} and its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variable of \\axiom{\\spad{b}}")) (|initiallyReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{initiallyReduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{initiallyReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{initiallyReduced?(a,{}\\spad{b})} returns \\spad{false} iff there exists an iterated initial of \\axiom{a} which is not reduced \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{b}}.")) (|headReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{headReduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{headReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{headReduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(head(a),{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|reduced?| (((|Boolean|) $ (|List| $)) "\\axiom{reduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{reduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{reduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(a,{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|supRittWu?| (((|Boolean|) $ $) "\\axiom{supRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is greater than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is less than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|RittWuCompare| (((|Union| (|Boolean|) "failed") $ $) "\\axiom{RittWuCompare(a,{}\\spad{b})} returns \\axiom{\"failed\"} if \\axiom{a} and \\axiom{\\spad{b}} have same rank \\spad{w}.\\spad{r}.\\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \\axiom{infRittWu?(a,{}\\spad{b})}.")) (|mainMonomials| (((|List| $) $) "\\axiom{mainMonomials(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [1],{} otherwise returns the list of the monomials of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainCoefficients| (((|List| $) $) "\\axiom{mainCoefficients(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [\\spad{p}],{} otherwise returns the list of the coefficients of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|leastMonomial| (($ $) "\\axiom{leastMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} the monomial of \\axiom{\\spad{p}} with lowest degree,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainMonomial| (($ $) "\\axiom{mainMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} \\axiom{mvar(\\spad{p})} raised to the power \\axiom{mdeg(\\spad{p})}.")) (|quasiMonic?| (((|Boolean|) $) "\\axiom{quasiMonic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff the initial of \\axiom{\\spad{p}} lies in the base ring \\axiom{\\spad{R}}.")) (|monic?| (((|Boolean|) $) "\\axiom{monic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff \\axiom{\\spad{p}} is monic as a univariate polynomial in its main variable.")) (|reductum| (($ $ |#4|) "\\axiom{reductum(\\spad{p},{}\\spad{v})} returns the reductum of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in \\axiom{\\spad{v}}.")) (|leadingCoefficient| (($ $ |#4|) "\\axiom{leadingCoefficient(\\spad{p},{}\\spad{v})} returns the leading coefficient of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as A univariate polynomial in \\axiom{\\spad{v}}.")) (|deepestInitial| (($ $) "\\axiom{deepestInitial(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the last term of \\axiom{iteratedInitials(\\spad{p})}.")) (|iteratedInitials| (((|List| $) $) "\\axiom{iteratedInitials(\\spad{p})} returns \\axiom{[]} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the list of the iterated initials of \\axiom{\\spad{p}}.")) (|deepestTail| (($ $) "\\axiom{deepestTail(\\spad{p})} returns \\axiom{0} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns tail(\\spad{p}),{} if \\axiom{tail(\\spad{p})} belongs to \\axiom{\\spad{R}} or \\axiom{mvar(tail(\\spad{p})) < mvar(\\spad{p})},{} otherwise returns \\axiom{deepestTail(tail(\\spad{p}))}.")) (|tail| (($ $) "\\axiom{tail(\\spad{p})} returns its reductum,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|head| (($ $) "\\axiom{head(\\spad{p})} returns \\axiom{\\spad{p}} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading term (monomial in the AXIOM sense),{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|init| (($ $) "\\axiom{init(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading coefficient,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mdeg| (((|NonNegativeInteger|) $) "\\axiom{mdeg(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{0},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{0},{} otherwise,{} returns the degree of \\axiom{\\spad{p}} in its main variable.")) (|mvar| ((|#4| $) "\\axiom{mvar(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in \\axiom{\\spad{V}}."))) NIL @@ -4300,11 +4300,11 @@ NIL ((|constructor| (NIL "This domain implements named rules")) (|name| (((|Symbol|) $) "\\spad{name(x)} returns the symbol"))) NIL NIL -(-1093 |Base| R -3029) +(-1093 |Base| R -3027) ((|constructor| (NIL "\\indented{1}{Rules for the pattern matcher} Author: Manuel Bronstein Date Created: 24 Oct 1988 Date Last Updated: 26 October 1993 Keywords: pattern,{} matching,{} rule.")) (|quotedOperators| (((|List| (|Symbol|)) $) "\\spad{quotedOperators(r)} returns the list of operators on the right hand side of \\spad{r} that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.")) (|elt| ((|#3| $ |#3| (|PositiveInteger|)) "\\spad{elt(r,f,n)} or \\spad{r}(\\spad{f},{} \\spad{n}) applies the rule \\spad{r} to \\spad{f} at most \\spad{n} times.")) (|rhs| ((|#3| $) "\\spad{rhs(r)} returns the right hand side of the rule \\spad{r}.")) (|lhs| ((|#3| $) "\\spad{lhs(r)} returns the left hand side of the rule \\spad{r}.")) (|pattern| (((|Pattern| |#1|) $) "\\spad{pattern(r)} returns the pattern corresponding to the left hand side of the rule \\spad{r}.")) (|suchThat| (($ $ (|List| (|Symbol|)) (|Mapping| (|Boolean|) (|List| |#3|))) "\\spad{suchThat(r, [a1,...,an], f)} returns the rewrite rule \\spad{r} with the predicate \\spad{f(a1,...,an)} attached to it.")) (|rule| (($ |#3| |#3| (|List| (|Symbol|))) "\\spad{rule(f, g, [f1,...,fn])} creates the rewrite rule \\spad{f == eval(eval(g, g is f), [f1,...,fn])},{} that is a rule with left-hand side \\spad{f} and right-hand side \\spad{g}; The symbols \\spad{f1},{}...,{}\\spad{fn} are the operators that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.") (($ |#3| |#3|) "\\spad{rule(f, g)} creates the rewrite rule: \\spad{f == eval(g, g is f)},{} with left-hand side \\spad{f} and right-hand side \\spad{g}."))) NIL NIL -(-1094 |Base| R -3029) +(-1094 |Base| R -3027) ((|constructor| (NIL "A ruleset is a set of pattern matching rules grouped together.")) (|elt| ((|#3| $ |#3| (|PositiveInteger|)) "\\spad{elt(r,f,n)} or \\spad{r}(\\spad{f},{} \\spad{n}) applies all the rules of \\spad{r} to \\spad{f} at most \\spad{n} times.")) (|rules| (((|List| (|RewriteRule| |#1| |#2| |#3|)) $) "\\spad{rules(r)} returns the rules contained in \\spad{r}.")) (|ruleset| (($ (|List| (|RewriteRule| |#1| |#2| |#3|))) "\\spad{ruleset([r1,...,rn])} creates the rule set \\spad{{r1,...,rn}}."))) NIL NIL @@ -4319,7 +4319,7 @@ NIL (-1097 R UP M) ((|constructor| (NIL "Domain which represents simple algebraic extensions of arbitrary rings. The first argument to the domain,{} \\spad{R},{} is the underlying ring,{} the second argument is a domain of univariate polynomials over \\spad{K},{} while the last argument specifies the defining minimal polynomial. The elements of the domain are canonically represented as polynomials of degree less than that of the minimal polynomial with coefficients in \\spad{R}. The second argument is both the type of the third argument and the underlying representation used by \\spadtype{SAE} itself."))) ((-4453 |has| |#1| (-373)) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-359))) (-3765 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-359)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-378))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-359)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-359)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-359)))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))))) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-359))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-359)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-378))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-359)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-359)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-359))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-359)))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194))))) (-12 (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))))) (-1098 UP SAE UPA) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of \\spadtype{Fraction Polynomial Integer}.")) (|factor| (((|Factored| |#3|) |#3|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) NIL @@ -4347,7 +4347,7 @@ NIL (-1104 R) ((|constructor| (NIL "\\spadtype{SequentialDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is sequential. \\blankline"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-924))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) +((|HasCategory| |#1| (QUOTE (-924))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (|HasCategory| |#1| (QUOTE (-463))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#1| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1105 (-1194)) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#1| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-238))) (|HasCategory| |#1| (QUOTE (-237))) (|HasCategory| |#1| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasAttribute| |#1| (QUOTE -4458)) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-924)))) (|HasCategory| |#1| (QUOTE (-146))))) (-1105 S) ((|constructor| (NIL "\\spadtype{OrderlyDifferentialVariable} adds a commonly used sequential ranking to the set of derivatives of an ordered list of differential indeterminates. A sequential ranking is a ranking \\spadfun{<} of the derivatives with the property that for any derivative \\spad{v},{} there are only a finite number of derivatives \\spad{u} with \\spad{u} \\spadfun{<} \\spad{v}. This domain belongs to \\spadtype{DifferentialVariableCategory}. It defines \\spadfun{weight} to be just \\spadfun{order},{} and it defines a sequential ranking \\spadfun{<} on derivatives \\spad{u} by the lexicographic order on the pair (\\spadfun{variable}(\\spad{u}),{} \\spadfun{order}(\\spad{u}))."))) NIL @@ -4407,7 +4407,7 @@ NIL (-1119 S) ((|constructor| (NIL "A set over a domain \\spad{D} models the usual mathematical notion of a finite set of elements from \\spad{D}. Sets are unordered collections of distinct elements (that is,{} order and duplication does not matter). The notation \\spad{set [a,b,c]} can be used to create a set and the usual operations such as union and intersection are available to form new sets. In our implementation,{} \\Language{} maintains the entries in sorted order. Specifically,{} the parts function returns the entries as a list in ascending order and the extract operation returns the maximum entry. Given two sets \\spad{s} and \\spad{t} where \\spad{\\#s = m} and \\spad{\\#t = n},{} the complexity of \\indented{2}{\\spad{s = t} is \\spad{O(min(n,m))}} \\indented{2}{\\spad{s < t} is \\spad{O(max(n,m))}} \\indented{2}{\\spad{union(s,t)},{} \\spad{intersect(s,t)},{} \\spad{minus(s,t)},{} \\spad{symmetricDifference(s,t)} is \\spad{O(max(n,m))}} \\indented{2}{\\spad{member(x,t)} is \\spad{O(n log n)}} \\indented{2}{\\spad{insert(x,t)} and \\spad{remove(x,t)} is \\spad{O(n)}}"))) ((-4460 . T) (-4450 . T) (-4461 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#1| (QUOTE (-378))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-1120 |Str| |Sym| |Int| |Flt| |Expr|) ((|constructor| (NIL "This category allows the manipulation of Lisp values while keeping the grunge fairly localized.")) (|#| (((|Integer|) $) "\\spad{\\#((a1,...,an))} returns \\spad{n}.")) (|cdr| (($ $) "\\spad{cdr((a1,...,an))} returns \\spad{(a2,...,an)}.")) (|car| (($ $) "\\spad{car((a1,...,an))} returns a1.")) (|expr| ((|#5| $) "\\spad{expr(s)} returns \\spad{s} as an element of Expr; Error: if \\spad{s} is not an atom that also belongs to Expr.")) (|float| ((|#4| $) "\\spad{float(s)} returns \\spad{s} as an element of \\spad{Flt}; Error: if \\spad{s} is not an atom that also belongs to \\spad{Flt}.")) (|integer| ((|#3| $) "\\spad{integer(s)} returns \\spad{s} as an element of Int. Error: if \\spad{s} is not an atom that also belongs to Int.")) (|symbol| ((|#2| $) "\\spad{symbol(s)} returns \\spad{s} as an element of \\spad{Sym}. Error: if \\spad{s} is not an atom that also belongs to \\spad{Sym}.")) (|string| ((|#1| $) "\\spad{string(s)} returns \\spad{s} as an element of \\spad{Str}. Error: if \\spad{s} is not an atom that also belongs to \\spad{Str}.")) (|destruct| (((|List| $) $) "\\spad{destruct((a1,...,an))} returns the list [a1,{}...,{}an].")) (|float?| (((|Boolean|) $) "\\spad{float?(s)} is \\spad{true} if \\spad{s} is an atom and belong to \\spad{Flt}.")) (|integer?| (((|Boolean|) $) "\\spad{integer?(s)} is \\spad{true} if \\spad{s} is an atom and belong to Int.")) (|symbol?| (((|Boolean|) $) "\\spad{symbol?(s)} is \\spad{true} if \\spad{s} is an atom and belong to \\spad{Sym}.")) (|string?| (((|Boolean|) $) "\\spad{string?(s)} is \\spad{true} if \\spad{s} is an atom and belong to \\spad{Str}.")) (|list?| (((|Boolean|) $) "\\spad{list?(s)} is \\spad{true} if \\spad{s} is a Lisp list,{} possibly ().")) (|pair?| (((|Boolean|) $) "\\spad{pair?(s)} is \\spad{true} if \\spad{s} has is a non-null Lisp list.")) (|atom?| (((|Boolean|) $) "\\spad{atom?(s)} is \\spad{true} if \\spad{s} is a Lisp atom.")) (|null?| (((|Boolean|) $) "\\spad{null?(s)} is \\spad{true} if \\spad{s} is the \\spad{S}-expression ().")) 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(|HasCategory| |#3| (QUOTE (-1066)))) (-12 (|HasCategory| |#3| (QUOTE (-1066))) (|HasCategory| |#3| (LIST (QUOTE -913) (QUOTE (-1194))))) (|HasCategory| |#3| (QUOTE (-861))) (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-23))) (|HasCategory| |#3| (QUOTE (-132))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#3| (QUOTE (-1117))) (|HasCategory| |#3| (LIST (QUOTE -318) (|devaluate| |#3|))))) (-1131 R |x|) ((|constructor| (NIL "This package produces functions for counting etc. real roots of univariate polynomials in \\spad{x} over \\spad{R},{} which must be an OrderedIntegralDomain")) (|countRealRootsMultiple| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{countRealRootsMultiple(p)} says how many real roots \\spad{p} has,{} counted with multiplicity")) (|SturmHabichtMultiple| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtMultiple(p1,p2)} computes \\spad{c_}{+}\\spad{-c_}{-} where \\spad{c_}{+} is the number of real roots of \\spad{p1} with p2>0 and \\spad{c_}{-} is the number of real roots of \\spad{p1} with p2<0. If p2=1 what you get is the number of real roots of \\spad{p1}.")) (|countRealRoots| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{countRealRoots(p)} says how many real roots \\spad{p} has")) (|SturmHabicht| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabicht(p1,p2)} computes \\spad{c_}{+}\\spad{-c_}{-} where \\spad{c_}{+} is the number of real roots of \\spad{p1} with p2>0 and \\spad{c_}{-} is the number of real roots of \\spad{p1} with p2<0. If p2=1 what you get is the number of real roots of \\spad{p1}.")) (|SturmHabichtCoefficients| (((|List| |#1|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtCoefficients(p1,p2)} computes the principal Sturm-Habicht coefficients of \\spad{p1} and \\spad{p2}")) (|SturmHabichtSequence| (((|List| (|UnivariatePolynomial| |#2| |#1|)) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtSequence(p1,p2)} computes the Sturm-Habicht sequence of \\spad{p1} and \\spad{p2}")) (|subresultantSequence| (((|List| (|UnivariatePolynomial| |#2| |#1|)) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{subresultantSequence(p1,p2)} computes the (standard) subresultant sequence of \\spad{p1} and \\spad{p2}"))) NIL @@ -4460,7 +4460,7 @@ NIL ((|constructor| (NIL "This domain represents a signature AST. A signature AST \\indented{2}{is a description of an exported operation,{} \\spadignore{e.g.} its name,{} result} \\indented{2}{type,{} and the list of its argument types.}")) (|signature| (((|Signature|) $) "\\spad{signature(s)} returns AST of the declared signature for \\spad{`s'}.")) (|name| (((|Identifier|) $) "\\spad{name(s)} returns the name of the signature \\spad{`s'}.")) (|signatureAst| (($ (|Identifier|) (|Signature|)) "\\spad{signatureAst(n,s,t)} builds the signature AST \\spad{n:} \\spad{s} \\spad{->} \\spad{t}"))) NIL NIL -(-1133 R -3029) +(-1133 R -3027) ((|constructor| (NIL "This package provides functions to determine the sign of an elementary function around a point or infinity.")) (|sign| (((|Union| (|Integer|) "failed") |#2| (|Symbol|) |#2| (|String|)) "\\spad{sign(f, x, a, s)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a} from below if \\spad{s} is \"left\",{} or above if \\spad{s} is \"right\".") 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A Taylor series is represented by a stream of polynomials from the polynomial domain \\spad{SMP}. The \\spad{n}th element of the stream is a form of degree \\spad{n}. SMTS is an internal domain.")) (|fintegrate| (($ (|Mapping| $) |#2| |#1|) "\\spad{fintegrate(f,v,c)} is the integral of \\spad{f()} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.} \\indented{1}{The evaluation of \\spad{f()} is delayed.}")) (|integrate| (($ $ |#2| |#1|) "\\spad{integrate(s,v,c)} is the integral of \\spad{s} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.}")) (|csubst| (((|Mapping| (|Stream| |#3|) |#3|) (|List| |#2|) (|List| (|Stream| |#3|))) "\\spad{csubst(a,b)} is for internal use only")) (* (($ |#3| $) "\\spad{smp*ts} multiplies a TaylorSeries by a monomial \\spad{SMP}.")) (|coerce| (($ |#3|) "\\spad{coerce(poly)} regroups the terms by total degree and forms a series.") (($ |#2|) "\\spad{coerce(var)} converts a variable to a Taylor series")) (|coefficient| ((|#3| $ (|NonNegativeInteger|)) "\\spad{coefficient(s, n)} gives the terms of total degree \\spad{n}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-373)))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-373)))) (-1144 R E V P) ((|constructor| (NIL "The category of square-free and normalized triangular sets. Thus,{} up to the primitivity axiom of [1],{} these sets are Lazard triangular sets.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991}"))) ((-4461 . T) (-4460 . T)) NIL -(-1145 UP -3029) +(-1145 UP -3027) ((|constructor| (NIL "This package factors the formulas out of the general solve code,{} allowing their recursive use over different domains. Care is taken to introduce few radicals so that radical extension domains can more easily simplify the results.")) (|aQuartic| ((|#2| |#2| |#2| |#2| |#2| |#2|) "\\spad{aQuartic(f,g,h,i,k)} \\undocumented")) (|aCubic| ((|#2| |#2| |#2| |#2| |#2|) "\\spad{aCubic(f,g,h,j)} \\undocumented")) (|aQuadratic| ((|#2| |#2| |#2| |#2|) "\\spad{aQuadratic(f,g,h)} \\undocumented")) (|aLinear| ((|#2| |#2| |#2|) "\\spad{aLinear(f,g)} \\undocumented")) (|quartic| (((|List| |#2|) |#2| |#2| |#2| |#2| |#2|) "\\spad{quartic(f,g,h,i,j)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{quartic(u)} \\undocumented")) (|cubic| (((|List| |#2|) |#2| |#2| |#2| |#2|) "\\spad{cubic(f,g,h,i)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{cubic(u)} \\undocumented")) (|quadratic| (((|List| |#2|) |#2| |#2| |#2|) "\\spad{quadratic(f,g,h)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{quadratic(u)} \\undocumented")) (|linear| (((|List| |#2|) |#2| |#2|) "\\spad{linear(f,g)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{linear(u)} \\undocumented")) (|mapSolve| (((|Record| (|:| |solns| (|List| |#2|)) (|:| |maps| (|List| (|Record| (|:| |arg| |#2|) (|:| |res| |#2|))))) |#1| (|Mapping| |#2| |#2|)) "\\spad{mapSolve(u,f)} \\undocumented")) (|particularSolution| ((|#2| |#1|) "\\spad{particularSolution(u)} \\undocumented")) (|solve| (((|List| |#2|) |#1|) "\\spad{solve(u)} \\undocumented"))) NIL NIL @@ -4563,11 +4563,11 @@ NIL (-1158 V C) ((|constructor| (NIL "This domain exports a modest implementation of splitting trees. Spliiting trees are needed when the evaluation of some quantity under some hypothesis requires to split the hypothesis into sub-cases. For instance by adding some new hypothesis on one hand and its negation on another hand. The computations are terminated is a splitting tree \\axiom{a} when \\axiom{status(value(a))} is \\axiom{\\spad{true}}. Thus,{} if for the splitting tree \\axiom{a} the flag \\axiom{status(value(a))} is \\axiom{\\spad{true}},{} then \\axiom{status(value(\\spad{d}))} is \\axiom{\\spad{true}} for any subtree \\axiom{\\spad{d}} of \\axiom{a}. This property of splitting trees is called the termination condition. If no vertex in a splitting tree \\axiom{a} is equal to another,{} \\axiom{a} is said to satisfy the no-duplicates condition. The splitting tree \\axiom{a} will satisfy this condition if nodes are added to \\axiom{a} by mean of \\axiom{splitNodeOf!