\documentclass{article} \usepackage{axiom} \begin{document} \title{\$SPAD/src/algebra fspace.spad} \author{Manuel Bronstein} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{category ES ExpressionSpace} <>= )abbrev category ES ExpressionSpace ++ Category for domains on which operators can be applied ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 27 May 1994 ++ Description: ++ An expression space is a set which is closed under certain operators; ++ Keywords: operator, kernel, expression, space. ExpressionSpace(): Category == Defn where N ==> NonNegativeInteger K ==> Kernel % OP ==> BasicOperator SY ==> Symbol Defn ==> Join(SetCategory, RetractableTo K, InnerEvalable(K, %), Evalable %) with elt : (OP, %) -> % ++ elt(op,x) or op(x) applies the unary operator op to x. elt : (OP, %, %) -> % ++ elt(op,x,y) or op(x, y) applies the binary operator op to x and y. elt : (OP, %, %, %) -> % ++ elt(op,x,y,z) or op(x, y, z) applies the ternary operator op to x, y and z. elt : (OP, %, %, %, %) -> % ++ elt(op,x,y,z,t) or op(x, y, z, t) applies the 4-ary operator op to x, y, z and t. elt : (OP, List %) -> % ++ elt(op,[x1,...,xn]) or op([x1,...,xn]) applies the n-ary operator op to x1,...,xn. subst : (%, Equation %) -> % ++ subst(f, k = g) replaces the kernel k by g formally in f. subst : (%, List Equation %) -> % ++ subst(f, [k1 = g1,...,kn = gn]) replaces the kernels k1,...,kn ++ by g1,...,gn formally in f. subst : (%, List K, List %) -> % ++ subst(f, [k1...,kn], [g1,...,gn]) replaces the kernels k1,...,kn ++ by g1,...,gn formally in f. box : % -> % ++ box(f) returns f with a 'box' around it that prevents f from ++ being evaluated when operators are applied to it. For example, ++ \spad{log(1)} returns 0, but \spad{log(box 1)} ++ returns the formal kernel log(1). box : List % -> % ++ box([f1,...,fn]) returns \spad{(f1,...,fn)} with a 'box' ++ around them that ++ prevents the fi from being evaluated when operators are applied to ++ them, and makes them applicable to a unary operator. For example, ++ \spad{atan(box [x, 2])} returns the formal kernel \spad{atan(x, 2)}. paren : % -> % ++ paren(f) returns (f). This prevents f from ++ being evaluated when operators are applied to it. For example, ++ \spad{log(1)} returns 0, but \spad{log(paren 1)} returns the ++ formal kernel log((1)). paren : List % -> % ++ paren([f1,...,fn]) returns \spad{(f1,...,fn)}. This ++ prevents the fi from being evaluated when operators are applied to ++ them, and makes them applicable to a unary operator. For example, ++ \spad{atan(paren [x, 2])} returns the formal ++ kernel \spad{atan((x, 2))}. distribute : % -> % ++ distribute(f) expands all the kernels in f that are ++ formally enclosed by a \spadfunFrom{box}{ExpressionSpace} ++ or \spadfunFrom{paren}{ExpressionSpace} expression. distribute : (%, %) -> % ++ distribute(f, g) expands all the kernels in f that contain g in their ++ arguments and that are formally ++ enclosed by a \spadfunFrom{box}{ExpressionSpace} ++ or a \spadfunFrom{paren}{ExpressionSpace} expression. height : % -> N ++ height(f) returns the highest nesting level appearing in f. ++ Constants have height 0. Symbols have height 1. For any ++ operator op and expressions f1,...,fn, \spad{op(f1,...,fn)} has ++ height equal to \spad{1 + max(height(f1),...,height(fn))}. mainKernel : % -> Union(K, "failed") ++ mainKernel(f) returns a kernel of f with maximum nesting level, or ++ if f has no kernels (i.e. f is a constant). kernels : % -> List K ++ kernels(f) returns the list of all the top-level kernels ++ appearing in f, but not the ones appearing in the arguments ++ of the top-level kernels. tower : % -> List K ++ tower(f) returns all the kernels appearing in f, no matter ++ what their levels are. operators : % -> List OP ++ operators(f) returns all the basic operators appearing in f, ++ no matter what their levels are. operator : OP -> OP ++ operator(op) returns a copy of op with the domain-dependent ++ properties appropriate for %. belong? : OP -> Boolean ++ belong?(op) tests if % accepts op as applicable to its ++ elements. is? : (%, OP) -> Boolean ++ is?(x, op) tests if x is a kernel and is its operator is op. is? : (%, SY) -> Boolean ++ is?(x, s) tests if x is a kernel and is the name of its ++ operator is s. kernel : (OP, %) -> % ++ kernel(op, x) constructs op(x) without evaluating it. kernel : (OP, List %) -> % ++ kernel(op, [f1,...,fn]) constructs \spad{op(f1,...,fn)} without ++ evaluating it. map : (% -> %, K) -> % ++ map(f, k) returns \spad{op(f(x1),...,f(xn))} where ++ \spad{k = op(x1,...,xn)}. freeOf? : (%, %) -> Boolean ++ freeOf?