} and if \\axiom{construct} is only used to create the root of \\axiom{a} with no children.")) (|splitNodeOf!| (($ $ $ (|List| (|SplittingNode| |#1| |#2|)) (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{splitNodeOf!(\\spad{l},{}a,{}\\spad{ls},{}sub?)} returns \\axiom{a} where the children list of \\axiom{\\spad{l}} has been set to \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in \\spad{ls} | not subNodeOf?(\\spad{s},{}a,{}sub?)]}. Thus,{} if \\axiom{\\spad{l}} is not a node of \\axiom{a},{} this latter splitting tree is unchanged.") (($ $ $ (|List| (|SplittingNode| |#1| |#2|))) "\\axiom{splitNodeOf!(\\spad{l},{}a,{}\\spad{ls})} returns \\axiom{a} where the children list of \\axiom{\\spad{l}} has been set to \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in \\spad{ls} | not nodeOf?(\\spad{s},{}a)]}. Thus,{} if \\axiom{\\spad{l}} is not a node of \\axiom{a},{} this latter splitting tree is unchanged.")) (|remove!| (($ (|SplittingNode| |#1| |#2|) $) "\\axiom{remove!(\\spad{s},{}a)} replaces a by remove(\\spad{s},{}a)")) (|remove| (($ (|SplittingNode| |#1| |#2|) $) "\\axiom{remove(\\spad{s},{}a)} returns the splitting tree obtained from a by removing every sub-tree \\axiom{\\spad{b}} such that \\axiom{value(\\spad{b})} and \\axiom{\\spad{s}} have the same value,{} condition and status.")) (|subNodeOf?| (((|Boolean|) (|SplittingNode| |#1| |#2|) $ (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{subNodeOf?(\\spad{s},{}a,{}sub?)} returns \\spad{true} iff for some node \\axiom{\\spad{n}} in \\axiom{a} we have \\axiom{\\spad{s} = \\spad{n}} or \\axiom{status(\\spad{n})} and \\axiom{subNode?(\\spad{s},{}\\spad{n},{}sub?)}.")) (|nodeOf?| (((|Boolean|) (|SplittingNode| |#1| |#2|) $) "\\axiom{nodeOf?(\\spad{s},{}a)} returns \\spad{true} iff some node of \\axiom{a} is equal to \\axiom{\\spad{s}}")) (|result| (((|List| (|Record| (|:| |val| |#1|) (|:| |tower| |#2|))) $) "\\axiom{result(a)} where \\axiom{\\spad{ls}} is the leaves list of \\axiom{a} returns \\axiom{[[value(\\spad{s}),{}condition(\\spad{s})]\\$\\spad{VT} for \\spad{s} in \\spad{ls}]} if the computations are terminated in \\axiom{a} else an error is produced.")) (|conditions| (((|List| |#2|) $) "\\axiom{conditions(a)} returns the list of the conditions of the leaves of a")) (|construct| (($ |#1| |#2| |#1| (|List| |#2|)) "\\axiom{construct(\\spad{v1},{}\\spad{t},{}\\spad{v2},{}\\spad{lt})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with children list given by \\axiom{[[[\\spad{v},{}\\spad{t}]\\$\\spad{S}]\\$\\% for \\spad{s} in \\spad{ls}]}.") (($ |#1| |#2| (|List| (|SplittingNode| |#1| |#2|))) "\\axiom{construct(\\spad{v},{}\\spad{t},{}\\spad{ls})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with children list given by \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in \\spad{ls}]}.") (($ |#1| |#2| (|List| $)) "\\axiom{construct(\\spad{v},{}\\spad{t},{}la)} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with \\axiom{la} as children list.") (($ (|SplittingNode| |#1| |#2|)) "\\axiom{construct(\\spad{s})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{\\spad{s}} and no children. Thus,{} if the status of \\axiom{\\spad{s}} is \\spad{false},{} \\axiom{[\\spad{s}]} represents the starting point of the evaluation \\axiom{value(\\spad{s})} under the hypothesis \\axiom{condition(\\spad{s})}.")) (|updateStatus!| (($ $) "\\axiom{updateStatus!(a)} returns a where the status of the vertices are updated to satisfy the \"termination condition\".")) (|extractSplittingLeaf| (((|Union| $ "failed") $) "\\axiom{extractSplittingLeaf(a)} returns the left most leaf (as a tree) whose status is \\spad{false} if any,{} else \"failed\" is returned."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -318) (LIST (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1157 |#1| |#2|) (QUOTE (-1117)))) (|HasCategory| (-1157 |#1| |#2|) (QUOTE (-1117))) (-3765 (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -318) (LIST (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1157 |#1| |#2|) (QUOTE (-1117))))) (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -318) (LIST (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1157 |#1| |#2|) (QUOTE (-1117)))) (|HasCategory| (-1157 |#1| |#2|) (QUOTE (-1117))) (-3763 (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -318) (LIST (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1157 |#1| |#2|) (QUOTE (-1117))))) (|HasCategory| (-1157 |#1| |#2|) (LIST (QUOTE -624) (QUOTE (-873))))) (-1159 |ndim| R) ((|constructor| (NIL "\\spadtype{SquareMatrix} is a matrix domain of square matrices,{} where the number of rows (= number of columns) is a parameter of the type.")) (|unitsKnown| ((|attribute|) "the invertible matrices are simply the matrices whose determinants are units in the Ring \\spad{R}.")) (|central| ((|attribute|) "the elements of the Ring \\spad{R},{} viewed as diagonal matrices,{} commute with all matrices and,{} indeed,{} are the only matrices which commute with all matrices.")) (|squareMatrix| (($ (|Matrix| |#2|)) "\\spad{squareMatrix(m)} converts a matrix of type \\spadtype{Matrix} to a matrix of type \\spadtype{SquareMatrix}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.")) (|new| (($ |#2|) "\\spad{new(c)} constructs a new \\spadtype{SquareMatrix} object of dimension \\spad{ndim} with initial entries equal to \\spad{c}."))) ((-4457 . T) (-4449 |has| |#2| (-6 (-4462 "*"))) (-4460 . T) (-4454 . T) (-4455 . T)) -((|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (QUOTE (-237))) (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (-12 (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-373))) (-3765 (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) +((|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (QUOTE (-237))) (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (-12 (|HasCategory| |#2| (QUOTE (-238))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (QUOTE (-316))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-373))) (-3763 (|HasAttribute| |#2| (QUOTE (-4462 "*"))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-238)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) (-1160 S) ((|constructor| (NIL "A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.")) (|elt| (($ $ $) "\\spad{elt(s,t)} returns the concatenation of \\spad{s} and \\spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \\axiom{\"smoo\" \"shed\"} returns \\axiom{\"smooshed\"}.")) (|rightTrim| (($ $ (|CharacterClass|)) "\\spad{rightTrim(s,cc)} returns \\spad{s} with all trailing occurences of characters in \\spad{cc} deleted. For example,{} \\axiom{rightTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"(abc\"}.") (($ $ (|Character|)) "\\spad{rightTrim(s,c)} returns \\spad{s} with all trailing occurrences of \\spad{c} deleted. For example,{} \\axiom{rightTrim(\" abc \",{} char \" \")} returns \\axiom{\" abc\"}.")) (|leftTrim| (($ $ (|CharacterClass|)) "\\spad{leftTrim(s,cc)} returns \\spad{s} with all leading characters in \\spad{cc} deleted. For example,{} \\axiom{leftTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc)\"}.") (($ $ (|Character|)) "\\spad{leftTrim(s,c)} returns \\spad{s} with all leading characters \\spad{c} deleted. For example,{} \\axiom{leftTrim(\" abc \",{} char \" \")} returns \\axiom{\"abc \"}.")) (|trim| (($ $ (|CharacterClass|)) "\\spad{trim(s,cc)} returns \\spad{s} with all characters in \\spad{cc} deleted from right and left ends. For example,{} \\axiom{trim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc\"}.") (($ $ (|Character|)) "\\spad{trim(s,c)} returns \\spad{s} with all characters \\spad{c} deleted from right and left ends. For example,{} \\axiom{trim(\" abc \",{} char \" \")} returns \\axiom{\"abc\"}.")) (|split| (((|List| $) $ (|CharacterClass|)) "\\spad{split(s,cc)} returns a list of substrings delimited by characters in \\spad{cc}.") (((|List| $) $ (|Character|)) "\\spad{split(s,c)} returns a list of substrings delimited by character \\spad{c}.")) (|coerce| (($ (|Character|)) "\\spad{coerce(c)} returns \\spad{c} as a string \\spad{s} with the character \\spad{c}.")) (|position| (((|Integer|) (|CharacterClass|) $ (|Integer|)) "\\spad{position(cc,t,i)} returns the position \\axiom{\\spad{j} \\spad{>=} \\spad{i}} in \\spad{t} of the first character belonging to \\spad{cc}.") (((|Integer|) $ $ (|Integer|)) "\\spad{position(s,t,i)} returns the position \\spad{j} of the substring \\spad{s} in string \\spad{t},{} where \\axiom{\\spad{j} \\spad{>=} \\spad{i}} is required.")) (|replace| (($ $ (|UniversalSegment| (|Integer|)) $) "\\spad{replace(s,i..j,t)} replaces the substring \\axiom{\\spad{s}(\\spad{i}..\\spad{j})} of \\spad{s} by string \\spad{t}.")) (|match?| (((|Boolean|) $ $ (|Character|)) "\\spad{match?(s,t,c)} tests if \\spad{s} matches \\spad{t} except perhaps for multiple and consecutive occurrences of character \\spad{c}. Typically \\spad{c} is the blank character.")) (|match| (((|NonNegativeInteger|) $ $ (|Character|)) "\\spad{match(p,s,wc)} tests if pattern \\axiom{\\spad{p}} matches subject \\axiom{\\spad{s}} where \\axiom{\\spad{wc}} is a wild card character. If no match occurs,{} the index \\axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \\axiom{match(\"*to*\",{}\"yorktown\",{}\\spad{\"*\"})} returns \\axiom{5} indicating a successful match starting at index \\axiom{5} of \\axiom{\"yorktown\"}.")) (|substring?| (((|Boolean|) $ $ (|Integer|)) "\\spad{substring?(s,t,i)} tests if \\spad{s} is a substring of \\spad{t} beginning at index \\spad{i}. Note: \\axiom{substring?(\\spad{s},{}\\spad{t},{}0) = prefix?(\\spad{s},{}\\spad{t})}.")) (|suffix?| (((|Boolean|) $ $) "\\spad{suffix?(s,t)} tests if the string \\spad{s} is the final substring of \\spad{t}. Note: \\axiom{suffix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.(\\spad{n} - \\spad{m} + \\spad{i}) for \\spad{i} in 0..maxIndex \\spad{s}])} where \\spad{m} and \\spad{n} denote the maxIndex of \\spad{s} and \\spad{t} respectively.")) (|prefix?| (((|Boolean|) $ $) "\\spad{prefix?(s,t)} tests if the string \\spad{s} is the initial substring of \\spad{t}. Note: \\axiom{prefix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.\\spad{i} for \\spad{i} in 0..maxIndex \\spad{s}])}.")) (|upperCase!| (($ $) "\\spad{upperCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by upper case characters.")) (|upperCase| (($ $) "\\spad{upperCase(s)} returns the string with all characters in upper case.")) (|lowerCase!| (($ $) "\\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by lower case.")) (|lowerCase| (($ $) "\\spad{lowerCase(s)} returns the string with all characters in lower case."))) NIL @@ -4587,7 +4587,7 @@ NIL (-1164 S) ((|constructor| (NIL "Linked List implementation of a Stack")) (|stack| (($ (|List| |#1|)) "\\spad{stack([x,y,...,z])} creates a stack with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-1165 A S) ((|constructor| (NIL "A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \\spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \\spadtype{DecimalExpansion}.")) (|possiblyInfinite?| (((|Boolean|) $) "\\spad{possiblyInfinite?(s)} tests if the stream \\spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \\axiom{possiblyInfinite?(\\spad{s}) = not explictlyFinite?(\\spad{s})}.")) (|explicitlyFinite?| (((|Boolean|) $) "\\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \\spad{false} otherwise. Note: for many datatypes,{} \\axiom{explicitlyFinite?(\\spad{s}) = not possiblyInfinite?(\\spad{s})}."))) NIL @@ -4599,7 +4599,7 @@ NIL (-1167 |Key| |Ent| |dent|) ((|constructor| (NIL "A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key."))) ((-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-861))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117)))) +((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-861))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117)))) (-1168) ((|constructor| (NIL "This domain represents an arithmetic progression iterator syntax.")) (|step| (((|SpadAst|) $) "\\spad{step(i)} returns the Spad AST denoting the step of the arithmetic progression represented by the iterator \\spad{i}.")) (|upperBound| (((|Maybe| (|SpadAst|)) $) "If the set of values assumed by the iteration variable is bounded from above,{} \\spad{upperBound(i)} returns the upper bound. Otherwise,{} its returns \\spad{nothing}.")) (|lowerBound| (((|SpadAst|) $) "\\spad{lowerBound(i)} returns the lower bound on the values assumed by the iteration variable.")) (|iterationVar| (((|Identifier|) $) "\\spad{iterationVar(i)} returns the name of the iterating variable of the arithmetic progression iterator \\spad{i}."))) NIL @@ -4627,7 +4627,7 @@ NIL (-1174 S) ((|constructor| (NIL "A stream is an implementation of an infinite sequence using a list of terms that have been computed and a function closure to compute additional terms when needed.")) (|filterUntil| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterUntil(p,s)} returns \\spad{[x0,x1,...,x(n)]} where \\spad{s = [x0,x1,x2,..]} and \\spad{n} is the smallest index such that \\spad{p(xn) = true}.")) (|filterWhile| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterWhile(p,s)} returns \\spad{[x0,x1,...,x(n-1)]} where \\spad{s = [x0,x1,x2,..]} and \\spad{n} is the smallest index such that \\spad{p(xn) = false}.")) (|generate| (($ (|Mapping| |#1| |#1|) |#1|) "\\spad{generate(f,x)} creates an infinite stream whose first element is \\spad{x} and whose \\spad{n}th element (\\spad{n > 1}) is \\spad{f} applied to the previous element. Note: \\spad{generate(f,x) = [x,f(x),f(f(x)),...]}.") (($ (|Mapping| |#1|)) "\\spad{generate(f)} creates an infinite stream all of whose elements are equal to \\spad{f()}. Note: \\spad{generate(f) = [f(),f(),f(),...]}.")) (|setrest!| (($ $ (|Integer|) $) "\\spad{setrest!(x,n,y)} sets rest(\\spad{x},{}\\spad{n}) to \\spad{y}. The function will expand cycles if necessary.")) (|showAll?| (((|Boolean|)) "\\spad{showAll?()} returns \\spad{true} if all computed entries of streams will be displayed.")) (|showAllElements| (((|OutputForm|) $) "\\spad{showAllElements(s)} creates an output form which displays all computed elements.")) (|output| (((|Void|) (|Integer|) $) "\\spad{output(n,st)} computes and displays the first \\spad{n} entries of \\spad{st}.")) (|cons| (($ |#1| $) "\\spad{cons(a,s)} returns a stream whose \\spad{first} is \\spad{a} and whose \\spad{rest} is \\spad{s}. Note: \\spad{cons(a,s) = concat(a,s)}.")) (|delay| (($ (|Mapping| $)) "\\spad{delay(f)} creates a stream with a lazy evaluation defined by function \\spad{f}. Caution: This function can only be called in compiled code.")) (|findCycle| (((|Record| (|:| |cycle?| (|Boolean|)) (|:| |prefix| (|NonNegativeInteger|)) (|:| |period| (|NonNegativeInteger|))) (|NonNegativeInteger|) $) "\\spad{findCycle(n,st)} determines if \\spad{st} is periodic within \\spad{n}.")) (|repeating?| (((|Boolean|) (|List| |#1|) $) "\\spad{repeating?(l,s)} returns \\spad{true} if a stream \\spad{s} is periodic with period \\spad{l},{} and \\spad{false} otherwise.")) (|repeating| (($ (|List| |#1|)) "\\spad{repeating(l)} is a repeating stream whose period is the list \\spad{l}.")) (|shallowlyMutable| ((|attribute|) "one may destructively alter a stream by assigning new values to its entries."))) ((-4461 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-1175) ((|constructor| (NIL "A category for string-like objects")) (|string| (($ (|Integer|)) "\\spad{string(i)} returns the decimal representation of \\spad{i} in a string"))) ((-4461 . T) (-4460 . T)) @@ -4635,11 +4635,11 @@ NIL (-1176) NIL ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) +((-3763 (-12 (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| (-145) (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| (-145) (QUOTE (-1117))) (|HasCategory| (-145) (LIST (QUOTE -318) (QUOTE (-145)))))) (-1177 |Entry|) ((|constructor| (NIL "This domain provides tables where the keys are strings. A specialized hash function for strings is used."))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1176))) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#1|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| (-1176) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (QUOTE (-1176))) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#1|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (QUOTE (-1117))) (|HasCategory| (-1176) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 (-1176)) (|:| -3179 |#1|)) (LIST (QUOTE -624) (QUOTE (-873))))) (-1178 A) ((|constructor| (NIL "StreamTaylorSeriesOperations implements Taylor series arithmetic,{} where a Taylor series is represented by a stream of its coefficients.")) (|power| (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{power(a,f)} returns the power series \\spad{f} raised to the power \\spad{a}.")) (|lazyGintegrate| (((|Stream| |#1|) (|Mapping| |#1| (|Integer|)) |#1| (|Mapping| (|Stream| |#1|))) "\\spad{lazyGintegrate(f,r,g)} is used for fixed point computations.")) (|mapdiv| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{mapdiv([a0,a1,..],[b0,b1,..])} returns \\spad{[a0/b0,a1/b1,..]}.")) (|powern| (((|Stream| |#1|) (|Fraction| (|Integer|)) (|Stream| |#1|)) "\\spad{powern(r,f)} raises power series \\spad{f} to the power \\spad{r}.")) (|nlde| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{nlde(u)} solves a first order non-linear differential equation described by \\spad{u} of the form \\spad{[[b<0,0>,b<0,1>,...],[b<1,0>,b<1,1>,.],...]}. the differential equation has the form \\spad{y' = sum(i=0 to infinity,j=0 to infinity,b<i,j>*(x**i)*(y**j))}.")) (|lazyIntegrate| (((|Stream| |#1|) |#1| (|Mapping| (|Stream| |#1|))) "\\spad{lazyIntegrate(r,f)} is a local function used for fixed point computations.")) (|integrate| (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{integrate(r,a)} returns the integral of the power series \\spad{a} with respect to the power series variableintegration where \\spad{r} denotes the constant of integration. Thus \\spad{integrate(a,[a0,a1,a2,...]) = [a,a0,a1/2,a2/3,...]}.")) (|invmultisect| (((|Stream| |#1|) (|Integer|) (|Integer|) (|Stream| |#1|)) "\\spad{invmultisect(a,b,st)} substitutes \\spad{x**((a+b)*n)} for \\spad{x**n} and multiplies by \\spad{x**b}.")) (|multisect| (((|Stream| |#1|) (|Integer|) (|Integer|) (|Stream| |#1|)) "\\spad{multisect(a,b,st)} selects the coefficients of \\spad{x**((a+b)*n+a)},{} and changes them to \\spad{x**n}.")) (|generalLambert| (((|Stream| |#1|) (|Stream| |#1|) (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x**a) + f(x**(a + d)) + f(x**(a + 2 d)) + ...}. \\spad{f(x)} should have zero constant coefficient and \\spad{a} and \\spad{d} should be positive.")) (|evenlambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{evenlambert(st)} computes \\spad{f(x**2) + f(x**4) + f(x**6) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f(x)} is a power series with constant coefficient 1,{} then \\spad{prod(f(x**(2*n)),n=1..infinity) = exp(evenlambert(log(f(x))))}.")) (|oddlambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{oddlambert(st)} computes \\spad{f(x) + f(x**3) + f(x**5) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f}(\\spad{x}) is a power series with constant coefficient 1 then \\spad{prod(f(x**(2*n-1)),n=1..infinity) = exp(oddlambert(log(f(x))))}.")) (|lambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{lambert(st)} computes \\spad{f(x) + f(x**2) + f(x**3) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f(x)} is a power series with constant coefficient 1 then \\spad{prod(f(x**n),n = 1..infinity) = exp(lambert(log(f(x))))}.")) (|addiag| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{addiag(x)} performs diagonal addition of a stream of streams. if \\spad{x} = \\spad{[[a<0,0>,a<0,1>,..],[a<1,0>,a<1,1>,..],[a<2,0>,a<2,1>,..],..]} and \\spad{addiag(x) = [b<0,b<1>,...], then b<k> = sum(i+j=k,a<i,j>)}.")) (|revert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{revert(a)} computes the inverse of a power series \\spad{a} with respect to composition. the series should have constant coefficient 0 and first order coefficient should be invertible.")) (|lagrange| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{lagrange(g)} produces the power series for \\spad{f} where \\spad{f} is implicitly defined as \\spad{f(z) = z*g(f(z))}.")) (|compose| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{compose(a,b)} composes the power series \\spad{a} with the power series \\spad{b}.")) (|eval| (((|Stream| |#1|) (|Stream| |#1|) |#1|) "\\spad{eval(a,r)} returns a stream of partial sums of the power series \\spad{a} evaluated at the power series variable equal to \\spad{r}.")) (|coerce| (((|Stream| |#1|) |#1|) "\\spad{coerce(r)} converts a ring element \\spad{r} to a stream with one element.")) (|gderiv| (((|Stream| |#1|) (|Mapping| |#1| (|Integer|)) (|Stream| |#1|)) "\\spad{gderiv(f,[a0,a1,a2,..])} returns \\spad{[f(0)*a0,f(1)*a1,f(2)*a2,..]}.")) (|deriv| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{deriv(a)} returns the derivative of the power series with respect to the power series variable. Thus \\spad{deriv([a0,a1,a2,...])} returns \\spad{[a1,2 a2,3 a3,...]}.")) (|mapmult| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{mapmult([a0,a1,..],[b0,b1,..])} returns \\spad{[a0*b0,a1*b1,..]}.")) (|int| (((|Stream| |#1|) |#1|) "\\spad{int(r)} returns [\\spad{r},{}\\spad{r+1},{}\\spad{r+2},{}...],{} where \\spad{r} is a ring element.")) (|oddintegers| (((|Stream| (|Integer|)) (|Integer|)) "\\spad{oddintegers(n)} returns \\spad{[n,n+2,n+4,...]}.")) (|integers| (((|Stream| (|Integer|)) (|Integer|)) "\\spad{integers(n)} returns \\spad{[n,n+1,n+2,...]}.")) (|monom| (((|Stream| |#1|) |#1| (|Integer|)) "\\spad{monom(deg,coef)} is a monomial of degree \\spad{deg} with coefficient \\spad{coef}.")) (|recip| (((|Union| (|Stream| |#1|) "failed") (|Stream| |#1|)) "\\spad{recip(a)} returns the power series reciprocal of \\spad{a},{} or \"failed\" if not possible.")) (/ (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a / b} returns the power series quotient of \\spad{a} by \\spad{b}. An error message is returned if \\spad{b} is not invertible. This function is used in fixed point computations.")) (|exquo| (((|Union| (|Stream| |#1|) "failed") (|Stream| |#1|) (|Stream| |#1|)) "\\spad{exquo(a,b)} returns the power series quotient of \\spad{a} by \\spad{b},{} if the quotient exists,{} and \"failed\" otherwise")) (* (((|Stream| |#1|) (|Stream| |#1|) |#1|) "\\spad{a * r} returns the power series scalar multiplication of \\spad{a} by \\spad{r:} \\spad{[a0,a1,...] * r = [a0 * r,a1 * r,...]}") (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{r * a} returns the power series scalar multiplication of \\spad{r} by \\spad{a}: \\spad{r * [a0,a1,...] = [r * a0,r * a1,...]}") (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a * b} returns the power series (Cauchy) product of \\spad{a} and \\spad{b:} \\spad{[a0,a1,...] * [b0,b1,...] = [c0,c1,...]} where \\spad{ck = sum(i + j = k,ai * bk)}.")) (- (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{- a} returns the power series negative of \\spad{a}: \\spad{- [a0,a1,...] = [- a0,- a1,...]}") (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a - b} returns the power series difference of \\spad{a} and \\spad{b}: \\spad{[a0,a1,..] - [b0,b1,..] = [a0 - b0,a1 - b1,..]}")) (+ (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a + b} returns the power series sum of \\spad{a} and \\spad{b}: \\spad{[a0,a1,..] + [b0,b1,..] = [a0 + b0,a1 + b1,..]}"))) NIL @@ -4669,10 +4669,10 @@ NIL NIL NIL (-1185 |Coef| |var| |cen|) -((|constructor| (NIL "Sparse Laurent series in one variable \\indented{2}{\\spadtype{SparseUnivariateLaurentSeries} is a domain representing Laurent} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{SparseUnivariateLaurentSeries(Integer,x,3)} represents Laurent} \\indented{2}{series in \\spad{(x - 3)} with integer coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. 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We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}."))) +((|constructor| (NIL "Sparse Puiseux series in one variable \\indented{2}{\\spadtype{SparseUnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{SparseUnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux} \\indented{2}{series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. 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We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} computes the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) +((|constructor| (NIL "Sparse Taylor series in one variable \\indented{2}{\\spadtype{SparseUnivariateTaylorSeries} is a domain representing Taylor} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spadtype{SparseUnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor} \\indented{2}{series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|)))) (|HasCategory| (-782) (QUOTE (-1129))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasSignature| |#1| (LIST (QUOTE -2883) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasCategory| |#1| (QUOTE (-373))) (-3765 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4413) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|)))) (|HasCategory| (-782) (QUOTE (-1129))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasSignature| |#1| (LIST (QUOTE -2882) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasCategory| |#1| (QUOTE (-373))) (-3763 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4388) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) (-1193) ((|constructor| (NIL "This domain builds representations of boolean expressions for use with the \\axiomType{FortranCode} domain.")) (NOT (($ $) "\\spad{NOT(x)} returns the \\axiomType{Switch} expression representing \\spad{\\~~x}.") (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{NOT(x)} returns the \\axiomType{Switch} expression representing \\spad{\\~~x}.")) (AND (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{AND(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x and y}.")) (EQ (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{EQ(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x = y}.")) (OR (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{OR(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x or y}.")) (GE (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{GE(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x>=y}.")) (LE (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{LE(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x<=y}.")) (GT (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{GT(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x>y}.")) (LT (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{LT(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x<y}.")) (|coerce| (($ (|Symbol|)) "\\spad{coerce(s)} \\undocumented{}"))) NIL @@ -4715,7 +4715,7 @@ NIL (-1196 R) ((|constructor| (NIL "This domain implements symmetric polynomial"))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-6 -4458)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3765 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| (-988) (QUOTE (-132))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasAttribute| |#1| (QUOTE -4458))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-3763 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-463))) (-12 (|HasCategory| (-988) (QUOTE (-132))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasAttribute| |#1| (QUOTE -4458))) (-1197) ((|constructor| (NIL "Creates and manipulates one global symbol table for FORTRAN code generation,{} containing details of types,{} dimensions,{} and argument lists.")) (|symbolTableOf| (((|SymbolTable|) (|Symbol|) $) "\\spad{symbolTableOf(f,tab)} returns the symbol table of \\spad{f}")) (|argumentListOf| (((|List| (|Symbol|)) (|Symbol|) $) "\\spad{argumentListOf(f,tab)} returns the argument list of \\spad{f}")) (|returnTypeOf| (((|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| "void")) (|Symbol|) $) "\\spad{returnTypeOf(f,tab)} returns the type of the object returned by \\spad{f}")) (|empty| (($) "\\spad{empty()} creates a new,{} empty symbol table.")) (|printTypes| (((|Void|) (|Symbol|)) "\\spad{printTypes(tab)} produces FORTRAN type declarations from \\spad{tab},{} on the current FORTRAN output stream")) (|printHeader| (((|Void|)) "\\spad{printHeader()} produces the FORTRAN header for the current subprogram in the global symbol table on the current FORTRAN output stream.") (((|Void|) (|Symbol|)) "\\spad{printHeader(f)} produces the FORTRAN header for subprogram \\spad{f} in the global symbol table on the current FORTRAN output stream.") (((|Void|) (|Symbol|) $) "\\spad{printHeader(f,tab)} produces the FORTRAN header for subprogram \\spad{f} in symbol table \\spad{tab} on the current FORTRAN output stream.")) (|returnType!| (((|Void|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| "void"))) "\\spad{returnType!(t)} declares that the return type of he current subprogram in the global symbol table is \\spad{t}.") (((|Void|) (|Symbol|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| "void"))) "\\spad{returnType!(f,t)} declares that the return type of subprogram \\spad{f} in the global symbol table is \\spad{t}.") (((|Void|) (|Symbol|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| "void")) $) "\\spad{returnType!(f,t,tab)} declares that the return type of subprogram \\spad{f} in symbol table \\spad{tab} is \\spad{t}.")) (|argumentList!| (((|Void|) (|List| (|Symbol|))) "\\spad{argumentList!(l)} declares that the argument list for the current subprogram in the global symbol table is \\spad{l}.") (((|Void|) (|Symbol|) (|List| (|Symbol|))) "\\spad{argumentList!(f,l)} declares that the argument list for subprogram \\spad{f} in the global symbol table is \\spad{l}.") (((|Void|) (|Symbol|) (|List| (|Symbol|)) $) "\\spad{argumentList!(f,l,tab)} declares that the argument list for subprogram \\spad{f} in symbol table \\spad{tab} is \\spad{l}.")) (|endSubProgram| (((|Symbol|)) "\\spad{endSubProgram()} asserts that we are no longer processing the current subprogram.")) (|currentSubProgram| (((|Symbol|)) "\\spad{currentSubProgram()} returns the name of the current subprogram being processed")) (|newSubProgram| (((|Void|) (|Symbol|)) "\\spad{newSubProgram(f)} asserts that from now on type declarations are part of subprogram \\spad{f}.")) (|declare!| (((|FortranType|) (|Symbol|) (|FortranType|) (|Symbol|)) "\\spad{declare!(u,t,asp)} declares the parameter \\spad{u} to have type \\spad{t} in \\spad{asp}.") (((|FortranType|) (|Symbol|) (|FortranType|)) "\\spad{declare!(u,t)} declares the parameter \\spad{u} to have type \\spad{t} in the current level of the symbol table.") (((|FortranType|) (|List| (|Symbol|)) (|FortranType|) (|Symbol|) $) "\\spad{declare!(u,t,asp,tab)} declares the parameters \\spad{u} of subprogram \\spad{asp} to have type \\spad{t} in symbol table \\spad{tab}.") (((|FortranType|) (|Symbol|) (|FortranType|) (|Symbol|) $) "\\spad{declare!(u,t,asp,tab)} declares the parameter \\spad{u} of subprogram \\spad{asp} to have type \\spad{t} in symbol table \\spad{tab}.")) (|clearTheSymbolTable| (((|Void|) (|Symbol|)) "\\spad{clearTheSymbolTable(x)} removes the symbol \\spad{x} from the table") (((|Void|)) "\\spad{clearTheSymbolTable()} clears the current symbol table.")) (|showTheSymbolTable| (($) "\\spad{showTheSymbolTable()} returns the current symbol table."))) NIL @@ -4759,7 +4759,7 @@ NIL (-1207 |Key| |Entry|) ((|constructor| (NIL "This is the general purpose table type. The keys are hashed to look up the entries. This creates a \\spadtype{HashTable} if equal for the Key domain is consistent with Lisp EQUAL otherwise an \\spadtype{AssociationList}"))) ((-4460 . T) (-4461 . T)) -((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3765 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -318) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4169) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -3179) (|devaluate| |#2|)))))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#2| (QUOTE (-1117)))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -625) (QUOTE (-547)))) (-12 (|HasCategory| |#2| (QUOTE (-1117))) (|HasCategory| |#2| (LIST (QUOTE -318) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#2| (QUOTE (-1117))) (-3763 (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#2| (LIST (QUOTE -624) (QUOTE (-873)))) (|HasCategory| (-2 (|:| -4169 |#1|) (|:| -3179 |#2|)) (LIST (QUOTE -624) (QUOTE (-873))))) (-1208 S) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: April 17,{} 2010 Date Last Modified: April 17,{} 2010")) (|operator| (($ |#1| (|Arity|)) "\\spad{operator(n,a)} returns an operator named \\spad{n} and with arity \\spad{a}."))) NIL @@ -4815,7 +4815,7 @@ NIL (-1221 S) ((|constructor| (NIL "\\spadtype{Tree(S)} is a basic domains of tree structures. Each tree is either empty or else is a {\\it node} consisting of a value and a list of (sub)trees.")) (|cyclicParents| (((|List| $) $) "\\spad{cyclicParents(t)} returns a list of cycles that are parents of \\spad{t}.")) (|cyclicEqual?| (((|Boolean|) $ $) "\\spad{cyclicEqual?(t1, t2)} tests of two cyclic trees have the same structure.")) (|cyclicEntries| (((|List| $) $) "\\spad{cyclicEntries(t)} returns a list of top-level cycles in tree \\spad{t}.")) (|cyclicCopy| (($ $) "\\spad{cyclicCopy(l)} makes a copy of a (possibly) cyclic tree \\spad{l}.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(t)} tests if \\spad{t} is a cyclic tree.")) (|tree| (($ |#1|) "\\spad{tree(nd)} creates a tree with value \\spad{nd},{} and no children") (($ (|List| |#1|)) "\\spad{tree(ls)} creates a tree from a list of elements of \\spad{s}.") (($ |#1| (|List| $)) "\\spad{tree(nd,ls)} creates a tree with value \\spad{nd},{} and children \\spad{ls}."))) ((-4461 . T) (-4460 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) +((-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1117))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (-1222 S) ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) NIL @@ -4824,7 +4824,7 @@ NIL ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) NIL NIL -(-1224 R -3029) +(-1224 R -3027) ((|constructor| (NIL "\\spadtype{TrigonometricManipulations} provides transformations from trigonometric functions to complex exponentials and logarithms,{} and back.")) (|complexForm| (((|Complex| |#2|) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| ((|#2| |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| ((|#2| |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) NIL NIL @@ -4832,7 +4832,7 @@ NIL ((|constructor| (NIL "This package provides functions that compute \"fraction-free\" inverses of upper and lower triangular matrices over a integral domain. By \"fraction-free inverses\" we mean the following: given a matrix \\spad{B} with entries in \\spad{R} and an element \\spad{d} of \\spad{R} such that \\spad{d} * inv(\\spad{B}) also has entries in \\spad{R},{} we return \\spad{d} * inv(\\spad{B}). Thus,{} it is not necessary to pass to the quotient field in any of our computations.")) (|LowTriBddDenomInv| ((|#4| |#4| |#1|) "\\spad{LowTriBddDenomInv(B,d)} returns \\spad{M},{} where \\spad{B} is a non-singular lower triangular matrix and \\spad{d} is an element of \\spad{R} such that \\spad{M = d * inv(B)} has entries in \\spad{R}.")) (|UpTriBddDenomInv| ((|#4| |#4| |#1|) "\\spad{UpTriBddDenomInv(B,d)} returns \\spad{M},{} where \\spad{B} is a non-singular upper triangular matrix and \\spad{d} is an element of \\spad{R} such that \\spad{M = d * inv(B)} has entries in \\spad{R}."))) NIL NIL -(-1226 R -3029) +(-1226 R -3027) ((|constructor| (NIL "TranscendentalManipulations provides functions to simplify and expand expressions involving transcendental operators.")) (|expandTrigProducts| ((|#2| |#2|) "\\spad{expandTrigProducts(e)} replaces \\axiom{sin(\\spad{x})*sin(\\spad{y})} by \\spad{(cos(x-y)-cos(x+y))/2},{} \\axiom{cos(\\spad{x})*cos(\\spad{y})} by \\spad{(cos(x-y)+cos(x+y))/2},{} and \\axiom{sin(\\spad{x})*cos(\\spad{y})} by \\spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses the pattern matcher and so is relatively expensive. To avoid getting into an infinite loop the transformations are applied at most ten times.")) (|removeSinhSq| ((|#2| |#2|) "\\spad{removeSinhSq(f)} converts every \\spad{sinh(u)**2} appearing in \\spad{f} into \\spad{1 - cosh(x)**2},{} and also reduces higher powers of \\spad{sinh(u)} with that formula.")) (|removeCoshSq| ((|#2| |#2|) "\\spad{removeCoshSq(f)} converts every \\spad{cosh(u)**2} appearing in \\spad{f} into \\spad{1 - sinh(x)**2},{} and also reduces higher powers of \\spad{cosh(u)} with that formula.")) (|removeSinSq| ((|#2| |#2|) "\\spad{removeSinSq(f)} converts every \\spad{sin(u)**2} appearing in \\spad{f} into \\spad{1 - cos(x)**2},{} and also reduces higher powers of \\spad{sin(u)} with that formula.")) (|removeCosSq| ((|#2| |#2|) "\\spad{removeCosSq(f)} converts every \\spad{cos(u)**2} appearing in \\spad{f} into \\spad{1 - sin(x)**2},{} and also reduces higher powers of \\spad{cos(u)} with that formula.")) (|coth2tanh| ((|#2| |#2|) "\\spad{coth2tanh(f)} converts every \\spad{coth(u)} appearing in \\spad{f} into \\spad{1/tanh(u)}.")) (|cot2tan| ((|#2| |#2|) "\\spad{cot2tan(f)} converts every \\spad{cot(u)} appearing in \\spad{f} into \\spad{1/tan(u)}.")) (|tanh2coth| ((|#2| |#2|) "\\spad{tanh2coth(f)} converts every \\spad{tanh(u)} appearing in \\spad{f} into \\spad{1/coth(u)}.")) (|tan2cot| ((|#2| |#2|) "\\spad{tan2cot(f)} converts every \\spad{tan(u)} appearing in \\spad{f} into \\spad{1/cot(u)}.")) (|tanh2trigh| ((|#2| |#2|) "\\spad{tanh2trigh(f)} converts every \\spad{tanh(u)} appearing in \\spad{f} into \\spad{sinh(u)/cosh(u)}.")) (|tan2trig| ((|#2| |#2|) "\\spad{tan2trig(f)} converts every \\spad{tan(u)} appearing in \\spad{f} into \\spad{sin(u)/cos(u)}.")) (|sinh2csch| ((|#2| |#2|) "\\spad{sinh2csch(f)} converts every \\spad{sinh(u)} appearing in \\spad{f} into \\spad{1/csch(u)}.")) (|sin2csc| ((|#2| |#2|) "\\spad{sin2csc(f)} converts every \\spad{sin(u)} appearing in \\spad{f} into \\spad{1/csc(u)}.")) (|sech2cosh| ((|#2| |#2|) "\\spad{sech2cosh(f)} converts every \\spad{sech(u)} appearing in \\spad{f} into \\spad{1/cosh(u)}.")) (|sec2cos| ((|#2| |#2|) "\\spad{sec2cos(f)} converts every \\spad{sec(u)} appearing in \\spad{f} into \\spad{1/cos(u)}.")) (|csch2sinh| ((|#2| |#2|) "\\spad{csch2sinh(f)} converts every \\spad{csch(u)} appearing in \\spad{f} into \\spad{1/sinh(u)}.")) (|csc2sin| ((|#2| |#2|) "\\spad{csc2sin(f)} converts every \\spad{csc(u)} appearing in \\spad{f} into \\spad{1/sin(u)}.")) (|coth2trigh| ((|#2| |#2|) "\\spad{coth2trigh(f)} converts every \\spad{coth(u)} appearing in \\spad{f} into \\spad{cosh(u)/sinh(u)}.")) (|cot2trig| ((|#2| |#2|) "\\spad{cot2trig(f)} converts every \\spad{cot(u)} appearing in \\spad{f} into \\spad{cos(u)/sin(u)}.")) (|cosh2sech| ((|#2| |#2|) "\\spad{cosh2sech(f)} converts every \\spad{cosh(u)} appearing in \\spad{f} into \\spad{1/sech(u)}.")) (|cos2sec| ((|#2| |#2|) "\\spad{cos2sec(f)} converts every \\spad{cos(u)} appearing in \\spad{f} into \\spad{1/sec(u)}.")) (|expandLog| ((|#2| |#2|) "\\spad{expandLog(f)} converts every \\spad{log(a/b)} appearing in \\spad{f} into \\spad{log(a) - log(b)},{} and every \\spad{log(a*b)} into \\spad{log(a) + log(b)}..")) (|expandPower| ((|#2| |#2|) "\\spad{expandPower(f)} converts every power \\spad{(a/b)**c} appearing in \\spad{f} into \\spad{a**c * b**(-c)}.")) (|simplifyLog| ((|#2| |#2|) "\\spad{simplifyLog(f)} converts every \\spad{log(a) - log(b)} appearing in \\spad{f} into \\spad{log(a/b)},{} every \\spad{log(a) + log(b)} into \\spad{log(a*b)} and every \\spad{n*log(a)} into \\spad{log(a^n)}.")) (|simplifyExp| ((|#2| |#2|) "\\spad{simplifyExp(f)} converts every product \\spad{exp(a)*exp(b)} appearing in \\spad{f} into \\spad{exp(a+b)}.")) (|htrigs| ((|#2| |#2|) "\\spad{htrigs(f)} converts all the exponentials in \\spad{f} into hyperbolic sines and cosines.")) (|simplify| ((|#2| |#2|) "\\spad{simplify(f)} performs the following simplifications on \\spad{f:}\\begin{items} \\item 1. rewrites trigs and hyperbolic trigs in terms of \\spad{sin} ,{}\\spad{cos},{} \\spad{sinh},{} \\spad{cosh}. \\item 2. rewrites \\spad{sin**2} and \\spad{sinh**2} in terms of \\spad{cos} and \\spad{cosh},{} \\item 3. rewrites \\spad{exp(a)*exp(b)} as \\spad{exp(a+b)}. \\item 4. rewrites \\spad{(a**(1/n))**m * (a**(1/s))**t} as a single power of a single radical of \\spad{a}. \\end{items}")) (|expand| ((|#2| |#2|) "\\spad{expand(f)} performs the following expansions on \\spad{f:}\\begin{items} \\item 1. logs of products are expanded into sums of logs,{} \\item 2. trigonometric and hyperbolic trigonometric functions of sums are expanded into sums of products of trigonometric and hyperbolic trigonometric functions. \\item 3. formal powers of the form \\spad{(a/b)**c} are expanded into \\spad{a**c * b**(-c)}. \\end{items}"))) NIL ((-12 (|HasCategory| |#1| (LIST (QUOTE -625) (LIST (QUOTE -904) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -898) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -898) (|devaluate| |#1|))))) @@ -4847,7 +4847,7 @@ NIL (-1229 |Coef|) ((|constructor| (NIL "\\spadtype{TaylorSeries} is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.")) (|fintegrate| (($ (|Mapping| $) (|Symbol|) |#1|) "\\spad{fintegrate(f,v,c)} is the integral of \\spad{f()} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.