(x, y) tests if x does not contain any occurrence of y, ++ where y is a single kernel. freeOf? : (%, SY) -> Boolean ++ freeOf?(x, s) tests if x does not contain any operator ++ whose name is s. eval : (%, List SY, List(% -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a)} in x by \spad{fi(a)} for any \spad{a}. eval : (%, List SY, List(List % -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a1,...,an)} in x by ++ \spad{fi(a1,...,an)} for any \spad{a1},...,\spad{an}. eval : (%, SY, List % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a1,..,am)} in x ++ by \spad{f(a1,..,am)} for any \spad{a1},...,\spad{am}. eval : (%, SY, % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a)} in x by \spad{f(a)} ++ for any \spad{a}. eval : (%, List OP, List(% -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a)} in x by \spad{fi(a)} for any \spad{a}. eval : (%, List OP, List(List % -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a1,...,an)} in x by ++ \spad{fi(a1,...,an)} for any \spad{a1},...,\spad{an}. eval : (%, OP, List % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a1,..,am)} in x ++ by \spad{f(a1,..,am)} for any \spad{a1},...,\spad{am}. eval : (%, OP, % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a)} in x by \spad{f(a)} ++ for any \spad{a}. if % has Ring then minPoly: K -> SparseUnivariatePolynomial % ++ minPoly(k) returns p such that \spad{p(k) = 0}. definingPolynomial: % -> % ++ definingPolynomial(x) returns an expression p such that ++ \spad{p(x) = 0}. if % has RetractableTo Integer then even?: % -> Boolean ++ even? x is true if x is an even integer. odd? : % -> Boolean ++ odd? x is true if x is an odd integer. add -- the 7 functions not provided are: -- kernels minPoly definingPolynomial -- coerce:K -> % eval:(%, List K, List %) -> % -- subst:(%, List K, List %) -> % -- eval:(%, List Symbol, List(List % -> %)) -> % macro PAREN == '%paren macro BOX == '%box macro DUMMYVAR == '%dummyVar allKernels: % -> List K allk : List % -> List K unwrap : (List K, %) -> % okkernel : (OP, List %) -> % mkKerLists: List Equation % -> Record(lstk: List K, lstv:List %) oppren := operator(PAREN)$CommonOperators() opbox := operator(BOX)$CommonOperators() box(x:%) == box [x] paren(x:%) == paren [x] belong? op == op = oppren or op = opbox tower f == sort! allKernels f allk l == reduce("setUnion", [allKernels f for f in l], nil$List(K)) operators f == [operator k for k in allKernels f] height f == reduce("max", [height k for k in kernels f], 0) freeOf?(x:%, s:SY) == not member?(s, [name operator k for k in allKernels x]) distribute x == unwrap([k for k in allKernels x | is?(k, oppren)], x) box(l:List %) == opbox l paren(l:List %) == oppren l freeOf?(x:%, k:%) == not member?(retract k, allKernels x) kernel(op:OP, arg:%) == kernel(op, [arg]) elt(op:OP, x:%) == op [x] elt(op:OP, x:%, y:%) == op [x, y] elt(op:OP, x:%, y:%, z:%) == op [x, y, z] elt(op:OP, x:%, y:%, z:%, t:%) == op [x, y, z, t] eval(x:%, s:SY, f:List % -> %) == eval(x, [s], [f]) eval(x:%, s:OP, f:List % -> %) == eval(x, [name s], [f]) eval(x:%, s:SY, f:% -> %) == eval(x, [s], [f first #1]) eval(x:%, s:OP, f:% -> %) == eval(x, [s], [f first #1]) subst(x:%, e:Equation %) == subst(x, [e]) eval(x:%, ls:List OP, lf:List(% -> %)) == eval(x, ls, [f first #1 for f in lf]$List(List % -> %)) eval(x:%, ls:List SY, lf:List(% -> %)) == eval(x, ls, [f first #1 for f in lf]$List(List % -> %)) eval(x:%, ls:List OP, lf:List(List % -> %)) == eval(x, [name s for s in ls]$List(SY), lf) map(fn, k) == (l := [fn x for x in argument k]$List(%)) = argument k => k::% (operator k) l operator op == is?(op, PAREN) => oppren is?(op, BOX) => opbox error "Unknown operator" mainKernel x == empty?(l := kernels x) => "failed" n := height(k := first l) for kk in rest l repeat if height(kk) > n then n := height kk k := kk k -- takes all the kernels except for the dummy variables, which are second -- arguments of rootOf's, integrals, sums and products which appear only in -- their first arguments allKernels f == s := removeDuplicates(l := kernels f) for k in l repeat t := (u := property(operator k, DUMMYVAR)) case None => arg := argument k s0 := remove!