} \\indented{1}{The evaluation of \\spad{f()} is delayed.}")) (|integrate| (($ $ (|Symbol|) |#1|) "\\spad{integrate(s,v,c)} is the integral of \\spad{s} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.}")) (|coerce| (($ (|Polynomial| |#1|)) "\\spad{coerce(s)} regroups terms of \\spad{s} by total degree \\indented{1}{and forms a series.}") (($ (|Symbol|)) "\\spad{coerce(s)} converts a variable to a Taylor series")) (|coefficient| (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{coefficient(s, n)} gives the terms of total degree \\spad{n}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-373)))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-373)))) (-1230 |Curve|) ((|constructor| (NIL "\\indented{2}{Package for constructing tubes around 3-dimensional parametric curves.} Domain of tubes around 3-dimensional parametric curves.")) (|tube| (($ |#1| (|List| (|List| (|Point| (|DoubleFloat|)))) (|Boolean|)) "\\spad{tube(c,ll,b)} creates a tube of the domain \\spadtype{TubePlot} from a space curve \\spad{c} of the category \\spadtype{PlottableSpaceCurveCategory},{} a list of lists of points (loops) \\spad{ll} and a boolean \\spad{b} which if \\spad{true} indicates a closed tube,{} or if \\spad{false} an open tube.")) (|setClosed| (((|Boolean|) $ (|Boolean|)) "\\spad{setClosed(t,b)} declares the given tube plot \\spad{t} to be closed if \\spad{b} is \\spad{true},{} or if \\spad{b} is \\spad{false},{} \\spad{t} is set to be open.")) (|open?| (((|Boolean|) $) "\\spad{open?(t)} tests whether the given tube plot \\spad{t} is open.")) (|closed?| (((|Boolean|) $) "\\spad{closed?(t)} tests whether the given tube plot \\spad{t} is closed.")) (|listLoops| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listLoops(t)} returns the list of lists of points,{} or the 'loops',{} of the given tube plot \\spad{t}.")) (|getCurve| ((|#1| $) "\\spad{getCurve(t)} returns the \\spadtype{PlottableSpaceCurveCategory} representing the parametric curve of the given tube plot \\spad{t}."))) NIL @@ -4860,7 +4860,7 @@ NIL ((|constructor| (NIL "\\indented{1}{This domain is used to interface with the interpreter\\spad{'s} notion} of comma-delimited sequences of values.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(x)} returns the number of elements in tuple \\spad{x}")) (|select| ((|#1| $ (|NonNegativeInteger|)) "\\spad{select(x,n)} returns the \\spad{n}-th element of tuple \\spad{x}. tuples are 0-based"))) NIL ((|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) -(-1233 -3029) +(-1233 -3027) ((|constructor| (NIL "A basic package for the factorization of bivariate polynomials over a finite field. The functions here represent the base step for the multivariate factorizer.")) (|twoFactor| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|)) (|Integer|)) "\\spad{twoFactor(p,n)} returns the factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}. Also,{} \\spad{p} is assumed primitive and square-free and \\spad{n} is the degree of the inner variable of \\spad{p} (maximum of the degrees of the coefficients of \\spad{p}).")) (|generalSqFr| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalSqFr(p)} returns the square-free factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}.")) (|generalTwoFactor| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalTwoFactor(p)} returns the factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}."))) NIL NIL @@ -4923,11 +4923,11 @@ NIL (-1248 |Coef| UTS) ((|constructor| (NIL "This package enables one to construct a univariate Laurent series domain from a univariate Taylor series domain. Univariate Laurent series are represented by a pair \\spad{[n,f(x)]},{} where \\spad{n} is an arbitrary integer and \\spad{f(x)} is a Taylor series. This pair represents the Laurent series \\spad{x**n * f(x)}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4454 . T) (-4455 . T) (-4457 . 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|#3|) (QUOTE (-924))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-174)))) (-12 (|HasCategory| (-1277 |#1| |#2| |#3|) (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| (-1277 |#1| |#2| |#3|) (QUOTE (-237))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| (-1277 |#1| |#2| |#3|) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-1277 |#1| |#2| |#3|) (QUOTE (-924))) (|HasCategory| |#1| (QUOTE (-373)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-1277 |#1| |#2| |#3|) (QUOTE (-924))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| (-1277 |#1| |#2| |#3|) (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-146))))) (-1250 ZP) ((|constructor| (NIL "Package for the factorization of univariate polynomials with integer coefficients. The factorization is done by \"lifting\" (HENSEL) the factorization over a finite field.")) (|henselFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|)) "\\spad{henselFact(m,flag)} returns the factorization of \\spad{m},{} FinalFact is a Record \\spad{s}.\\spad{t}. FinalFact.contp=content \\spad{m},{} FinalFact.factors=List of irreducible factors of \\spad{m} with exponent ,{} if \\spad{flag} =true the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(m)} returns the factorization of \\spad{m} square free polynomial")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(m)} returns the factorization of \\spad{m}"))) NIL @@ -4963,7 +4963,7 @@ NIL (-1258 |x| R) ((|constructor| (NIL "This domain represents univariate polynomials in some symbol over arbitrary (not necessarily commutative) coefficient rings. The representation is sparse in the sense that only non-zero terms are represented.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#2| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}"))) (((-4462 "*") |has| |#2| (-174)) (-4453 |has| |#2| (-567)) (-4456 |has| |#2| (-373)) (-4458 |has| |#2| (-6 -4458)) (-4455 . T) (-4454 . T) (-4457 . T)) -((|HasCategory| |#2| (QUOTE (-924))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3765 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3765 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-1169))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-237))) (|HasCategory| |#2| (QUOTE (-238))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3765 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) +((|HasCategory| |#2| (QUOTE (-924))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-174))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-567)))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-389)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-389))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -898) (QUOTE (-575)))) (|HasCategory| |#2| (LIST (QUOTE -898) (QUOTE (-575))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-389)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -625) (LIST (QUOTE -904) (QUOTE (-575)))))) (-12 (|HasCategory| (-1099) (LIST (QUOTE -625) (QUOTE (-547)))) (|HasCategory| |#2| (LIST (QUOTE -625) (QUOTE (-547))))) (|HasCategory| |#2| (LIST (QUOTE -650) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (QUOTE (-575)))) (-3763 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| |#2| (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (-3763 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-567))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-463))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-1169))) (|HasCategory| |#2| (LIST (QUOTE -915) (QUOTE (-1194)))) (|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasCategory| |#2| (QUOTE (-237))) (|HasCategory| |#2| (QUOTE (-238))) (|HasAttribute| |#2| (QUOTE -4458)) (|HasCategory| |#2| (QUOTE (-463))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (-3763 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-924)))) (|HasCategory| |#2| (QUOTE (-146))))) (-1259 R PR S PS) ((|constructor| (NIL "Mapping from polynomials over \\spad{R} to polynomials over \\spad{S} given a map from \\spad{R} to \\spad{S} assumed to send zero to zero.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f, p)} takes a function \\spad{f} from \\spad{R} to \\spad{S},{} and applies it to each (non-zero) coefficient of a polynomial \\spad{p} over \\spad{R},{} getting a new polynomial over \\spad{S}. Note: since the map is not applied to zero elements,{} it may map zero to zero."))) NIL @@ -4979,7 +4979,7 @@ NIL (-1262 S |Coef| |Expon|) ((|constructor| (NIL "\\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.")) (|eval| (((|Stream| |#2|) $ |#2|) "\\spad{eval(f,a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.")) (|extend| (($ $ |#3|) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree \\spad{<=} \\spad{n} to be computed.")) (|approximate| ((|#2| $ |#3|) "\\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.")) (|truncate| (($ $ |#3| |#3|) "\\spad{truncate(f,k1,k2)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (($ $ |#3|) "\\spad{truncate(f,k)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| ((|#3| $ |#3|) "\\spad{order(f,n) = min(m,n)},{} where \\spad{m} is the degree of the lowest order non-zero term in \\spad{f}.") ((|#3| $) "\\spad{order(f)} is the degree of the lowest order non-zero term in \\spad{f}. This will result in an infinite loop if \\spad{f} has no non-zero terms.")) (|multiplyExponents| (($ $ (|PositiveInteger|)) "\\spad{multiplyExponents(f,n)} multiplies all exponents of the power series \\spad{f} by the positive integer \\spad{n}.")) (|center| ((|#2| $) "\\spad{center(f)} returns the point about which the series \\spad{f} is expanded.")) (|variable| (((|Symbol|) $) "\\spad{variable(f)} returns the (unique) power series variable of the power series \\spad{f}.")) (|terms| (((|Stream| (|Record| (|:| |k| |#3|) (|:| |c| |#2|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) NIL -((|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#2| (LIST (QUOTE *) (LIST (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1129))) (|HasSignature| |#2| (LIST (QUOTE **) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (LIST (QUOTE -2883) (LIST (|devaluate| |#2|) (QUOTE (-1194)))))) +((|HasCategory| |#2| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#2| (LIST (QUOTE *) (LIST (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1129))) (|HasSignature| |#2| (LIST (QUOTE **) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (LIST (QUOTE -2882) (LIST (|devaluate| |#2|) (QUOTE (-1194)))))) (-1263 |Coef| |Expon|) ((|constructor| (NIL "\\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.")) (|eval| (((|Stream| |#1|) $ |#1|) "\\spad{eval(f,a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.")) (|extend| (($ $ |#2|) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree \\spad{<=} \\spad{n} to be computed.")) (|approximate| ((|#1| $ |#2|) "\\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.")) (|truncate| (($ $ |#2| |#2|) "\\spad{truncate(f,k1,k2)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (($ $ |#2|) "\\spad{truncate(f,k)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| ((|#2| $ |#2|) "\\spad{order(f,n) = min(m,n)},{} where \\spad{m} is the degree of the lowest order non-zero term in \\spad{f}.") ((|#2| $) "\\spad{order(f)} is the degree of the lowest order non-zero term in \\spad{f}. This will result in an infinite loop if \\spad{f} has no non-zero terms.")) (|multiplyExponents| (($ $ (|PositiveInteger|)) "\\spad{multiplyExponents(f,n)} multiplies all exponents of the power series \\spad{f} by the positive integer \\spad{n}.")) (|center| ((|#1| $) "\\spad{center(f)} returns the point about which the series \\spad{f} is expanded.")) (|variable| (((|Symbol|) $) "\\spad{variable(f)} returns the (unique) power series variable of the power series \\spad{f}.")) (|terms| (((|Stream| (|Record| (|:| |k| |#2|) (|:| |c| |#1|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) @@ -5007,15 +5007,15 @@ NIL (-1269 |Coef| ULS) ((|constructor| (NIL "This package enables one to construct a univariate Puiseux series domain from a univariate Laurent series domain. Univariate Puiseux series are represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|)))) (|HasCategory| (-418 (-575)) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-3765 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasSignature| |#1| (LIST (QUOTE -2883) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3765 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4413) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) +((|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|)))) (|HasCategory| (-418 (-575)) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasSignature| |#1| (LIST (QUOTE -2882) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3763 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4388) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-1270 |Coef| |var| |cen|) -((|constructor| (NIL "Dense Puiseux series in one variable \\indented{2}{\\spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{UnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux series in} \\indented{2}{\\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}."))) +((|constructor| (NIL "Dense Puiseux series in one variable \\indented{2}{\\spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{UnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux series in} \\indented{2}{\\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4458 |has| |#1| (-373)) (-4452 |has| |#1| (-373)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|)))) (|HasCategory| (-418 (-575)) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-3765 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasSignature| |#1| (LIST (QUOTE -2883) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3765 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4413) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (|HasCategory| |#1| (QUOTE (-174))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575))) (|devaluate| |#1|)))) (|HasCategory| (-418 (-575)) (QUOTE (-1129))) (|HasCategory| |#1| (QUOTE (-373))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-3763 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-567)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasSignature| |#1| (LIST (QUOTE -2882) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -418) (QUOTE (-575)))))) (-3763 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4388) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) (-1271 R FE |var| |cen|) ((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent functions with essential singularities. Objects in this domain are sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series. Thus,{} the elements of this domain are sums of expressions of the form \\spad{g(x) * exp(f(x))},{} where \\spad{g}(\\spad{x}) is a univariate Puiseux series and \\spad{f}(\\spad{x}) is a univariate Puiseux series with no terms of non-negative degree.")) (|dominantTerm| (((|Union| (|Record| (|:| |%term| (|Record| (|:| |%coef| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expon| (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expTerms| (|List| (|Record| (|:| |k| (|Fraction| (|Integer|))) (|:| |c| |#2|)))))) (|:| |%type| (|String|))) "failed") $) "\\spad{dominantTerm(f(var))} returns the term that dominates the limiting behavior of \\spad{f(var)} as \\spad{var -> cen+} together with a \\spadtype{String} which briefly describes that behavior. The value of the \\spadtype{String} will be \\spad{\"zero\"} (resp. \\spad{\"infinity\"}) if the term tends to zero (resp. infinity) exponentially and will \\spad{\"series\"} if the term is a Puiseux series.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> cen+,f(var))}."))) (((-4462 "*") |has| (-1270 |#2| |#3| |#4|) (-174)) (-4453 |has| (-1270 |#2| |#3| |#4|) (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-146))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-148))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-174))) (-3765 (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-373))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-463))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-567)))) +((|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-146))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-148))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-174))) (-3763 (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575)))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -1055) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| (-1270 |#2| |#3| |#4|) (LIST (QUOTE -1055) (QUOTE (-575)))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-373))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-463))) (|HasCategory| (-1270 |#2| |#3| |#4|) (QUOTE (-567)))) (-1272 A S) ((|constructor| (NIL "A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \\spadfun{value} of the list designating the head,{} or \\spadfun{first},{} of the list,{} and the child designating the tail,{} or \\spadfun{rest},{} of the list. A node with no child then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.")) (|split!| (($ $ (|Integer|)) "\\spad{split!(u,n)} splits \\spad{u} into two aggregates: \\axiom{\\spad{v} = rest(\\spad{u},{}\\spad{n})} and \\axiom{\\spad{w} = first(\\spad{u},{}\\spad{n})},{} returning \\axiom{\\spad{v}}. Note: afterwards \\axiom{rest(\\spad{u},{}\\spad{n})} returns \\axiom{empty()}.")) (|setlast!| ((|#2| $ |#2|) "\\spad{setlast!(u,x)} destructively changes the last element of \\spad{u} to \\spad{x}.")) (|setrest!| (($ $ $) "\\spad{setrest!(u,v)} destructively changes the rest of \\spad{u} to \\spad{v}.")) (|setelt| ((|#2| $ "last" |#2|) "\\spad{setelt(u,\"last\",x)} (also written: \\axiom{\\spad{u}.last \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setlast!(\\spad{u},{}\\spad{v})}.") (($ $ "rest" $) "\\spad{setelt(u,\"rest\",v)} (also written: \\axiom{\\spad{u}.rest \\spad{:=} \\spad{v}}) is equivalent to \\axiom{setrest!(\\spad{u},{}\\spad{v})}.") ((|#2| $ "first" |#2|) "\\spad{setelt(u,\"first\",x)} (also written: \\axiom{\\spad{u}.first \\spad{:=} \\spad{x}}) is equivalent to \\axiom{setfirst!(\\spad{u},{}\\spad{x})}.")) (|setfirst!| ((|#2| $ |#2|) "\\spad{setfirst!(u,x)} destructively changes the first element of a to \\spad{x}.")) (|cycleSplit!| (($ $) "\\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} is the cyclic list where \\spad{v} is the head of the cycle,{} \\axiom{cycleSplit!(\\spad{w})} will drop \\spad{v} off \\spad{w} thus destructively changing \\spad{w} to \\spad{u},{} and returning \\spad{v}.")) (|concat!| (($ $ |#2|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}. Note: \\axiom{concat!(a,{}\\spad{x}) = setlast!(a,{}[\\spad{x}])}.") (($ $ $) "\\spad{concat!(u,v)} destructively concatenates \\spad{v} to the end of \\spad{u}. Note: \\axiom{concat!(\\spad{u},{}\\spad{v}) = setlast!(\\spad{u},{}\\spad{v})}.")) (|cycleTail| (($ $) "\\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.")) (|cycleLength| (((|NonNegativeInteger|) $) "\\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \\spad{u},{} or 0 is \\spad{u} has no such cycle.")) (|cycleEntry| (($ $) "\\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \\spad{u},{} or \\axiom{empty()} if none exists.")) (|third| ((|#2| $) "\\spad{third(u)} returns the third element of \\spad{u}. Note: \\axiom{third(\\spad{u}) = first(rest(rest(\\spad{u})))}.")) (|second| ((|#2| $) "\\spad{second(u)} returns the second element of \\spad{u}. Note: \\axiom{second(\\spad{u}) = first(rest(\\spad{u}))}.")) (|tail| (($ $) "\\spad{tail(u)} returns the last node of \\spad{u}. Note: if \\spad{u} is \\axiom{shallowlyMutable},{} \\axiom{setrest(tail(\\spad{u}),{}\\spad{v}) = concat(\\spad{u},{}\\spad{v})}.")) (|last| (($ $ (|NonNegativeInteger|)) "\\spad{last(u,n)} returns a copy of the last \\spad{n} (\\axiom{\\spad{n} \\spad{>=} 0}) nodes of \\spad{u}. Note: \\axiom{last(\\spad{u},{}\\spad{n})} is a list of \\spad{n} elements.") ((|#2| $) "\\spad{last(u)} resturn the last element of \\spad{u}. Note: for lists,{} \\axiom{last(\\spad{u}) = \\spad{u} . (maxIndex \\spad{u}) = \\spad{u} . (\\# \\spad{u} - 1)}.")) (|rest| (($ $ (|NonNegativeInteger|)) "\\spad{rest(u,n)} returns the \\axiom{\\spad{n}}th (\\spad{n} \\spad{>=} 0) node of \\spad{u}. Note: \\axiom{rest(\\spad{u},{}0) = \\spad{u}}.") (($ $) "\\spad{rest(u)} returns an aggregate consisting of all but the first element of \\spad{u} (equivalently,{} the next node of \\spad{u}).")) (|elt| ((|#2| $ "last") "\\spad{elt(u,\"last\")} (also written: \\axiom{\\spad{u} . last}) is equivalent to last \\spad{u}.") (($ $ "rest") "\\spad{elt(\\%,\"rest\")} (also written: \\axiom{\\spad{u}.rest}) is equivalent to \\axiom{rest \\spad{u}}.") ((|#2| $ "first") "\\spad{elt(u,\"first\")} (also written: \\axiom{\\spad{u} . first}) is equivalent to first \\spad{u}.")) (|first| (($ $ (|NonNegativeInteger|)) "\\spad{first(u,n)} returns a copy of the first \\spad{n} (\\axiom{\\spad{n} \\spad{>=} 0}) elements of \\spad{u}.") ((|#2| $) "\\spad{first(u)} returns the first element of \\spad{u} (equivalently,{} the value at the current node).")) (|concat| (($ |#2| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) NIL @@ -5031,20 +5031,20 @@ NIL (-1275 S |Coef|) ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (** (($ $ |#2|) "\\spad{f(x) ** a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.")) (|polynomial| (((|Polynomial| |#2|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#2|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|multiplyCoefficients| (($ (|Mapping| |#2| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = 0..infinity,a[n] * x**n))} returns \\spad{sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Laurent series are represented by a Taylor series and an order.")) (|quoByVar| (($ $) "\\spad{quoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.")) (|coefficients| (((|Stream| |#2|) $) "\\spad{coefficients(a0 + a1 x + a2 x**2 + ...)} returns a stream of coefficients: \\spad{[a0,a1,a2,...]}. The entries of the stream may be zero.")) (|series| (($ (|Stream| |#2|)) "\\spad{series([a0,a1,a2,...])} is the Taylor series \\spad{a0 + a1 x + a2 x**2 + ...}.") (($ (|Stream| (|Record| (|:| |k| (|NonNegativeInteger|)) (|:| |c| |#2|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) NIL -((|HasCategory| |#2| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-974))) (|HasCategory| |#2| (QUOTE (-1220))) (|HasSignature| |#2| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#2|)))) (|HasSignature| |#2| (LIST (QUOTE -4413) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1194))))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373)))) +((|HasCategory| |#2| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#2| (QUOTE (-974))) (|HasCategory| |#2| (QUOTE (-1220))) (|HasSignature| |#2| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#2|)))) (|HasSignature| |#2| (LIST (QUOTE -4388) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1194))))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#2| (QUOTE (-373)))) (-1276 |Coef|) ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (** (($ $ |#1|) "\\spad{f(x) ** a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.")) (|polynomial| (((|Polynomial| |#1|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = 0..infinity,a[n] * x**n))} returns \\spad{sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Laurent series are represented by a Taylor series and an order.")) (|quoByVar| (($ $) "\\spad{quoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.")) (|coefficients| (((|Stream| |#1|) $) "\\spad{coefficients(a0 + a1 x + a2 x**2 + ...)} returns a stream of coefficients: \\spad{[a0,a1,a2,...]}. The entries of the stream may be zero.")) (|series| (($ (|Stream| |#1|)) "\\spad{series([a0,a1,a2,...])} is the Taylor series \\spad{a0 + a1 x + a2 x**2 + ...}.") (($ (|Stream| (|Record| (|:| |k| (|NonNegativeInteger|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) NIL (-1277 |Coef| |var| |cen|) -((|constructor| (NIL "Dense Taylor series in one variable \\spadtype{UnivariateTaylorSeries} is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring,{} the power series variable,{} and the center of the power series expansion. For example,{} \\spadtype{UnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|invmultisect| (($ (|Integer|) (|Integer|) $) "\\spad{invmultisect(a,b,f(x))} substitutes \\spad{x^((a+b)*n)} \\indented{1}{for \\spad{x^n} and multiples by \\spad{x^b}.}")) (|multisect| (($ (|Integer|) (|Integer|) $) "\\spad{multisect(a,b,f(x))} selects the coefficients of \\indented{1}{\\spad{x^((a+b)*n+a)},{} and changes this monomial to \\spad{x^n}.}")) (|revert| (($ $) "\\spad{revert(f(x))} returns a Taylor series \\spad{g(x)} such that \\spad{f(g(x)) = g(f(x)) = x}. Series \\spad{f(x)} should have constant coefficient 0 and invertible 1st order coefficient.")) (|generalLambert| (($ $ (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x^a) + f(x^(a + d)) + \\indented{1}{f(x^(a + 2 d)) + ... }. \\spad{f(x)} should have zero constant} \\indented{1}{coefficient and \\spad{a} and \\spad{d} should be positive.}")) (|evenlambert| (($ $) "\\spad{evenlambert(f(x))} returns \\spad{f(x^2) + f(x^4) + f(x^6) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n))) = exp(log(evenlambert(f(x))))}.}")) (|oddlambert| (($ $) "\\spad{oddlambert(f(x))} returns \\spad{f(x) + f(x^3) + f(x^5) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.}")) (|lambert| (($ $) "\\spad{lambert(f(x))} returns \\spad{f(x) + f(x^2) + f(x^3) + ...}. \\indented{1}{This function is used for computing infinite products.} \\indented{1}{\\spad{f(x)} should have zero constant coefficient.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n = 1..infinity,f(x^n)) = exp(log(lambert(f(x))))}.}")) (|lagrange| (($ $) "\\spad{lagrange(g(x))} produces the Taylor series for \\spad{f(x)} \\indented{1}{where \\spad{f(x)} is implicitly defined as \\spad{f(x) = x*g(f(x))}.}")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} computes the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) +((|constructor| (NIL "Dense Taylor series in one variable \\spadtype{UnivariateTaylorSeries} is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring,{} the power series variable,{} and the center of the power series expansion. For example,{} \\spadtype{UnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|invmultisect| (($ (|Integer|) (|Integer|) $) "\\spad{invmultisect(a,b,f(x))} substitutes \\spad{x^((a+b)*n)} \\indented{1}{for \\spad{x^n} and multiples by \\spad{x^b}.}")) (|multisect| (($ (|Integer|) (|Integer|) $) "\\spad{multisect(a,b,f(x))} selects the coefficients of \\indented{1}{\\spad{x^((a+b)*n+a)},{} and changes this monomial to \\spad{x^n}.}")) (|revert| (($ $) "\\spad{revert(f(x))} returns a Taylor series \\spad{g(x)} such that \\spad{f(g(x)) = g(f(x)) = x}. Series \\spad{f(x)} should have constant coefficient 0 and invertible 1st order coefficient.")) (|generalLambert| (($ $ (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x^a) + f(x^(a + d)) + \\indented{1}{f(x^(a + 2 d)) + ... }. \\spad{f(x)} should have zero constant} \\indented{1}{coefficient and \\spad{a} and \\spad{d} should be positive.}")) (|evenlambert| (($ $) "\\spad{evenlambert(f(x))} returns \\spad{f(x^2) + f(x^4) + f(x^6) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n))) = exp(log(evenlambert(f(x))))}.}")) (|oddlambert| (($ $) "\\spad{oddlambert(f(x))} returns \\spad{f(x) + f(x^3) + f(x^5) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.}")) (|lambert| (($ $) "\\spad{lambert(f(x))} returns \\spad{f(x) + f(x^2) + f(x^3) + ...}. \\indented{1}{This function is used for computing infinite products.} \\indented{1}{\\spad{f(x)} should have zero constant coefficient.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n = 1..infinity,f(x^n)) = exp(log(lambert(f(x))))}.}")) (|lagrange| (($ $) "\\spad{lagrange(g(x))} produces the Taylor series for \\spad{f(x)} \\indented{1}{where \\spad{f(x)} is implicitly defined as \\spad{f(x) = x*g(f(x))}.}")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) (((-4462 "*") |has| |#1| (-174)) (-4453 |has| |#1| (-567)) (-4454 . T) (-4455 . T) (-4457 . T)) -((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3765 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|)))) (|HasCategory| (-782) (QUOTE (-1129))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasSignature| |#1| (LIST (QUOTE -2883) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasCategory| |#1| (QUOTE (-373))) (-3765 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4413) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) +((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasCategory| |#1| (QUOTE (-567))) (-3763 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-567)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -913) (QUOTE (-1194)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-782)) (|devaluate| |#1|)))) (|HasCategory| (-782) (QUOTE (-1129))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasSignature| |#1| (LIST (QUOTE -2882) (LIST (|devaluate| |#1|) (QUOTE (-1194)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-782))))) (|HasCategory| |#1| (QUOTE (-373))) (-3763 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-575)))) (|HasCategory| |#1| (QUOTE (-974))) (|HasCategory| |#1| (QUOTE (-1220))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -418) (QUOTE (-575))))) (|HasSignature| |#1| (LIST (QUOTE -4388) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1194))))) (|HasSignature| |#1| (LIST (QUOTE -1606) (LIST (LIST (QUOTE -655) (QUOTE (-1194))) (|devaluate| |#1|))))))) (-1278 |Coef| UTS) ((|constructor| (NIL "\\indented{1}{This package provides Taylor series solutions to regular} linear or non-linear ordinary differential equations of arbitrary order.")) (|mpsode| (((|List| |#2|) (|List| |#1|) (|List| (|Mapping| |#2| (|List| |#2|)))) "\\spad{mpsode(r,f)} solves the system of differential equations \\spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]},{} \\spad{y[i](a) = r[i]} for \\spad{i} in 1..\\spad{n}.")) (|ode| ((|#2| (|Mapping| |#2| (|List| |#2|)) (|List| |#1|)) "\\spad{ode(f,cl)} is the solution to \\spad{y<n>=f(y,y',..,y<n-1>)} such that \\spad{y<i>(a) = cl.i} for \\spad{i} in 1..\\spad{n}.")) (|ode2| ((|#2| (|Mapping| |#2| |#2| |#2|) |#1| |#1|) "\\spad{ode2(f,c0,c1)} is the solution to \\spad{y'' = f(y,y')} such that \\spad{y(a) = c0} and \\spad{y'(a) = c1}.")) (|ode1| ((|#2| (|Mapping| |#2| |#2|) |#1|) "\\spad{ode1(f,c)} is the solution to \\spad{y' = f(y)} such that \\spad{y(a) = c}.")) (|fixedPointExquo| ((|#2| |#2| |#2|) "\\spad{fixedPointExquo(f,g)} computes the exact quotient of \\spad{f} and \\spad{g} using a fixed point computation.")) (|stFuncN| (((|Mapping| (|Stream| |#1|) (|List| (|Stream| |#1|))) (|Mapping| |#2| (|List| |#2|))) "\\spad{stFuncN(f)} is a local function xported due to compiler problem. This function is of no interest to the top-level user.")) (|stFunc2| (((|Mapping| (|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) (|Mapping| |#2| |#2| |#2|)) "\\spad{stFunc2(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user.")) (|stFunc1| (((|Mapping| (|Stream| |#1|) (|Stream| |#1|)) (|Mapping| |#2| |#2|)) "\\spad{stFunc1(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user."))) NIL NIL -(-1279 -3029 UP L UTS) +(-1279 -3027 UP L UTS) ((|constructor| (NIL "\\spad{RUTSodetools} provides tools to interface with the series \\indented{1}{ODE solver when presented with linear ODEs.}")) (RF2UTS ((|#4| (|Fraction| |#2|)) "\\spad{RF2UTS(f)} converts \\spad{f} to a Taylor series.")) (LODO2FUN (((|Mapping| |#4| (|List| |#4|)) |#3|) "\\spad{LODO2FUN(op)} returns the function to pass to the series ODE solver in order to solve \\spad{op y = 0}.")) (UTS2UP ((|#2| |#4| (|NonNegativeInteger|)) "\\spad{UTS2UP(s, n)} converts the first \\spad{n} terms of \\spad{s} to a univariate polynomial.")) (UP2UTS ((|#4| |#2|) "\\spad{UP2UTS(p)} converts \\spad{p} to a Taylor series."))) NIL ((|HasCategory| |#1| (QUOTE (-567)))) @@ -5071,7 +5071,7 @@ NIL (-1285 R) ((|constructor| (NIL "This type represents vector like objects with varying lengths and indexed by a finite segment of integers starting at 1.")) (|vector| (($ (|List| |#1|)) "\\spad{vector(l)} converts the list \\spad{l} to a vector."))) ((-4461 . T) (-4460 . T)) -((-3765 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3765 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3765 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1019))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) +((-3763 (-12 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-3763 (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873))))) (|HasCategory| |#1| (LIST (QUOTE -625) (QUOTE (-547)))) (-3763 (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117)))) (|HasCategory| |#1| (QUOTE (-861))) (|HasCategory| (-575) (QUOTE (-861))) (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-737))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1019))) (|HasCategory| |#1| (QUOTE (-1066)))) (|HasCategory| |#1| (LIST (QUOTE -624) (QUOTE (-873)))) (-12 (|HasCategory| |#1| (QUOTE (-1117))) (|HasCategory| |#1| (LIST (QUOTE -318) (|devaluate| |#1|))))) (-1286) ((|constructor| (NIL "TwoDimensionalViewport creates viewports to display graphs.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(v)} returns the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport} as output of the domain \\spadtype{OutputForm}.")) (|key| (((|Integer|) $) "\\spad{key(v)} returns the process ID number of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport}.")) (|reset| (((|Void|) $) "\\spad{reset(v)} sets the current state of the graph characteristics of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} back to their initial settings.")) (|write| (((|String|) $ (|String|) (|List| (|String|))) "\\spad{write(v,s,lf)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v} and the optional file types indicated by the list \\spad{lf}.") (((|String|) $ (|String|) (|String|)) "\\spad{write(v,s,f)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v} and an optional file type \\spad{f}.") (((|String|) $ (|String|)) "\\spad{write(v,s)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v}.")) (|resize| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{resize(v,w,h)} displays the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with a width of \\spad{w} and a height of \\spad{h},{} keeping the upper left-hand corner position unchanged.")) (|update| (((|Void|) $ (|GraphImage|) (|PositiveInteger|)) "\\spad{update(v,gr,n)} drops the graph \\spad{gr} in slot \\spad{n} of viewport \\spad{v}. The graph \\spad{gr} must have been transmitted already and acquired an integer key.")) (|move| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{move(v,x,y)} displays the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \\spad{x},{} \\spad{y}.")) (|show| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{show(v,n,s)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the graph if \\spad{s} is \"off\".")) (|translate| (((|Void|) $ (|PositiveInteger|) (|Float|) (|Float|)) "\\spad{translate(v,n,dx,dy)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} translated by \\spad{dx} in the \\spad{x}-coordinate direction from the center of the viewport,{} and by \\spad{dy} in the \\spad{y}-coordinate direction from the center. Setting \\spad{dx} and \\spad{dy} to \\spad{0} places the center of the graph at the center of the viewport.")) (|scale| (((|Void|) $ (|PositiveInteger|) (|Float|) (|Float|)) "\\spad{scale(v,n,sx,sy)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} scaled by the factor \\spad{sx} in the \\spad{x}-coordinate direction and by the factor \\spad{sy} in the \\spad{y}-coordinate direction.")) (|dimensions| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{dimensions(v,x,y,width,height)} sets the position of the upper left-hand corner of the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} to the window coordinate \\spad{x},{} \\spad{y},{} and sets the dimensions of the window to that of \\spad{width},{} \\spad{height}. The new dimensions are not displayed until the function \\spadfun{makeViewport2D} is executed again for \\spad{v}.")) (|close| (((|Void|) $) "\\spad{close(v)} closes the viewport window of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and terminates the corresponding process ID.")) (|controlPanel| (((|Void|) $ (|String|)) "\\spad{controlPanel(v,s)} displays the control panel of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or hides the control panel if \\spad{s} is \"off\".")) (|connect| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{connect(v,n,s)} displays the lines connecting the graph points in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the lines if \\spad{s} is \"off\".")) (|region| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{region(v,n,s)} displays the bounding box of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the bounding box if \\spad{s} is \"off\".")) (|points| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{points(v,n,s)} displays the points of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the points if \\spad{s} is \"off\".")) (|units| (((|Void|) $ (|PositiveInteger|) (|Palette|)) "\\spad{units(v,n,c)} displays the units of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the units color set to the given palette color \\spad{c}.") (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{units(v,n,s)} displays the units of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the units if \\spad{s} is \"off\".")) (|axes| (((|Void|) $ (|PositiveInteger|) (|Palette|)) "\\spad{axes(v,n,c)} displays the axes of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the axes color set to the given palette color \\spad{c}.") (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{axes(v,n,s)} displays the axes of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the axes if \\spad{s} is \"off\".")) (|getGraph| (((|GraphImage|) $ (|PositiveInteger|)) "\\spad{getGraph(v,n)} returns the graph which is of the domain \\spadtype{GraphImage} which is located in graph field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of the domain \\spadtype{TwoDimensionalViewport}.")) (|putGraph| (((|Void|) $ (|GraphImage|) (|PositiveInteger|)) "\\spad{putGraph(v,gi,n)} sets the graph field indicated by \\spad{n},{} of the indicated two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} to be the graph,{} \\spad{gi} of domain \\spadtype{GraphImage}. The contents of viewport,{} \\spad{v},{} will contain \\spad{gi} when the function \\spadfun{makeViewport2D} is called to create the an updated viewport \\spad{v}.")) (|title| (((|Void|) $ (|String|)) "\\spad{title(v,s)} changes the title which is shown in the two-dimensional viewport window,{} \\spad{v} of domain \\spadtype{TwoDimensionalViewport}.")) (|graphs| (((|Vector| (|Union| (|GraphImage|) "undefined")) $) "\\spad{graphs(v)} returns a vector,{} or list,{} which is a union of all the graphs,{} of the domain \\spadtype{GraphImage},{} which are allocated for the two-dimensional viewport,{} \\spad{v},{} of domain \\spadtype{TwoDimensionalViewport}. Those graphs which have no data are labeled \"undefined\",{} otherwise their contents are shown.")) (|graphStates| (((|Vector| (|Record| (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)) (|:| |points| (|Integer|)) (|:| |connect| (|Integer|)) (|:| |spline| (|Integer|)) (|:| |axes| (|Integer|)) (|:| |axesColor| (|Palette|)) (|:| |units| (|Integer|)) (|:| |unitsColor| (|Palette|)) (|:| |showing| (|Integer|)))) $) "\\spad{graphStates(v)} returns and shows a listing of a record containing the current state of the characteristics of each of the ten graph records in the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport}.")) (|graphState| (((|Void|) $ (|PositiveInteger|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Palette|) (|Integer|) (|Palette|) (|Integer|)) "\\spad{graphState(v,num,sX,sY,dX,dY,pts,lns,box,axes,axesC,un,unC,cP)} sets the state of the characteristics for the graph indicated by \\spad{num} in the given two-dimensional viewport \\spad{v},{} of domain \\spadtype{TwoDimensionalViewport},{} to the values given as parameters. The scaling of the graph in the \\spad{x} and \\spad{y} component directions is set to be \\spad{sX} and \\spad{sY}; the window translation in the \\spad{x} and \\spad{y} component directions is set to be \\spad{dX} and \\spad{dY}; The graph points,{} lines,{} bounding \\spad{box},{} \\spad{axes},{} or units will be shown in the viewport if their given parameters \\spad{pts},{} \\spad{lns},{} \\spad{box},{} \\spad{axes} or \\spad{un} are set to be \\spad{1},{} but will not be shown if they are set to \\spad{0}. The color of the \\spad{axes} and the color of the units are indicated by the palette colors \\spad{axesC} and \\spad{unC} respectively. To display the control panel when the viewport window is displayed,{} set \\spad{cP} to \\spad{1},{} otherwise set it to \\spad{0}.")) (|options| (($ $ (|List| (|DrawOption|))) "\\spad{options(v,lopt)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and returns \\spad{v} with it\\spad{'s} draw options modified to be those which are indicated in the given list,{} \\spad{lopt} of domain \\spadtype{DrawOption}.") (((|List| (|DrawOption|)) $) "\\spad{options(v)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and returns a list containing the draw options from the domain \\spadtype{DrawOption} for \\spad{v}.")) (|makeViewport2D| (($ (|GraphImage|) (|List| (|DrawOption|))) "\\spad{makeViewport2D(gi,lopt)} creates and displays a viewport window of the domain \\spadtype{TwoDimensionalViewport} whose graph field is assigned to be the given graph,{} \\spad{gi},{} of domain \\spadtype{GraphImage},{} and whose options field is set to be the list of options,{} \\spad{lopt} of domain \\spadtype{DrawOption}.") (($ $) "\\spad{makeViewport2D(v)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and displays a viewport window on the screen which contains the contents of \\spad{v}.")) (|viewport2D| (($) "\\spad{viewport2D()} returns an undefined two-dimensional viewport of the domain \\spadtype{TwoDimensionalViewport} whose contents are empty.")) (|getPickedPoints| (((|List| (|Point| (|DoubleFloat|))) $) "\\spad{getPickedPoints(x)} returns a list of small floats for the points the user interactively picked on the viewport for full integration into the system,{} some design issues need to be addressed: \\spadignore{e.g.} how to go through the GraphImage interface,{} how to default to graphs,{} etc."))) NIL @@ -5104,7 +5104,7 @@ NIL ((|constructor| (NIL "This package implements the Weierstrass preparation theorem \\spad{f} or multivariate power series. weierstrass(\\spad{v},{}\\spad{p}) where \\spad{v} is a variable,{} and \\spad{p} is a TaylorSeries(\\spad{R}) in which the terms of lowest degree \\spad{s} must include c*v**s where \\spad{c} is a constant,{}\\spad{s>0},{} is a list of TaylorSeries coefficients A[\\spad{i}] of the equivalent polynomial A = A[0] + A[1]\\spad{*v} + A[2]*v**2 + ... + A[\\spad{s}-1]*v**(\\spad{s}-1) + v**s such that p=A*B ,{} \\spad{B} being a TaylorSeries of minimum degree 0")) (|qqq| (((|Mapping| (|Stream| (|TaylorSeries| |#1|)) (|Stream| (|TaylorSeries| |#1|))) (|NonNegativeInteger|) (|TaylorSeries| |#1|) (|Stream| (|TaylorSeries| |#1|))) "\\spad{qqq(n,s,st)} is used internally.")) (|weierstrass| (((|List| (|TaylorSeries| |#1|)) (|Symbol|) (|TaylorSeries| |#1|)) "\\spad{weierstrass(v,ts)} where \\spad{v} is a variable and \\spad{ts} is \\indented{1}{a TaylorSeries,{} impements the Weierstrass Preparation} \\indented{1}{Theorem. The result is a list of TaylorSeries that} \\indented{1}{are the coefficients of the equivalent series.}")) (|clikeUniv| (((|Mapping| (|SparseUnivariatePolynomial| (|Polynomial| |#1|)) (|Polynomial| |#1|)) (|Symbol|)) "\\spad{clikeUniv(v)} is used internally.")) (|sts2stst| (((|Stream| (|Stream| (|Polynomial| |#1|))) (|Symbol|) (|Stream| (|Polynomial| |#1|))) "\\spad{sts2stst(v,s)} is used internally.")) (|cfirst| (((|Mapping| (|Stream| (|Polynomial| |#1|)) (|Stream| (|Polynomial| |#1|))) (|NonNegativeInteger|)) "\\spad{cfirst n} is used internally.")) (|crest| (((|Mapping| (|Stream| (|Polynomial| |#1|)) (|Stream| (|Polynomial| |#1|))) (|NonNegativeInteger|)) "\\spad{crest n} is used internally."))) NIL NIL -(-1294 K R UP -3029) +(-1294 K R UP -3027) ((|constructor| (NIL "In this package \\spad{K} is a finite field,{} \\spad{R} is a ring of univariate polynomials over \\spad{K},{} and \\spad{F} is a framed algebra over \\spad{R}. The package provides a function to compute the integral closure of \\spad{R} in the quotient field of \\spad{F} as well as a function to compute a \"local integral basis\" at a specific prime.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) |#2|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}."))) NIL NIL @@ -5140,11 +5140,11 @@ NIL ((|constructor| (NIL "This category specifies opeations for polynomials and formal series with non-commutative variables.")) (|varList| (((|List| |#1|) $) "\\spad{varList(x)} returns the list of variables which appear in \\spad{x}.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,x)} returns \\spad{Sum(fn(r_i) w_i)} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|sh| (($ $ (|NonNegativeInteger|)) "\\spad{sh(x,n)} returns the shuffle power of \\spad{x} to the \\spad{n}.") (($ $ $) "\\spad{sh(x,y)} returns the shuffle-product of \\spad{x} by \\spad{y}. This multiplication is associative and commutative.")) (|quasiRegular| (($ $) "\\spad{quasiRegular(x)} return \\spad{x} minus its constant term.")) (|quasiRegular?| (((|Boolean|) $) "\\spad{quasiRegular?(x)} return \\spad{true} if \\spad{constant(x)} is zero.")) (|constant| ((|#2| $) "\\spad{constant(x)} returns the constant term of \\spad{x}.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(x)} returns \\spad{true} if \\spad{x} is constant.")) (|coerce| (($ |#1|) "\\spad{coerce(v)} returns \\spad{v}.")) (|mirror| (($ $) "\\spad{mirror(x)} returns \\spad{Sum(r_i mirror(w_i))} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} returns \\spad{true} if \\spad{x} is a monomial")) (|monom| (($ (|OrderedFreeMonoid| |#1|) |#2|) "\\spad{monom(w,r)} returns the product of the word \\spad{w} by the coefficient \\spad{r}.")) (|rquo| (($ $ $) "\\spad{rquo(x,y)} returns the right simplification of \\spad{x} by \\spad{y}.") (($ $ (|OrderedFreeMonoid| |#1|)) "\\spad{rquo(x,w)} returns the right simplification of \\spad{x} by \\spad{w}.") (($ $ |#1|) "\\spad{rquo(x,v)} returns the right simplification of \\spad{x} by the variable \\spad{v}.")) (|lquo| (($ $ $) "\\spad{lquo(x,y)} returns the left simplification of \\spad{x} by \\spad{y}.") (($ $ (|OrderedFreeMonoid| |#1|)) "\\spad{lquo(x,w)} returns the left simplification of \\spad{x} by the word \\spad{w}.") (($ $ |#1|) "\\spad{lquo(x,v)} returns the left simplification of \\spad{x} by the variable \\spad{v}.")) (|coef| ((|#2| $ $) "\\spad{coef(x,y)} returns scalar product of \\spad{x} by \\spad{y},{} the set of words being regarded as an orthogonal basis.") ((|#2| $ (|OrderedFreeMonoid| |#1|)) "\\spad{coef(x,w)} returns the coefficient of the word \\spad{w} in \\spad{x}.")) (|mindegTerm| (((|Record| (|:| |k| (|OrderedFreeMonoid| |#1|)) (|:| |c| |#2|)) $) "\\spad{mindegTerm(x)} returns the term whose word is \\spad{mindeg(x)}.")) (|mindeg| (((|OrderedFreeMonoid| |#1|) $) "\\spad{mindeg(x)} returns the little word which appears in \\spad{x}. Error if \\spad{x=0}.")) (* (($ $ |#2|) "\\spad{x * r} returns the product of \\spad{x} by \\spad{r}. Usefull if \\spad{R} is a non-commutative Ring.") (($ |#1| $) "\\spad{v * x} returns the product of a variable \\spad{x} by \\spad{x}."))) ((-4453 |has| |#2| (-6 -4453)) (-4455 . T) (-4454 . T) (-4457 . T)) NIL -(-1303 S -3029) +(-1303 S -3027) ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|NonNegativeInteger|)) "\\spad{Frobenius(a,s)} returns \\spad{a**(q**s)} where \\spad{q} is the size()\\$\\spad{F}.") (($ $) "\\spad{Frobenius(a)} returns \\spad{a ** q} where \\spad{q} is the \\spad{size()\\$F}.")) (|transcendenceDegree| (((|NonNegativeInteger|)) "\\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.")) (|extensionDegree| (((|OnePointCompletion| (|PositiveInteger|))) "\\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \\spad{infinity} if it is not.")) (|degree| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{degree(a)} returns the degree of minimal polynomial of an element \\spad{a} if \\spad{a} is algebraic with respect to the ground field \\spad{F},{} and \\spad{infinity} otherwise.")) (|inGroundField?| (((|Boolean|) $) "\\spad{inGroundField?(a)} tests whether an element \\spad{a} is already in the ground field \\spad{F}.")) (|transcendent?| (((|Boolean|) $) "\\spad{transcendent?(a)} tests whether an element \\spad{a} is transcendent with respect to the ground field \\spad{F}.")) (|algebraic?| (((|Boolean|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) NIL ((|HasCategory| |#2| (QUOTE (-378))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148)))) -(-1304 -3029) +(-1304 -3027) ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|NonNegativeInteger|)) "\\spad{Frobenius(a,s)} returns \\spad{a**(q**s)} where \\spad{q} is the size()\\$\\spad{F}.") (($ $) "\\spad{Frobenius(a)} returns \\spad{a ** q} where \\spad{q} is the \\spad{size()\\$F}.")) (|transcendenceDegree| (((|NonNegativeInteger|)) "\\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.")) (|extensionDegree| (((|OnePointCompletion| (|PositiveInteger|))) "\\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \\spad{infinity} if it is not.")) (|degree| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{degree(a)} returns the degree of minimal polynomial of an element \\spad{a} if \\spad{a} is algebraic with respect to the ground field \\spad{F},{} and \\spad{infinity} otherwise.")) (|inGroundField?| (((|Boolean|) $) "\\spad{inGroundField?(a)} tests whether an element \\spad{a} is already in the ground field \\spad{F}.")) (|transcendent?| (((|Boolean|) $) "\\spad{transcendent?(a)} tests whether an element \\spad{a} is transcendent with respect to the ground field \\spad{F}.")) (|algebraic?| (((|Boolean|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) ((-4452 . T) (-4458 . T) (-4453 . T) ((-4462 "*") . T) (-4454 . T) (-4455 . T) (-4457 . T)) NIL @@ -5204,4 +5204,4 @@ NIL NIL NIL NIL -((-3 NIL 2280931 2280936 2280941 2280946) (-2 NIL 2280911 2280916 2280921 2280926) (-1 NIL 2280891 2280896 2280901 2280906) (0 NIL 2280871 2280876 2280881 2280886) (-1314 "ZMOD.spad" 2280680 2280693 2280809 2280866) (-1313 "ZLINDEP.spad" 2279746 2279757 2280670 2280675) (-1312 "ZDSOLVE.spad" 2269691 2269713 2279736 2279741) (-1311 "YSTREAM.spad" 2269186 2269197 2269681 2269686) (-1310 "YDIAGRAM.spad" 2268820 2268829 2269176 2269181) (-1309 "XRPOLY.spad" 2268040 2268060 2268676 2268745) (-1308 "XPR.spad" 2265835 2265848 2267758 2267857) (-1307 "XPOLY.spad" 2265390 2265401 2265691 2265760) (-1306 "XPOLYC.spad" 2264709 2264725 2265316 2265385) (-1305 "XPBWPOLY.spad" 2263146 2263166 2264489 2264558) (-1304 "XF.spad" 2261609 2261624 2263048 2263141) (-1303 "XF.spad" 2260052 2260069 2261493 2261498) (-1302 "XFALG.spad" 2257100 2257116 2259978 2260047) (-1301 "XEXPPKG.spad" 2256351 2256377 2257090 2257095) (-1300 "XDPOLY.spad" 2255965 2255981 2256207 2256276) (-1299 "XALG.spad" 2255625 2255636 2255921 2255960) (-1298 "WUTSET.spad" 2251464 2251481 2255271 2255298) (-1297 "WP.spad" 2250663 2250707 2251322 2251389) (-1296 "WHILEAST.spad" 2250461 2250470 2250653 2250658) (-1295 "WHEREAST.spad" 2250132 2250141 2250451 2250456) (-1294 "WFFINTBS.spad" 2247795 2247817 2250122 2250127) (-1293 "WEIER.spad" 2246017 2246028 2247785 2247790) (-1292 "VSPACE.spad" 2245690 2245701 2245985 2246012) (-1291 "VSPACE.spad" 2245383 2245396 2245680 2245685) (-1290 "VOID.spad" 2245060 2245069 2245373 2245378) (-1289 "VIEW.spad" 2242740 2242749 2245050 2245055) (-1288 "VIEWDEF.spad" 2237941 2237950 2242730 2242735) (-1287 "VIEW3D.spad" 2221902 2221911 2237931 2237936) (-1286 "VIEW2D.spad" 2209793 2209802 2221892 2221897) (-1285 "VECTOR.spad" 2208467 2208478 2208718 2208745) (-1284 "VECTOR2.spad" 2207106 2207119 2208457 2208462) (-1283 "VECTCAT.spad" 2205010 2205021 2207074 2207101) (-1282 "VECTCAT.spad" 2202721 2202734 2204787 2204792) (-1281 "VARIABLE.spad" 2202501 2202516 2202711 2202716) (-1280 "UTYPE.spad" 2202145 2202154 2202491 2202496) (-1279 "UTSODETL.spad" 2201440 2201464 2202101 2202106) (-1278 "UTSODE.spad" 2199656 2199676 2201430 2201435) (-1277 "UTS.spad" 2194460 2194488 2198123 2198220) (-1276 "UTSCAT.spad" 2191939 2191955 2194358 2194455) (-1275 "UTSCAT.spad" 2189062 2189080 2191483 2191488) (-1274 "UTS2.spad" 2188657 2188692 2189052 2189057) (-1273 "URAGG.spad" 2183330 2183341 2188647 2188652) (-1272 "URAGG.spad" 2177967 2177980 2183286 2183291) (-1271 "UPXSSING.spad" 2175612 2175638 2177048 2177181) (-1270 "UPXS.spad" 2172766 2172794 2173744 2173893) (-1269 "UPXSCONS.spad" 2170525 2170545 2170898 2171047) (-1268 "UPXSCCA.spad" 2169096 2169116 2170371 2170520) (-1267 "UPXSCCA.spad" 2167809 2167831 2169086 2169091) (-1266 "UPXSCAT.spad" 2166398 2166414 2167655 2167804) (-1265 "UPXS2.spad" 2165941 2165994 2166388 2166393) (-1264 "UPSQFREE.spad" 2164355 2164369 2165931 2165936) (-1263 "UPSCAT.spad" 2162142 2162166 2164253 2164350) (-1262 "UPSCAT.spad" 2159635 2159661 2161748 2161753) (-1261 "UPOLYC.spad" 2154675 2154686 2159477 2159630) (-1260 "UPOLYC.spad" 2149607 2149620 2154411 2154416) (-1259 "UPOLYC2.spad" 2149078 2149097 2149597 2149602) (-1258 "UP.spad" 2146184 2146199 2146571 2146724) (-1257 "UPMP.spad" 2145084 2145097 2146174 2146179) (-1256 "UPDIVP.spad" 2144649 2144663 2145074 2145079) (-1255 "UPDECOMP.spad" 2142894 2142908 2144639 2144644) (-1254 "UPCDEN.spad" 2142103 2142119 2142884 2142889) (-1253 "UP2.spad" 2141467 2141488 2142093 2142098) (-1252 "UNISEG.spad" 2140820 2140831 2141386 2141391) (-1251 "UNISEG2.spad" 2140317 2140330 2140776 2140781) (-1250 "UNIFACT.spad" 2139420 2139432 2140307 2140312) (-1249 "ULS.spad" 2129062 2129090 2130149 2130578) (-1248 "ULSCONS.spad" 2120196 2120216 2120566 2120715) (-1247 "ULSCCAT.spad" 2117933 2117953 2120042 2120191) (-1246 "ULSCCAT.spad" 2115778 2115800 2117889 2117894) (-1245 "ULSCAT.spad" 2114010 2114026 2115624 2115773) (-1244 "ULS2.spad" 2113524 2113577 2114000 2114005) (-1243 "UINT8.spad" 2113401 2113410 2113514 2113519) (-1242 "UINT64.spad" 2113277 2113286 2113391 2113396) (-1241 "UINT32.spad" 2113153 2113162 2113267 2113272) (-1240 "UINT16.spad" 2113029 2113038 2113143 2113148) (-1239 "UFD.spad" 2112094 2112103 2112955 2113024) (-1238 "UFD.spad" 2111221 2111232 2112084 2112089) (-1237 "UDVO.spad" 2110102 2110111 2111211 2111216) (-1236 "UDPO.spad" 2107595 2107606 2110058 2110063) (-1235 "TYPE.spad" 2107527 2107536 2107585 2107590) (-1234 "TYPEAST.spad" 2107446 2107455 2107517 2107522) (-1233 "TWOFACT.spad" 2106098 2106113 2107436 2107441) (-1232 "TUPLE.spad" 2105584 2105595 2105997 2106002) (-1231 "TUBETOOL.spad" 2102451 2102460 2105574 2105579) (-1230 "TUBE.spad" 2101098 2101115 2102441 2102446) (-1229 "TS.spad" 2099697 2099713 2100663 2100760) (-1228 "TSETCAT.spad" 2086824 2086841 2099665 2099692) (-1227 "TSETCAT.spad" 2073937 2073956 2086780 2086785) (-1226 "TRMANIP.spad" 2068303 2068320 2073643 2073648) (-1225 "TRIMAT.spad" 2067266 2067291 2068293 2068298) (-1224 "TRIGMNIP.spad" 2065793 2065810 2067256 2067261) (-1223 "TRIGCAT.spad" 2065305 2065314 2065783 2065788) (-1222 "TRIGCAT.spad" 2064815 2064826 2065295 2065300) (-1221 "TREE.spad" 2063390 2063401 2064422 2064449) (-1220 "TRANFUN.spad" 2063229 2063238 2063380 2063385) (-1219 "TRANFUN.spad" 2063066 2063077 2063219 2063224) (-1218 "TOPSP.spad" 2062740 2062749 2063056 2063061) (-1217 "TOOLSIGN.spad" 2062403 2062414 2062730 2062735) (-1216 "TEXTFILE.spad" 2060964 2060973 2062393 2062398) (-1215 "TEX.spad" 2058110 2058119 2060954 2060959) (-1214 "TEX1.spad" 2057666 2057677 2058100 2058105) (-1213 "TEMUTL.spad" 2057221 2057230 2057656 2057661) (-1212 "TBCMPPK.spad" 2055314 2055337 2057211 2057216) (-1211 "TBAGG.spad" 2054364 2054387 2055294 2055309) (-1210 "TBAGG.spad" 2053422 2053447 2054354 2054359) (-1209 "TANEXP.spad" 2052830 2052841 2053412 2053417) (-1208 "TALGOP.spad" 2052554 2052565 2052820 2052825) (-1207 "TABLE.spad" 2050965 2050988 2051235 2051262) (-1206 "TABLEAU.spad" 2050446 2050457 2050955 2050960) (-1205 "TABLBUMP.spad" 2047249 2047260 2050436 2050441) (-1204 "SYSTEM.spad" 2046477 2046486 2047239 2047244) (-1203 "SYSSOLP.spad" 2043960 2043971 2046467 2046472) (-1202 "SYSPTR.spad" 2043859 2043868 2043950 2043955) (-1201 "SYSNNI.spad" 2043041 2043052 2043849 2043854) (-1200 "SYSINT.spad" 2042445 2042456 2043031 2043036) (-1199 "SYNTAX.spad" 2038651 2038660 2042435 2042440) (-1198 "SYMTAB.spad" 2036719 2036728 2038641 2038646) (-1197 "SYMS.spad" 2032742 2032751 2036709 2036714) (-1196 "SYMPOLY.spad" 2031749 2031760 2031831 2031958) (-1195 "SYMFUNC.spad" 2031250 2031261 2031739 2031744) (-1194 "SYMBOL.spad" 2028753 2028762 2031240 2031245) (-1193 "SWITCH.spad" 2025524 2025533 2028743 2028748) (-1192 "SUTS.spad" 2022429 2022457 2023991 2024088) (-1191 "SUPXS.spad" 2019570 2019598 2020561 2020710) (-1190 "SUP.spad" 2016290 2016301 2017063 2017216) (-1189 "SUPFRACF.spad" 2015395 2015413 2016280 2016285) (-1188 "SUP2.spad" 2014787 2014800 2015385 2015390) (-1187 "SUMRF.spad" 2013761 2013772 2014777 2014782) (-1186 "SUMFS.spad" 2013398 2013415 2013751 2013756) (-1185 "SULS.spad" 2003027 2003055 2004127 2004556) (-1184 "SUCHTAST.spad" 2002796 2002805 2003017 2003022) (-1183 "SUCH.spad" 2002478 2002493 2002786 2002791) (-1182 "SUBSPACE.spad" 1994593 1994608 2002468 2002473) (-1181 "SUBRESP.spad" 1993763 1993777 1994549 1994554) (-1180 "STTF.spad" 1989862 1989878 1993753 1993758) (-1179 "STTFNC.spad" 1986330 1986346 1989852 1989857) (-1178 "STTAYLOR.spad" 1978965 1978976 1986211 1986216) (-1177 "STRTBL.spad" 1977470 1977487 1977619 1977646) (-1176 "STRING.spad" 1976879 1976888 1976893 1976920) (-1175 "STRICAT.spad" 1976667 1976676 1976847 1976874) (-1174 "STREAM.spad" 1973585 1973596 1976192 1976207) (-1173 "STREAM3.spad" 1973158 1973173 1973575 1973580) (-1172 "STREAM2.spad" 1972286 1972299 1973148 1973153) (-1171 "STREAM1.spad" 1971992 1972003 1972276 1972281) (-1170 "STINPROD.spad" 1970928 1970944 1971982 1971987) (-1169 "STEP.spad" 1970129 1970138 1970918 1970923) (-1168 "STEPAST.spad" 1969363 1969372 1970119 1970124) (-1167 "STBL.spad" 1967889 1967917 1968056 1968071) (-1166 "STAGG.spad" 1966964 1966975 1967879 1967884) (-1165 "STAGG.spad" 1966037 1966050 1966954 1966959) (-1164 "STACK.spad" 1965394 1965405 1965644 1965671) (-1163 "SREGSET.spad" 1963098 1963115 1965040 1965067) (-1162 "SRDCMPK.spad" 1961659 1961679 1963088 1963093) (-1161 "SRAGG.spad" 1956802 1956811 1961627 1961654) (-1160 "SRAGG.spad" 1951965 1951976 1956792 1956797) (-1159 "SQMATRIX.spad" 1949544 1949562 1950460 1950547) (-1158 "SPLTREE.spad" 1944096 1944109 1948980 1949007) (-1157 "SPLNODE.spad" 1940684 1940697 1944086 1944091) (-1156 "SPFCAT.spad" 1939493 1939502 1940674 1940679) (-1155 "SPECOUT.spad" 1938045 1938054 1939483 1939488) (-1154 "SPADXPT.spad" 1929640 1929649 1938035 1938040) (-1153 "spad-parser.spad" 1929105 1929114 1929630 1929635) (-1152 "SPADAST.spad" 1928806 1928815 1929095 1929100) (-1151 "SPACEC.spad" 1913005 1913016 1928796 1928801) (-1150 "SPACE3.spad" 1912781 1912792 1912995 1913000) (-1149 "SORTPAK.spad" 1912330 1912343 1912737 1912742) (-1148 "SOLVETRA.spad" 1910093 1910104 1912320 1912325) (-1147 "SOLVESER.spad" 1908621 1908632 1910083 1910088) (-1146 "SOLVERAD.spad" 1904647 1904658 1908611 1908616) (-1145 "SOLVEFOR.spad" 1903109 1903127 1904637 1904642) (-1144 "SNTSCAT.spad" 1902709 1902726 1903077 1903104) (-1143 "SMTS.spad" 1900981 1901007 1902274 1902371) (-1142 "SMP.spad" 1898456 1898476 1898846 1898973) (-1141 "SMITH.spad" 1897301 1897326 1898446 1898451) (-1140 "SMATCAT.spad" 1895411 1895441 1897245 1897296) (-1139 "SMATCAT.spad" 1893453 1893485 1895289 1895294) (-1138 "SKAGG.spad" 1892416 1892427 1893421 1893448) (-1137 "SINT.spad" 1891356 1891365 1892282 1892411) (-1136 "SIMPAN.spad" 1891084 1891093 1891346 1891351) (-1135 "SIG.spad" 1890414 1890423 1891074 1891079) (-1134 "SIGNRF.spad" 1889532 1889543 1890404 1890409) (-1133 "SIGNEF.spad" 1888811 1888828 1889522 1889527) (-1132 "SIGAST.spad" 1888196 1888205 1888801 1888806) (-1131 "SHP.spad" 1886124 1886139 1888152 1888157) (-1130 "SHDP.spad" 1874327 1874354 1874836 1874935) (-1129 "SGROUP.spad" 1873935 1873944 1874317 1874322) (-1128 "SGROUP.spad" 1873541 1873552 1873925 1873930) (-1127 "SGCF.spad" 1866680 1866689 1873531 1873536) (-1126 "SFRTCAT.spad" 1865610 1865627 1866648 1866675) (-1125 "SFRGCD.spad" 1864673 1864693 1865600 1865605) (-1124 "SFQCMPK.spad" 1859310 1859330 1864663 1864668) (-1123 "SFORT.spad" 1858749 1858763 1859300 1859305) (-1122 "SEXOF.spad" 1858592 1858632 1858739 1858744) (-1121 "SEX.spad" 1858484 1858493 1858582 1858587) (-1120 "SEXCAT.spad" 1856265 1856305 1858474 1858479) (-1119 "SET.spad" 1854589 1854600 1855686 1855725) (-1118 "SETMN.spad" 1853039 1853056 1854579 1854584) (-1117 "SETCAT.spad" 1852361 1852370 1853029 1853034) (-1116 "SETCAT.spad" 1851681 1851692 1852351 1852356) (-1115 "SETAGG.spad" 1848230 1848241 1851661 1851676) (-1114 "SETAGG.spad" 1844787 1844800 1848220 1848225) (-1113 "SEQAST.spad" 1844490 1844499 1844777 1844782) (-1112 "SEGXCAT.spad" 1843646 1843659 1844480 1844485) (-1111 "SEG.spad" 1843459 1843470 1843565 1843570) (-1110 "SEGCAT.spad" 1842384 1842395 1843449 1843454) (-1109 "SEGBIND.spad" 1842142 1842153 1842331 1842336) (-1108 "SEGBIND2.spad" 1841840 1841853 1842132 1842137) (-1107 "SEGAST.spad" 1841554 1841563 1841830 1841835) (-1106 "SEG2.spad" 1840989 1841002 1841510 1841515) (-1105 "SDVAR.spad" 1840265 1840276 1840979 1840984) (-1104 "SDPOL.spad" 1837598 1837609 1837889 1838016) (-1103 "SCPKG.spad" 1835687 1835698 1837588 1837593) (-1102 "SCOPE.spad" 1834840 1834849 1835677 1835682) (-1101 "SCACHE.spad" 1833536 1833547 1834830 1834835) (-1100 "SASTCAT.spad" 1833445 1833454 1833526 1833531) (-1099 "SAOS.spad" 1833317 1833326 1833435 1833440) (-1098 "SAERFFC.spad" 1833030 1833050 1833307 1833312) (-1097 "SAE.spad" 1830500 1830516 1831111 1831246) (-1096 "SAEFACT.spad" 1830201 1830221 1830490 1830495) (-1095 "RURPK.spad" 1827860 1827876 1830191 1830196) (-1094 "RULESET.spad" 1827313 1827337 1827850 1827855) (-1093 "RULE.spad" 1825553 1825577 1827303 1827308) (-1092 "RULECOLD.spad" 1825405 1825418 1825543 1825548) (-1091 "RTVALUE.spad" 1825140 1825149 1825395 1825400) (-1090 "RSTRCAST.spad" 1824857 1824866 1825130 1825135) (-1089 "RSETGCD.spad" 1821235 1821255 1824847 1824852) (-1088 "RSETCAT.spad" 1811171 1811188 1821203 1821230) (-1087 "RSETCAT.spad" 1801127 1801146 1811161 1811166) (-1086 "RSDCMPK.spad" 1799579 1799599 1801117 1801122) (-1085 "RRCC.spad" 1797963 1797993 1799569 1799574) (-1084 "RRCC.spad" 1796345 1796377 1797953 1797958) (-1083 "RPTAST.spad" 1796047 1796056 1796335 1796340) (-1082 "RPOLCAT.spad" 1775407 1775422 1795915 1796042) (-1081 "RPOLCAT.spad" 1754480 1754497 1774990 1774995) (-1080 "ROUTINE.spad" 1750363 1750372 1753127 1753154) (-1079 "ROMAN.spad" 1749691 1749700 1750229 1750358) (-1078 "ROIRC.spad" 1748771 1748803 1749681 1749686) (-1077 "RNS.spad" 1747674 1747683 1748673 1748766) (-1076 "RNS.spad" 1746663 1746674 1747664 1747669) (-1075 "RNG.spad" 1746398 1746407 1746653 1746658) (-1074 "RNGBIND.spad" 1745558 1745572 1746353 1746358) (-1073 "RMODULE.spad" 1745323 1745334 1745548 1745553) (-1072 "RMCAT2.spad" 1744743 1744800 1745313 1745318) (-1071 "RMATRIX.spad" 1743567 1743586 1743910 1743949) (-1070 "RMATCAT.spad" 1739146 1739177 1743523 1743562) (-1069 "RMATCAT.spad" 1734615 1734648 1738994 1738999) (-1068 "RLINSET.spad" 1734170 1734181 1734605 1734610) (-1067 "RINTERP.spad" 1734058 1734078 1734160 1734165) (-1066 "RING.spad" 1733528 1733537 1734038 1734053) (-1065 "RING.spad" 1733006 1733017 1733518 1733523) (-1064 "RIDIST.spad" 1732398 1732407 1732996 1733001) (-1063 "RGCHAIN.spad" 1730981 1730997 1731883 1731910) (-1062 "RGBCSPC.spad" 1730762 1730774 1730971 1730976) (-1061 "RGBCMDL.spad" 1730292 1730304 1730752 1730757) (-1060 "RF.spad" 1727934 1727945 1730282 1730287) (-1059 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1568802 1568929) (-966 "POLYLIFT.spad" 1567552 1567575 1568277 1568282) (-965 "POLYCATQ.spad" 1565670 1565692 1567542 1567547) (-964 "POLYCAT.spad" 1559140 1559161 1565538 1565665) (-963 "POLYCAT.spad" 1551948 1551971 1558348 1558353) (-962 "POLY2UP.spad" 1551400 1551414 1551938 1551943) (-961 "POLY2.spad" 1550997 1551009 1551390 1551395) (-960 "POLUTIL.spad" 1549938 1549967 1550953 1550958) (-959 "POLTOPOL.spad" 1548686 1548701 1549928 1549933) (-958 "POINT.spad" 1547524 1547534 1547611 1547638) (-957 "PNTHEORY.spad" 1544226 1544234 1547514 1547519) (-956 "PMTOOLS.spad" 1543001 1543015 1544216 1544221) (-955 "PMSYM.spad" 1542550 1542560 1542991 1542996) (-954 "PMQFCAT.spad" 1542141 1542155 1542540 1542545) (-953 "PMPRED.spad" 1541620 1541634 1542131 1542136) (-952 "PMPREDFS.spad" 1541074 1541096 1541610 1541615) (-951 "PMPLCAT.spad" 1540154 1540172 1541006 1541011) (-950 "PMLSAGG.spad" 1539739 1539753 1540144 1540149) (-949 "PMKERNEL.spad" 1539318 1539330 1539729 1539734) (-948 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(-835 "OMERRK.spad" 1365915 1365923 1366871 1366876) (-834 "OMENC.spad" 1365259 1365267 1365905 1365910) (-833 "OMDEV.spad" 1359568 1359576 1365249 1365254) (-832 "OMCONN.spad" 1358977 1358985 1359558 1359563) (-831 "OINTDOM.spad" 1358740 1358748 1358903 1358972) (-830 "OFMONOID.spad" 1356863 1356873 1358696 1358701) (-829 "ODVAR.spad" 1356124 1356134 1356853 1356858) (-828 "ODR.spad" 1355768 1355794 1355936 1356085) (-827 "ODPOL.spad" 1353057 1353067 1353397 1353524) (-826 "ODP.spad" 1341396 1341416 1341769 1341868) (-825 "ODETOOLS.spad" 1340045 1340064 1341386 1341391) (-824 "ODESYS.spad" 1337739 1337756 1340035 1340040) (-823 "ODERTRIC.spad" 1333748 1333765 1337696 1337701) (-822 "ODERED.spad" 1333147 1333171 1333738 1333743) (-821 "ODERAT.spad" 1330762 1330779 1333137 1333142) (-820 "ODEPRRIC.spad" 1327799 1327821 1330752 1330757) (-819 "ODEPROB.spad" 1327056 1327064 1327789 1327794) (-818 "ODEPRIM.spad" 1324390 1324412 1327046 1327051) (-817 "ODEPAL.spad" 1323776 1323800 1324380 1324385) (-816 "ODEPACK.spad" 1310442 1310450 1323766 1323771) (-815 "ODEINT.spad" 1309877 1309893 1310432 1310437) (-814 "ODEIFTBL.spad" 1307272 1307280 1309867 1309872) (-813 "ODEEF.spad" 1302763 1302779 1307262 1307267) (-812 "ODECONST.spad" 1302300 1302318 1302753 1302758) (-811 "ODECAT.spad" 1300898 1300906 1302290 1302295) (-810 "OCT.spad" 1299034 1299044 1299748 1299787) (-809 "OCTCT2.spad" 1298680 1298701 1299024 1299029) (-808 "OC.spad" 1296476 1296486 1298636 1298675) (-807 "OC.spad" 1293997 1294009 1296159 1296164) (-806 "OCAMON.spad" 1293845 1293853 1293987 1293992) (-805 "OASGP.spad" 1293660 1293668 1293835 1293840) (-804 "OAMONS.spad" 1293182 1293190 1293650 1293655) (-803 "OAMON.spad" 1293043 1293051 1293172 1293177) (-802 "OAGROUP.spad" 1292905 1292913 1293033 1293038) (-801 "NUMTUBE.spad" 1292496 1292512 1292895 1292900) (-800 "NUMQUAD.spad" 1280472 1280480 1292486 1292491) (-799 "NUMODE.spad" 1271826 1271834 1280462 1280467) (-798 "NUMINT.spad" 1269392 1269400 1271816 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(-159 "COMBF.spad" 178393 178409 181001 181006) (-158 "COLOR.spad" 177230 177238 178383 178388) (-157 "COLONAST.spad" 176896 176904 177220 177225) (-156 "CMPLXRT.spad" 176607 176624 176886 176891) (-155 "CLLCTAST.spad" 176269 176277 176597 176602) (-154 "CLIP.spad" 172377 172385 176259 176264) (-153 "CLIF.spad" 171032 171048 172333 172372) (-152 "CLAGG.spad" 167537 167547 171022 171027) (-151 "CLAGG.spad" 163913 163925 167400 167405) (-150 "CINTSLPE.spad" 163244 163257 163903 163908) (-149 "CHVAR.spad" 161382 161404 163234 163239) (-148 "CHARZ.spad" 161297 161305 161362 161377) (-147 "CHARPOL.spad" 160807 160817 161287 161292) (-146 "CHARNZ.spad" 160560 160568 160787 160802) (-145 "CHAR.spad" 158434 158442 160550 160555) (-144 "CFCAT.spad" 157762 157770 158424 158429) (-143 "CDEN.spad" 156958 156972 157752 157757) (-142 "CCLASS.spad" 155107 155115 156369 156408) (-141 "CATEGORY.spad" 154149 154157 155097 155102) (-140 "CATCTOR.spad" 154040 154048 154139 154144) (-139 "CATAST.spad" 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133011) (-118 "BPADICRT.spad" 129828 129840 130083 130176) (-117 "BPADIC.spad" 129492 129504 129754 129823) (-116 "BOUNDZRO.spad" 129148 129165 129482 129487) (-115 "BOP.spad" 124330 124338 129138 129143) (-114 "BOP1.spad" 121796 121806 124320 124325) (-113 "BOOLE.spad" 121446 121454 121786 121791) (-112 "BOOLEAN.spad" 120884 120892 121436 121441) (-111 "BMODULE.spad" 120596 120608 120852 120879) (-110 "BITS.spad" 120017 120025 120232 120259) (-109 "BINDING.spad" 119430 119438 120007 120012) (-108 "BINARY.spad" 117444 117452 117800 117893) (-107 "BGAGG.spad" 116649 116659 117424 117439) (-106 "BGAGG.spad" 115862 115874 116639 116644) (-105 "BFUNCT.spad" 115426 115434 115842 115857) (-104 "BEZOUT.spad" 114566 114593 115376 115381) (-103 "BBTREE.spad" 111411 111421 114173 114200) (-102 "BASTYPE.spad" 111083 111091 111401 111406) (-101 "BASTYPE.spad" 110753 110763 111073 111078) (-100 "BALFACT.spad" 110212 110225 110743 110748) (-99 "AUTOMOR.spad" 109663 109672 110192 110207) (-98 "ATTREG.spad" 106386 106393 109415 109658) (-97 "ATTRBUT.spad" 102409 102416 106366 106381) (-96 "ATTRAST.spad" 102126 102133 102399 102404) (-95 "ATRIG.spad" 101596 101603 102116 102121) (-94 "ATRIG.spad" 101064 101073 101586 101591) (-93 "ASTCAT.spad" 100968 100975 101054 101059) (-92 "ASTCAT.spad" 100870 100879 100958 100963) (-91 "ASTACK.spad" 100209 100218 100477 100504) (-90 "ASSOCEQ.spad" 99035 99046 100165 100170) (-89 "ASP9.spad" 98116 98129 99025 99030) (-88 "ASP8.spad" 97159 97172 98106 98111) (-87 "ASP80.spad" 96481 96494 97149 97154) (-86 "ASP7.spad" 95641 95654 96471 96476) (-85 "ASP78.spad" 95092 95105 95631 95636) (-84 "ASP77.spad" 94461 94474 95082 95087) (-83 "ASP74.spad" 93553 93566 94451 94456) (-82 "ASP73.spad" 92824 92837 93543 93548) (-81 "ASP6.spad" 91691 91704 92814 92819) (-80 "ASP55.spad" 90200 90213 91681 91686) (-79 "ASP50.spad" 88017 88030 90190 90195) (-78 "ASP4.spad" 87312 87325 88007 88012) (-77 "ASP49.spad" 86311 86324 87302 87307) (-76 "ASP42.spad" 84718 84757 86301 86306) (-75 "ASP41.spad" 83297 83336 84708 84713) (-74 "ASP35.spad" 82285 82298 83287 83292) (-73 "ASP34.spad" 81586 81599 82275 82280) (-72 "ASP33.spad" 81146 81159 81576 81581) (-71 "ASP31.spad" 80286 80299 81136 81141) (-70 "ASP30.spad" 79178 79191 80276 80281) (-69 "ASP29.spad" 78644 78657 79168 79173) (-68 "ASP28.spad" 69917 69930 78634 78639) (-67 "ASP27.spad" 68814 68827 69907 69912) (-66 "ASP24.spad" 67901 67914 68804 68809) (-65 "ASP20.spad" 67365 67378 67891 67896) (-64 "ASP1.spad" 66746 66759 67355 67360) (-63 "ASP19.spad" 61432 61445 66736 66741) (-62 "ASP12.spad" 60846 60859 61422 61427) (-61 "ASP10.spad" 60117 60130 60836 60841) (-60 "ARRAY2.spad" 59477 59486 59724 59751) (-59 "ARRAY1.spad" 58314 58323 58660 58687) (-58 "ARRAY12.spad" 57027 57038 58304 58309) (-57 "ARR2CAT.spad" 52801 52822 56995 57022) (-56 "ARR2CAT.spad" 48595 48618 52791 52796) (-55 "ARITY.spad" 47967 47974 48585 48590) (-54 "APPRULE.spad" 47227 47249 47957 47962) (-53 "APPLYORE.spad" 46846 46859 47217 47222) (-52 "ANY.spad" 45705 45712 46836 46841) (-51 "ANY1.spad" 44776 44785 45695 45700) (-50 "ANTISYM.spad" 43221 43237 44756 44771) (-49 "ANON.spad" 42914 42921 43211 43216) (-48 "AN.spad" 41223 41230 42730 42823) (-47 "AMR.spad" 39408 39419 41121 41218) (-46 "AMR.spad" 37430 37443 39145 39150) (-45 "ALIST.spad" 34842 34863 35192 35219) (-44 "ALGSC.spad" 33977 34003 34714 34767) (-43 "ALGPKG.spad" 29760 29771 33933 33938) (-42 "ALGMFACT.spad" 28953 28967 29750 29755) (-41 "ALGMANIP.spad" 26427 26442 28786 28791) (-40 "ALGFF.spad" 24068 24095 24285 24441) (-39 "ALGFACT.spad" 23195 23205 24058 24063) (-38 "ALGEBRA.spad" 23028 23037 23151 23190) (-37 "ALGEBRA.spad" 22893 22904 23018 23023) (-36 "ALAGG.spad" 22405 22426 22861 22888) (-35 "AHYP.spad" 21786 21793 22395 22400) (-34 "AGG.spad" 20103 20110 21776 21781) (-33 "AGG.spad" 18384 18393 20059 20064) (-32 "AF.spad" 16815 16830 18319 18324) (-31 "ADDAST.spad" 16493 16500 16805 16810) (-30 "ACPLOT.spad" 15084 15091 16483 16488) (-29 "ACFS.spad" 12893 12902 14986 15079) (-28 "ACFS.spad" 10788 10799 12883 12888) (-27 "ACF.spad" 7470 7477 10690 10783) (-26 "ACF.spad" 4238 4247 7460 7465) (-25 "ABELSG.spad" 3779 3786 4228 4233) (-24 "ABELSG.spad" 3318 3327 3769 3774) (-23 "ABELMON.spad" 2861 2868 3308 3313) (-22 "ABELMON.spad" 2402 2411 2851 2856) (-21 "ABELGRP.spad" 2067 2074 2392 2397) (-20 "ABELGRP.spad" 1730 1739 2057 2062) (-19 "A1AGG.spad" 870 879 1698 1725) (-18 "A1AGG.spad" 30 41 860 865))
\ No newline at end of file +((-3 NIL 2279571 2279576 2279581 2279586) (-2 NIL 2279551 2279556 2279561 2279566) (-1 NIL 2279531 2279536 2279541 2279546) (0 NIL 2279511 2279516 2279521 2279526) (-1314 "ZMOD.spad" 2279320 2279333 2279449 2279506) (-1313 "ZLINDEP.spad" 2278386 2278397 2279310 2279315) (-1312 "ZDSOLVE.spad" 2268331 2268353 2278376 2278381) (-1311 "YSTREAM.spad" 2267826 2267837 2268321 2268326) (-1310 "YDIAGRAM.spad" 2267460 2267469 2267816 2267821) (-1309 "XRPOLY.spad" 2266680 2266700 2267316 2267385) (-1308 "XPR.spad" 2264475 2264488 2266398 2266497) (-1307 "XPOLY.spad" 2264030 2264041 2264331 2264400) (-1306 "XPOLYC.spad" 2263349 2263365 2263956 2264025) (-1305 "XPBWPOLY.spad" 2261786 2261806 2263129 2263198) (-1304 "XF.spad" 2260249 2260264 2261688 2261781) (-1303 "XF.spad" 2258692 2258709 2260133 2260138) (-1302 "XFALG.spad" 2255740 2255756 2258618 2258687) (-1301 "XEXPPKG.spad" 2254991 2255017 2255730 2255735) (-1300 "XDPOLY.spad" 2254605 2254621 2254847 2254916) (-1299 "XALG.spad" 2254265 2254276 2254561 2254600) (-1298 "WUTSET.spad" 2250104 2250121 2253911 2253938) (-1297 "WP.spad" 2249303 2249347 2249962 2250029) (-1296 "WHILEAST.spad" 2249101 2249110 2249293 2249298) (-1295 "WHEREAST.spad" 2248772 2248781 2249091 2249096) (-1294 "WFFINTBS.spad" 2246435 2246457 2248762 2248767) (-1293 "WEIER.spad" 2244657 2244668 2246425 2246430) (-1292 "VSPACE.spad" 2244330 2244341 2244625 2244652) (-1291 "VSPACE.spad" 2244023 2244036 2244320 2244325) (-1290 "VOID.spad" 2243700 2243709 2244013 2244018) (-1289 "VIEW.spad" 2241380 2241389 2243690 2243695) (-1288 "VIEWDEF.spad" 2236581 2236590 2241370 2241375) (-1287 "VIEW3D.spad" 2220542 2220551 2236571 2236576) (-1286 "VIEW2D.spad" 2208433 2208442 2220532 2220537) (-1285 "VECTOR.spad" 2207107 2207118 2207358 2207385) (-1284 "VECTOR2.spad" 2205746 2205759 2207097 2207102) (-1283 "VECTCAT.spad" 2203650 2203661 2205714 2205741) (-1282 "VECTCAT.spad" 2201361 2201374 2203427 2203432) (-1281 "VARIABLE.spad" 2201141 2201156 2201351 2201356) (-1280 "UTYPE.spad" 2200785 2200794 2201131 2201136) (-1279 "UTSODETL.spad" 2200080 2200104 2200741 2200746) (-1278 "UTSODE.spad" 2198296 2198316 2200070 2200075) (-1277 "UTS.spad" 2193243 2193271 2196763 2196860) (-1276 "UTSCAT.spad" 2190722 2190738 2193141 2193238) (-1275 "UTSCAT.spad" 2187845 2187863 2190266 2190271) (-1274 "UTS2.spad" 2187440 2187475 2187835 2187840) (-1273 "URAGG.spad" 2182113 2182124 2187430 2187435) (-1272 "URAGG.spad" 2176750 2176763 2182069 2182074) (-1271 "UPXSSING.spad" 2174395 2174421 2175831 2175964) (-1270 "UPXS.spad" 2171691 2171719 2172527 2172676) (-1269 "UPXSCONS.spad" 2169450 2169470 2169823 2169972) (-1268 "UPXSCCA.spad" 2168021 2168041 2169296 2169445) (-1267 "UPXSCCA.spad" 2166734 2166756 2168011 2168016) (-1266 "UPXSCAT.spad" 2165323 2165339 2166580 2166729) (-1265 "UPXS2.spad" 2164866 2164919 2165313 2165318) (-1264 "UPSQFREE.spad" 2163280 2163294 2164856 2164861) (-1263 "UPSCAT.spad" 2161067 2161091 2163178 2163275) (-1262 "UPSCAT.spad" 2158560 2158586 2160673 2160678) (-1261 "UPOLYC.spad" 2153600 2153611 2158402 2158555) (-1260 "UPOLYC.spad" 2148532 2148545 2153336 2153341) (-1259 "UPOLYC2.spad" 2148003 2148022 2148522 2148527) (-1258 "UP.spad" 2145109 2145124 2145496 2145649) (-1257 "UPMP.spad" 2144009 2144022 2145099 2145104) (-1256 "UPDIVP.spad" 2143574 2143588 2143999 2144004) (-1255 "UPDECOMP.spad" 2141819 2141833 2143564 2143569) (-1254 "UPCDEN.spad" 2141028 2141044 2141809 2141814) (-1253 "UP2.spad" 2140392 2140413 2141018 2141023) (-1252 "UNISEG.spad" 2139745 2139756 2140311 2140316) (-1251 "UNISEG2.spad" 2139242 2139255 2139701 2139706) (-1250 "UNIFACT.spad" 2138345 2138357 2139232 2139237) (-1249 "ULS.spad" 2128129 2128157 2129074 2129503) (-1248 "ULSCONS.spad" 2119263 2119283 2119633 2119782) (-1247 "ULSCCAT.spad" 2117000 2117020 2119109 2119258) (-1246 "ULSCCAT.spad" 2114845 2114867 2116956 2116961) (-1245 "ULSCAT.spad" 2113077 2113093 2114691 2114840) (-1244 "ULS2.spad" 2112591 2112644 2113067 2113072) (-1243 "UINT8.spad" 2112468 2112477 2112581 2112586) (-1242 "UINT64.spad" 2112344 2112353 2112458 2112463) (-1241 "UINT32.spad" 2112220 2112229 2112334 2112339) (-1240 "UINT16.spad" 2112096 2112105 2112210 2112215) (-1239 "UFD.spad" 2111161 2111170 2112022 2112091) (-1238 "UFD.spad" 2110288 2110299 2111151 2111156) (-1237 "UDVO.spad" 2109169 2109178 2110278 2110283) (-1236 "UDPO.spad" 2106662 2106673 2109125 2109130) (-1235 "TYPE.spad" 2106594 2106603 2106652 2106657) (-1234 "TYPEAST.spad" 2106513 2106522 2106584 2106589) (-1233 "TWOFACT.spad" 2105165 2105180 2106503 2106508) (-1232 "TUPLE.spad" 2104651 2104662 2105064 2105069) (-1231 "TUBETOOL.spad" 2101518 2101527 2104641 2104646) (-1230 "TUBE.spad" 2100165 2100182 2101508 2101513) (-1229 "TS.spad" 2098764 2098780 2099730 2099827) (-1228 "TSETCAT.spad" 2085891 2085908 2098732 2098759) (-1227 "TSETCAT.spad" 2073004 2073023 2085847 2085852) (-1226 "TRMANIP.spad" 2067370 2067387 2072710 2072715) (-1225 "TRIMAT.spad" 2066333 2066358 2067360 2067365) (-1224 "TRIGMNIP.spad" 2064860 2064877 2066323 2066328) (-1223 "TRIGCAT.spad" 2064372 2064381 2064850 2064855) (-1222 "TRIGCAT.spad" 2063882 2063893 2064362 2064367) (-1221 "TREE.spad" 2062457 2062468 2063489 2063516) (-1220 "TRANFUN.spad" 2062296 2062305 2062447 2062452) (-1219 "TRANFUN.spad" 2062133 2062144 2062286 2062291) (-1218 "TOPSP.spad" 2061807 2061816 2062123 2062128) (-1217 "TOOLSIGN.spad" 2061470 2061481 2061797 2061802) (-1216 "TEXTFILE.spad" 2060031 2060040 2061460 2061465) (-1215 "TEX.spad" 2057177 2057186 2060021 2060026) (-1214 "TEX1.spad" 2056733 2056744 2057167 2057172) (-1213 "TEMUTL.spad" 2056288 2056297 2056723 2056728) (-1212 "TBCMPPK.spad" 2054381 2054404 2056278 2056283) (-1211 "TBAGG.spad" 2053431 2053454 2054361 2054376) (-1210 "TBAGG.spad" 2052489 2052514 2053421 2053426) (-1209 "TANEXP.spad" 2051897 2051908 2052479 2052484) (-1208 "TALGOP.spad" 2051621 2051632 2051887 2051892) (-1207 "TABLE.spad" 2050032 2050055 2050302 2050329) (-1206 "TABLEAU.spad" 2049513 2049524 2050022 2050027) (-1205 "TABLBUMP.spad" 2046316 2046327 2049503 2049508) (-1204 "SYSTEM.spad" 2045544 2045553 2046306 2046311) (-1203 "SYSSOLP.spad" 2043027 2043038 2045534 2045539) (-1202 "SYSPTR.spad" 2042926 2042935 2043017 2043022) (-1201 "SYSNNI.spad" 2042108 2042119 2042916 2042921) (-1200 "SYSINT.spad" 2041512 2041523 2042098 2042103) (-1199 "SYNTAX.spad" 2037718 2037727 2041502 2041507) (-1198 "SYMTAB.spad" 2035786 2035795 2037708 2037713) (-1197 "SYMS.spad" 2031809 2031818 2035776 2035781) (-1196 "SYMPOLY.spad" 2030816 2030827 2030898 2031025) (-1195 "SYMFUNC.spad" 2030317 2030328 2030806 2030811) (-1194 "SYMBOL.spad" 2027820 2027829 2030307 2030312) (-1193 "SWITCH.spad" 2024591 2024600 2027810 2027815) (-1192 "SUTS.spad" 2021639 2021667 2023058 2023155) (-1191 "SUPXS.spad" 2018922 2018950 2019771 2019920) (-1190 "SUP.spad" 2015642 2015653 2016415 2016568) (-1189 "SUPFRACF.spad" 2014747 2014765 2015632 2015637) (-1188 "SUP2.spad" 2014139 2014152 2014737 2014742) (-1187 "SUMRF.spad" 2013113 2013124 2014129 2014134) (-1186 "SUMFS.spad" 2012750 2012767 2013103 2013108) (-1185 "SULS.spad" 2002521 2002549 2003479 2003908) (-1184 "SUCHTAST.spad" 2002290 2002299 2002511 2002516) (-1183 "SUCH.spad" 2001972 2001987 2002280 2002285) (-1182 "SUBSPACE.spad" 1994087 1994102 2001962 2001967) (-1181 "SUBRESP.spad" 1993257 1993271 1994043 1994048) (-1180 "STTF.spad" 1989356 1989372 1993247 1993252) (-1179 "STTFNC.spad" 1985824 1985840 1989346 1989351) (-1178 "STTAYLOR.spad" 1978459 1978470 1985705 1985710) (-1177 "STRTBL.spad" 1976964 1976981 1977113 1977140) (-1176 "STRING.spad" 1976373 1976382 1976387 1976414) (-1175 "STRICAT.spad" 1976161 1976170 1976341 1976368) (-1174 "STREAM.spad" 1973079 1973090 1975686 1975701) (-1173 "STREAM3.spad" 1972652 1972667 1973069 1973074) (-1172 "STREAM2.spad" 1971780 1971793 1972642 1972647) (-1171 "STREAM1.spad" 1971486 1971497 1971770 1971775) (-1170 "STINPROD.spad" 1970422 1970438 1971476 1971481) (-1169 "STEP.spad" 1969623 1969632 1970412 1970417) (-1168 "STEPAST.spad" 1968857 1968866 1969613 1969618) (-1167 "STBL.spad" 1967383 1967411 1967550 1967565) (-1166 "STAGG.spad" 1966458 1966469 1967373 1967378) (-1165 "STAGG.spad" 1965531 1965544 1966448 1966453) (-1164 "STACK.spad" 1964888 1964899 1965138 1965165) (-1163 "SREGSET.spad" 1962592 1962609 1964534 1964561) (-1162 "SRDCMPK.spad" 1961153 1961173 1962582 1962587) (-1161 "SRAGG.spad" 1956296 1956305 1961121 1961148) (-1160 "SRAGG.spad" 1951459 1951470 1956286 1956291) (-1159 "SQMATRIX.spad" 1949038 1949056 1949954 1950041) (-1158 "SPLTREE.spad" 1943590 1943603 1948474 1948501) (-1157 "SPLNODE.spad" 1940178 1940191 1943580 1943585) (-1156 "SPFCAT.spad" 1938987 1938996 1940168 1940173) (-1155 "SPECOUT.spad" 1937539 1937548 1938977 1938982) (-1154 "SPADXPT.spad" 1929134 1929143 1937529 1937534) (-1153 "spad-parser.spad" 1928599 1928608 1929124 1929129) (-1152 "SPADAST.spad" 1928300 1928309 1928589 1928594) (-1151 "SPACEC.spad" 1912499 1912510 1928290 1928295) (-1150 "SPACE3.spad" 1912275 1912286 1912489 1912494) (-1149 "SORTPAK.spad" 1911824 1911837 1912231 1912236) (-1148 "SOLVETRA.spad" 1909587 1909598 1911814 1911819) (-1147 "SOLVESER.spad" 1908115 1908126 1909577 1909582) (-1146 "SOLVERAD.spad" 1904141 1904152 1908105 1908110) (-1145 "SOLVEFOR.spad" 1902603 1902621 1904131 1904136) (-1144 "SNTSCAT.spad" 1902203 1902220 1902571 1902598) (-1143 "SMTS.spad" 1900475 1900501 1901768 1901865) (-1142 "SMP.spad" 1897950 1897970 1898340 1898467) (-1141 "SMITH.spad" 1896795 1896820 1897940 1897945) (-1140 "SMATCAT.spad" 1894905 1894935 1896739 1896790) (-1139 "SMATCAT.spad" 1892947 1892979 1894783 1894788) (-1138 "SKAGG.spad" 1891910 1891921 1892915 1892942) (-1137 "SINT.spad" 1890850 1890859 1891776 1891905) (-1136 "SIMPAN.spad" 1890578 1890587 1890840 1890845) (-1135 "SIG.spad" 1889908 1889917 1890568 1890573) (-1134 "SIGNRF.spad" 1889026 1889037 1889898 1889903) (-1133 "SIGNEF.spad" 1888305 1888322 1889016 1889021) (-1132 "SIGAST.spad" 1887690 1887699 1888295 1888300) (-1131 "SHP.spad" 1885618 1885633 1887646 1887651) (-1130 "SHDP.spad" 1873821 1873848 1874330 1874429) (-1129 "SGROUP.spad" 1873429 1873438 1873811 1873816) (-1128 "SGROUP.spad" 1873035 1873046 1873419 1873424) (-1127 "SGCF.spad" 1866174 1866183 1873025 1873030) (-1126 "SFRTCAT.spad" 1865104 1865121 1866142 1866169) (-1125 "SFRGCD.spad" 1864167 1864187 1865094 1865099) (-1124 "SFQCMPK.spad" 1858804 1858824 1864157 1864162) (-1123 "SFORT.spad" 1858243 1858257 1858794 1858799) (-1122 "SEXOF.spad" 1858086 1858126 1858233 1858238) (-1121 "SEX.spad" 1857978 1857987 1858076 1858081) (-1120 "SEXCAT.spad" 1855759 1855799 1857968 1857973) (-1119 "SET.spad" 1854083 1854094 1855180 1855219) (-1118 "SETMN.spad" 1852533 1852550 1854073 1854078) (-1117 "SETCAT.spad" 1851855 1851864 1852523 1852528) (-1116 "SETCAT.spad" 1851175 1851186 1851845 1851850) (-1115 "SETAGG.spad" 1847724 1847735 1851155 1851170) (-1114 "SETAGG.spad" 1844281 1844294 1847714 1847719) (-1113 "SEQAST.spad" 1843984 1843993 1844271 1844276) (-1112 "SEGXCAT.spad" 1843140 1843153 1843974 1843979) (-1111 "SEG.spad" 1842953 1842964 1843059 1843064) (-1110 "SEGCAT.spad" 1841878 1841889 1842943 1842948) (-1109 "SEGBIND.spad" 1841636 1841647 1841825 1841830) (-1108 "SEGBIND2.spad" 1841334 1841347 1841626 1841631) (-1107 "SEGAST.spad" 1841048 1841057 1841324 1841329) (-1106 "SEG2.spad" 1840483 1840496 1841004 1841009) (-1105 "SDVAR.spad" 1839759 1839770 1840473 1840478) (-1104 "SDPOL.spad" 1837092 1837103 1837383 1837510) (-1103 "SCPKG.spad" 1835181 1835192 1837082 1837087) (-1102 "SCOPE.spad" 1834334 1834343 1835171 1835176) (-1101 "SCACHE.spad" 1833030 1833041 1834324 1834329) (-1100 "SASTCAT.spad" 1832939 1832948 1833020 1833025) (-1099 "SAOS.spad" 1832811 1832820 1832929 1832934) (-1098 "SAERFFC.spad" 1832524 1832544 1832801 1832806) (-1097 "SAE.spad" 1829994 1830010 1830605 1830740) (-1096 "SAEFACT.spad" 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(-1040 "REALSOLV.spad" 1695544 1695553 1696394 1696399) (-1039 "REAL.spad" 1695416 1695425 1695534 1695539) (-1038 "REAL0Q.spad" 1692714 1692729 1695406 1695411) (-1037 "REAL0.spad" 1689558 1689573 1692704 1692709) (-1036 "RDUCEAST.spad" 1689279 1689288 1689548 1689553) (-1035 "RDIV.spad" 1688934 1688959 1689269 1689274) (-1034 "RDIST.spad" 1688501 1688512 1688924 1688929) (-1033 "RDETRS.spad" 1687365 1687383 1688491 1688496) (-1032 "RDETR.spad" 1685504 1685522 1687355 1687360) (-1031 "RDEEFS.spad" 1684603 1684620 1685494 1685499) (-1030 "RDEEF.spad" 1683613 1683630 1684593 1684598) (-1029 "RCFIELD.spad" 1680799 1680808 1683515 1683608) (-1028 "RCFIELD.spad" 1678071 1678082 1680789 1680794) (-1027 "RCAGG.spad" 1675999 1676010 1678061 1678066) (-1026 "RCAGG.spad" 1673854 1673867 1675918 1675923) (-1025 "RATRET.spad" 1673214 1673225 1673844 1673849) (-1024 "RATFACT.spad" 1672906 1672918 1673204 1673209) (-1023 "RANDSRC.spad" 1672225 1672234 1672896 1672901) (-1022 "RADUTIL.spad" 1671981 1671990 1672215 1672220) (-1021 "RADIX.spad" 1668805 1668819 1670351 1670444) (-1020 "RADFF.spad" 1666544 1666581 1666663 1666819) (-1019 "RADCAT.spad" 1666139 1666148 1666534 1666539) (-1018 "RADCAT.spad" 1665732 1665743 1666129 1666134) (-1017 "QUEUE.spad" 1665080 1665091 1665339 1665366) (-1016 "QUAT.spad" 1663568 1663579 1663911 1663976) (-1015 "QUATCT2.spad" 1663188 1663207 1663558 1663563) (-1014 "QUATCAT.spad" 1661358 1661369 1663118 1663183) (-1013 "QUATCAT.spad" 1659279 1659292 1661041 1661046) (-1012 "QUAGG.spad" 1658106 1658117 1659247 1659274) (-1011 "QQUTAST.spad" 1657874 1657883 1658096 1658101) (-1010 "QFORM.spad" 1657492 1657507 1657864 1657869) (-1009 "QFCAT.spad" 1656194 1656205 1657394 1657487) (-1008 "QFCAT.spad" 1654487 1654500 1655689 1655694) (-1007 "QFCAT2.spad" 1654179 1654196 1654477 1654482) (-1006 "QEQUAT.spad" 1653737 1653746 1654169 1654174) (-1005 "QCMPACK.spad" 1648483 1648503 1653727 1653732) (-1004 "QALGSET.spad" 1644561 1644594 1648397 1648402) (-1003 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1587628 1587655) (-984 "PROPLOG.spad" 1586667 1586675 1587085 1587090) (-983 "PROPFUN2.spad" 1586290 1586303 1586657 1586662) (-982 "PROPFUN1.spad" 1585688 1585699 1586280 1586285) (-981 "PROPFRML.spad" 1584256 1584267 1585678 1585683) (-980 "PROPERTY.spad" 1583744 1583752 1584246 1584251) (-979 "PRODUCT.spad" 1581426 1581438 1581710 1581765) (-978 "PR.spad" 1579818 1579830 1580517 1580644) (-977 "PRINT.spad" 1579570 1579578 1579808 1579813) (-976 "PRIMES.spad" 1577823 1577833 1579560 1579565) (-975 "PRIMELT.spad" 1575904 1575918 1577813 1577818) (-974 "PRIMCAT.spad" 1575531 1575539 1575894 1575899) (-973 "PRIMARR.spad" 1574536 1574546 1574714 1574741) (-972 "PRIMARR2.spad" 1573303 1573315 1574526 1574531) (-971 "PREASSOC.spad" 1572685 1572697 1573293 1573298) (-970 "PPCURVE.spad" 1571822 1571830 1572675 1572680) (-969 "PORTNUM.spad" 1571597 1571605 1571812 1571817) (-968 "POLYROOT.spad" 1570446 1570468 1571553 1571558) (-967 "POLY.spad" 1567781 1567791 1568296 1568423) (-966 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1538802 1538807) (-947 "PMFS.spad" 1537969 1537987 1538382 1538387) (-946 "PMDOWN.spad" 1537259 1537273 1537959 1537964) (-945 "PMASS.spad" 1536269 1536277 1537249 1537254) (-944 "PMASSFS.spad" 1535236 1535252 1536259 1536264) (-943 "PLOTTOOL.spad" 1535016 1535024 1535226 1535231) (-942 "PLOT.spad" 1529939 1529947 1535006 1535011) (-941 "PLOT3D.spad" 1526403 1526411 1529929 1529934) (-940 "PLOT1.spad" 1525560 1525570 1526393 1526398) (-939 "PLEQN.spad" 1512850 1512877 1525550 1525555) (-938 "PINTERP.spad" 1512472 1512491 1512840 1512845) (-937 "PINTERPA.spad" 1512256 1512272 1512462 1512467) (-936 "PI.spad" 1511865 1511873 1512230 1512251) (-935 "PID.spad" 1510835 1510843 1511791 1511860) (-934 "PICOERCE.spad" 1510492 1510502 1510825 1510830) (-933 "PGROEB.spad" 1509093 1509107 1510482 1510487) (-932 "PGE.spad" 1500710 1500718 1509083 1509088) (-931 "PGCD.spad" 1499600 1499617 1500700 1500705) (-930 "PFRPAC.spad" 1498749 1498759 1499590 1499595) (-929 "PFR.spad" 1495412 1495422 1498651 1498744) (-928 "PFOTOOLS.spad" 1494670 1494686 1495402 1495407) (-927 "PFOQ.spad" 1494040 1494058 1494660 1494665) (-926 "PFO.spad" 1493459 1493486 1494030 1494035) (-925 "PF.spad" 1493033 1493045 1493264 1493357) (-924 "PFECAT.spad" 1490715 1490723 1492959 1493028) (-923 "PFECAT.spad" 1488425 1488435 1490671 1490676) (-922 "PFBRU.spad" 1486313 1486325 1488415 1488420) (-921 "PFBR.spad" 1483873 1483896 1486303 1486308) (-920 "PERM.spad" 1479680 1479690 1483703 1483718) (-919 "PERMGRP.spad" 1474450 1474460 1479670 1479675) (-918 "PERMCAT.spad" 1473111 1473121 1474430 1474445) (-917 "PERMAN.spad" 1471643 1471657 1473101 1473106) (-916 "PENDTREE.spad" 1470984 1470994 1471272 1471277) (-915 "PDSPC.spad" 1469797 1469807 1470974 1470979) (-914 "PDSPC.spad" 1468608 1468620 1469787 1469792) (-913 "PDRING.spad" 1468450 1468460 1468588 1468603) (-912 "PDEPROB.spad" 1467465 1467473 1468440 1468445) (-911 "PDEPACK.spad" 1461505 1461513 1467455 1467460) (-910 "PDECOMP.spad" 1460975 1460992 1461495 1461500) (-909 "PDECAT.spad" 1459331 1459339 1460965 1460970) (-908 "PDDOM.spad" 1458769 1458782 1459321 1459326) (-907 "PDDOM.spad" 1458205 1458220 1458759 1458764) (-906 "PCOMP.spad" 1458058 1458071 1458195 1458200) (-905 "PBWLB.spad" 1456646 1456663 1458048 1458053) (-904 "PATTERN.spad" 1451185 1451195 1456636 1456641) (-903 "PATTERN2.spad" 1450923 1450935 1451175 1451180) (-902 "PATTERN1.spad" 1449259 1449275 1450913 1450918) (-901 "PATRES.spad" 1446834 1446846 1449249 1449254) (-900 "PATRES2.spad" 1446506 1446520 1446824 1446829) (-899 "PATMATCH.spad" 1444703 1444734 1446214 1446219) (-898 "PATMAB.spad" 1444132 1444142 1444693 1444698) (-897 "PATLRES.spad" 1443218 1443232 1444122 1444127) (-896 "PATAB.spad" 1442982 1442992 1443208 1443213) (-895 "PARTPERM.spad" 1440990 1440998 1442972 1442977) (-894 "PARSURF.spad" 1440424 1440452 1440980 1440985) (-893 "PARSU2.spad" 1440221 1440237 1440414 1440419) (-892 "script-parser.spad" 1439741 1439749 1440211 1440216) (-891 "PARSCURV.spad" 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1388236) (-853 "ORDCOMP2.spad" 1386418 1386430 1387115 1387120) (-852 "OPTPROB.spad" 1385056 1385064 1386408 1386413) (-851 "OPTPACK.spad" 1377465 1377473 1385046 1385051) (-850 "OPTCAT.spad" 1375144 1375152 1377455 1377460) (-849 "OPSIG.spad" 1374798 1374806 1375134 1375139) (-848 "OPQUERY.spad" 1374347 1374355 1374788 1374793) (-847 "OP.spad" 1374089 1374099 1374169 1374236) (-846 "OPERCAT.spad" 1373555 1373565 1374079 1374084) (-845 "OPERCAT.spad" 1373019 1373031 1373545 1373550) (-844 "ONECOMP.spad" 1371764 1371774 1372566 1372595) (-843 "ONECOMP2.spad" 1371188 1371200 1371754 1371759) (-842 "OMSERVER.spad" 1370194 1370202 1371178 1371183) (-841 "OMSAGG.spad" 1369982 1369992 1370150 1370189) (-840 "OMPKG.spad" 1368598 1368606 1369972 1369977) (-839 "OM.spad" 1367571 1367579 1368588 1368593) (-838 "OMLO.spad" 1366996 1367008 1367457 1367496) (-837 "OMEXPR.spad" 1366830 1366840 1366986 1366991) (-836 "OMERR.spad" 1366375 1366383 1366820 1366825) (-835 "OMERRK.spad" 1365409 1365417 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228753 228758) (-197 "D01ALFA.spad" 227813 227821 228263 228268) (-196 "D01AKFA.spad" 227339 227347 227803 227808) (-195 "D01AJFA.spad" 226862 226870 227329 227334) (-194 "D01AGNT.spad" 222929 222937 226852 226857) (-193 "CYCLOTOM.spad" 222435 222443 222919 222924) (-192 "CYCLES.spad" 219227 219235 222425 222430) (-191 "CVMP.spad" 218644 218654 219217 219222) (-190 "CTRIGMNP.spad" 217144 217160 218634 218639) (-189 "CTOR.spad" 216835 216843 217134 217139) (-188 "CTORKIND.spad" 216438 216446 216825 216830) (-187 "CTORCAT.spad" 215687 215695 216428 216433) (-186 "CTORCAT.spad" 214934 214944 215677 215682) (-185 "CTORCALL.spad" 214523 214533 214924 214929) (-184 "CSTTOOLS.spad" 213768 213781 214513 214518) (-183 "CRFP.spad" 207492 207505 213758 213763) (-182 "CRCEAST.spad" 207212 207220 207482 207487) (-181 "CRAPACK.spad" 206263 206273 207202 207207) (-180 "CPMATCH.spad" 205767 205782 206188 206193) (-179 "CPIMA.spad" 205472 205491 205757 205762) (-178 "COORDSYS.spad" 200481 200491 205462 205467) (-177 "CONTOUR.spad" 199892 199900 200471 200476) (-176 "CONTFRAC.spad" 195642 195652 199794 199887) (-175 "CONDUIT.spad" 195400 195408 195632 195637) (-174 "COMRING.spad" 195074 195082 195338 195395) (-173 "COMPPROP.spad" 194592 194600 195064 195069) (-172 "COMPLPAT.spad" 194359 194374 194582 194587) (-171 "COMPLEX.spad" 189736 189746 189980 190241) (-170 "COMPLEX2.spad" 189451 189463 189726 189731) (-169 "COMPILER.spad" 189000 189008 189441 189446) (-168 "COMPFACT.spad" 188602 188616 188990 188995) (-167 "COMPCAT.spad" 186674 186684 188336 188597) (-166 "COMPCAT.spad" 184474 184486 186138 186143) (-165 "COMMUPC.spad" 184222 184240 184464 184469) (-164 "COMMONOP.spad" 183755 183763 184212 184217) (-163 "COMM.spad" 183566 183574 183745 183750) (-162 "COMMAAST.spad" 183329 183337 183556 183561) (-161 "COMBOPC.spad" 182244 182252 183319 183324) (-160 "COMBINAT.spad" 181011 181021 182234 182239) (-159 "COMBF.spad" 178393 178409 181001 181006) (-158 "COLOR.spad" 177230 177238 178383 178388) (-157 "COLONAST.spad" 176896 176904 177220 177225) (-156 "CMPLXRT.spad" 176607 176624 176886 176891) (-155 "CLLCTAST.spad" 176269 176277 176597 176602) (-154 "CLIP.spad" 172377 172385 176259 176264) (-153 "CLIF.spad" 171032 171048 172333 172372) (-152 "CLAGG.spad" 167537 167547 171022 171027) (-151 "CLAGG.spad" 163913 163925 167400 167405) (-150 "CINTSLPE.spad" 163244 163257 163903 163908) (-149 "CHVAR.spad" 161382 161404 163234 163239) (-148 "CHARZ.spad" 161297 161305 161362 161377) (-147 "CHARPOL.spad" 160807 160817 161287 161292) (-146 "CHARNZ.spad" 160560 160568 160787 160802) (-145 "CHAR.spad" 158434 158442 160550 160555) (-144 "CFCAT.spad" 157762 157770 158424 158429) (-143 "CDEN.spad" 156958 156972 157752 157757) (-142 "CCLASS.spad" 155107 155115 156369 156408) (-141 "CATEGORY.spad" 154149 154157 155097 155102) (-140 "CATCTOR.spad" 154040 154048 154139 154144) (-139 "CATAST.spad" 153658 153666 154030 154035) (-138 "CASEAST.spad" 153372 153380 153648 153653) (-137 "CARTEN.spad" 148739 148763 153362 153367) (-136 "CARTEN2.spad" 148129 148156 148729 148734) (-135 "CARD.spad" 145424 145432 148103 148124) (-134 "CAPSLAST.spad" 145198 145206 145414 145419) (-133 "CACHSET.spad" 144822 144830 145188 145193) (-132 "CABMON.spad" 144377 144385 144812 144817) (-131 "BYTEORD.spad" 144052 144060 144367 144372) (-130 "BYTE.spad" 143479 143487 144042 144047) (-129 "BYTEBUF.spad" 141338 141346 142648 142675) (-128 "BTREE.spad" 140411 140421 140945 140972) (-127 "BTOURN.spad" 139416 139426 140018 140045) (-126 "BTCAT.spad" 138808 138818 139384 139411) (-125 "BTCAT.spad" 138220 138232 138798 138803) (-124 "BTAGG.spad" 137686 137694 138188 138215) (-123 "BTAGG.spad" 137172 137182 137676 137681) (-122 "BSTREE.spad" 135913 135923 136779 136806) (-121 "BRILL.spad" 134110 134121 135903 135908) (-120 "BRAGG.spad" 133050 133060 134100 134105) (-119 "BRAGG.spad" 131954 131966 133006 133011) (-118 "BPADICRT.spad" 129828 129840 130083 130176) (-117 "BPADIC.spad" 129492 129504 129754 129823) (-116 "BOUNDZRO.spad" 129148 129165 129482 129487) (-115 "BOP.spad" 124330 124338 129138 129143) (-114 "BOP1.spad" 121796 121806 124320 124325) (-113 "BOOLE.spad" 121446 121454 121786 121791) (-112 "BOOLEAN.spad" 120884 120892 121436 121441) (-111 "BMODULE.spad" 120596 120608 120852 120879) (-110 "BITS.spad" 120017 120025 120232 120259) (-109 "BINDING.spad" 119430 119438 120007 120012) (-108 "BINARY.spad" 117444 117452 117800 117893) (-107 "BGAGG.spad" 116649 116659 117424 117439) (-106 "BGAGG.spad" 115862 115874 116639 116644) (-105 "BFUNCT.spad" 115426 115434 115842 115857) (-104 "BEZOUT.spad" 114566 114593 115376 115381) (-103 "BBTREE.spad" 111411 111421 114173 114200) (-102 "BASTYPE.spad" 111083 111091 111401 111406) (-101 "BASTYPE.spad" 110753 110763 111073 111078) (-100 "BALFACT.spad" 110212 110225 110743 110748) (-99 "AUTOMOR.spad" 109663 109672 110192 110207) (-98 "ATTREG.spad" 106386 106393 109415 109658) (-97 "ATTRBUT.spad" 102409 102416 106366 106381) (-96 "ATTRAST.spad" 102126 102133 102399 102404) (-95 "ATRIG.spad" 101596 101603 102116 102121) (-94 "ATRIG.spad" 101064 101073 101586 101591) (-93 "ASTCAT.spad" 100968 100975 101054 101059) (-92 "ASTCAT.spad" 100870 100879 100958 100963) (-91 "ASTACK.spad" 100209 100218 100477 100504) (-90 "ASSOCEQ.spad" 99035 99046 100165 100170) (-89 "ASP9.spad" 98116 98129 99025 99030) (-88 "ASP8.spad" 97159 97172 98106 98111) (-87 "ASP80.spad" 96481 96494 97149 97154) (-86 "ASP7.spad" 95641 95654 96471 96476) (-85 "ASP78.spad" 95092 95105 95631 95636) (-84 "ASP77.spad" 94461 94474 95082 95087) (-83 "ASP74.spad" 93553 93566 94451 94456) (-82 "ASP73.spad" 92824 92837 93543 93548) (-81 "ASP6.spad" 91691 91704 92814 92819) (-80 "ASP55.spad" 90200 90213 91681 91686) (-79 "ASP50.spad" 88017 88030 90190 90195) (-78 "ASP4.spad" 87312 87325 88007 88012) (-77 "ASP49.spad" 86311 86324 87302 87307) (-76 "ASP42.spad" 84718 84757 86301 86306) (-75 "ASP41.spad" 83297 83336 84708 84713) (-74 "ASP35.spad" 82285 82298 83287 83292) (-73 "ASP34.spad" 81586 81599 82275 82280) (-72 "ASP33.spad" 81146 81159 81576 81581) (-71 "ASP31.spad" 80286 80299 81136 81141) (-70 "ASP30.spad" 79178 79191 80276 80281) (-69 "ASP29.spad" 78644 78657 79168 79173) (-68 "ASP28.spad" 69917 69930 78634 78639) (-67 "ASP27.spad" 68814 68827 69907 69912) (-66 "ASP24.spad" 67901 67914 68804 68809) (-65 "ASP20.spad" 67365 67378 67891 67896) (-64 "ASP1.spad" 66746 66759 67355 67360) (-63 "ASP19.spad" 61432 61445 66736 66741) (-62 "ASP12.spad" 60846 60859 61422 61427) (-61 "ASP10.spad" 60117 60130 60836 60841) (-60 "ARRAY2.spad" 59477 59486 59724 59751) (-59 "ARRAY1.spad" 58314 58323 58660 58687) (-58 "ARRAY12.spad" 57027 57038 58304 58309) (-57 "ARR2CAT.spad" 52801 52822 56995 57022) (-56 "ARR2CAT.spad" 48595 48618 52791 52796) (-55 "ARITY.spad" 47967 47974 48585 48590) (-54 "APPRULE.spad" 47227 47249 47957 47962) (-53 "APPLYORE.spad" 46846 46859 47217 47222) (-52 "ANY.spad" 45705 45712 46836 46841) (-51 "ANY1.spad" 44776 44785 45695 45700) (-50 "ANTISYM.spad" 43221 43237 44756 44771) (-49 "ANON.spad" 42914 42921 43211 43216) (-48 "AN.spad" 41223 41230 42730 42823) (-47 "AMR.spad" 39408 39419 41121 41218) (-46 "AMR.spad" 37430 37443 39145 39150) (-45 "ALIST.spad" 34842 34863 35192 35219) (-44 "ALGSC.spad" 33977 34003 34714 34767) (-43 "ALGPKG.spad" 29760 29771 33933 33938) (-42 "ALGMFACT.spad" 28953 28967 29750 29755) (-41 "ALGMANIP.spad" 26427 26442 28786 28791) (-40 "ALGFF.spad" 24068 24095 24285 24441) (-39 "ALGFACT.spad" 23195 23205 24058 24063) (-38 "ALGEBRA.spad" 23028 23037 23151 23190) (-37 "ALGEBRA.spad" 22893 22904 23018 23023) (-36 "ALAGG.spad" 22405 22426 22861 22888) (-35 "AHYP.spad" 21786 21793 22395 22400) (-34 "AGG.spad" 20103 20110 21776 21781) (-33 "AGG.spad" 18384 18393 20059 20064) (-32 "AF.spad" 16815 16830 18319 18324) (-31 "ADDAST.spad" 16493 16500 16805 16810) (-30 "ACPLOT.spad" 15084 15091 16483 16488) (-29 "ACFS.spad" 12893 12902 14986 15079) (-28 "ACFS.spad" 10788 10799 12883 12888) (-27 "ACF.spad" 7470 7477 10690 10783) (-26 "ACF.spad" 4238 4247 7460 7465) (-25 "ABELSG.spad" 3779 3786 4228 4233) (-24 "ABELSG.spad" 3318 3327 3769 3774) (-23 "ABELMON.spad" 2861 2868 3308 3313) (-22 "ABELMON.spad" 2402 2411 2851 2856) (-21 "ABELGRP.spad" 2067 2074 2392 2397) (-20 "ABELGRP.spad" 1730 1739 2057 2062) (-19 "A1AGG.spad" 870 879 1698 1725) (-18 "A1AGG.spad" 30 41 860 865))
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