(retract(second arg)@K, allKernels first arg) arg := rest rest arg n := (u::None) pretend N if n > 1 then arg := rest arg setUnion(s0, allk arg) allk argument k s := setUnion(s, t) s kernel(op:OP, args:List %) == not belong? op => error "Unknown operator" okkernel(op, args) okkernel(op, l) == kernel(op, l, 1 + reduce("max", [height f for f in l], 0))$K :: % elt(op:OP, args:List %) == not belong? op => error "Unknown operator" (#args)::Arity ~= arity op and (arity op ~= arbitrary()) => error "Wrong number of arguments" (v := evaluate(op,args)$BasicOperatorFunctions1(%)) case % => v::% okkernel(op, args) retract f == (k := mainKernel f) case "failed" => error "not a kernel" k::K::% ~= f => error "not a kernel" k::K retractIfCan f == (k := mainKernel f) case "failed" => "failed" k::K::% ~= f => "failed" k is?(f:%, s:SY) == (k := retractIfCan f) case "failed" => false is?(k::K, s) is?(f:%, op:OP) == (k := retractIfCan f) case "failed" => false is?(k::K, op) unwrap(l, x) == for k in reverse_! l repeat x := eval(x, k, first argument k) x distribute(x, y) == ky := retract y unwrap([k for k in allKernels x | is?(k, '%paren) and member?(ky, allKernels(k::%))], x) -- in case of conflicting substitutions e.g. [x = a, x = b], -- the first one prevails. -- this is not part of the semantics of the function, but just -- a feature of this implementation. eval(f:%, leq:List Equation %) == rec := mkKerLists leq eval(f, rec.lstk, rec.lstv) subst(f:%, leq:List Equation %) == rec := mkKerLists leq subst(f, rec.lstk, rec.lstv) mkKerLists leq == lk := empty()$List(K) lv := empty()$List(%) for eq in leq repeat (k := retractIfCan(lhs eq)@Union(K, "failed")) case "failed" => error "left hand side must be a single kernel" if not member?(k::K, lk) then lk := concat(k::K, lk) lv := concat(rhs eq, lv) [lk, lv] if % has RetractableTo Integer then intpred?: (%, Integer -> Boolean) -> Boolean even? x == intpred?(x, even?) odd? x == intpred?(x, odd?) intpred?(x, pred?) == (u := retractIfCan(x)@Union(Integer, "failed")) case Integer and pred?(u::Integer) @ \section{package ES1 ExpressionSpaceFunctions1} <>= )abbrev package ES1 ExpressionSpaceFunctions1 ++ Lifting of maps from expression spaces to kernels over them ++ Author: Manuel Bronstein ++ Date Created: 23 March 1988 ++ Date Last Updated: 19 April 1991 ++ Description: ++ This package allows a map from any expression space into any object ++ to be lifted to a kernel over the expression set, using a given ++ property of the operator of the kernel. -- should not be exposed ExpressionSpaceFunctions1(F:ExpressionSpace, S:Type): with map: (F -> S, String, Kernel F) -> S ++ map(f, p, k) uses the property p of the operator ++ of k, in order to lift f and apply it to k. == add -- prop contains an evaluation function List S -> S map(F2S, prop, k) == args := [F2S x for x in argument k]$List(S) (p := property(operator k, prop)) case None => ((p::None) pretend (List S -> S)) args error "Operator does not have required property" @ \section{package ES2 ExpressionSpaceFunctions2} <>= )abbrev package ES2 ExpressionSpaceFunctions2 ++ Lifting of maps from expression spaces to kernels over them ++ Author: Manuel Bronstein ++ Date Created: 23 March 1988 ++ Date Last Updated: 19 April 1991 ++ Description: ++ This package allows a mapping E -> F to be lifted to a kernel over E; ++ This lifting can fail if the operator of the kernel cannot be applied ++ in F; Do not use this package with E = F, since this may ++ drop some properties of the operators. ExpressionSpaceFunctions2(E:ExpressionSpace, F:ExpressionSpace): with map: (E -> F, Kernel E) -> F ++ map(f, k) returns \spad{g = op(f(a1),...,f(an))} where ++ \spad{k = op(a1,...,an)}. == add map(f, k) == (operator(operator k)$F) [f x for x in argument k]$List(F) @ \section{category FS FunctionSpace} <>= )abbrev category FS FunctionSpace ++ Category for formal functions ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 14 February 1994 ++ Description: ++ A space of formal functions with arguments in an arbitrary ++ ordered set. ++ Keywords: operator, kernel, function. FunctionSpace(R: SetCategory): Category == Definition where OP ==> BasicOperator O ==> OutputForm SY ==> Symbol N ==> NonNegativeInteger Z ==> Integer K ==> Kernel % Q ==> Fraction R PR ==> Polynomial R MP ==> SparseMultivariatePolynomial(R, K) QF==> PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,MP,%) Definition ==> Join(ExpressionSpace, RetractableTo SY, Patternable R, FullyPatternMatchable R, FullyRetractableTo R) with ground? : % -> Boolean ++ ground?(f) tests if f is an element of R. ground : % -> R ++ ground(f) returns f as an element of R. ++ An error occurs if f is not an element of R. variables : % -> List SY ++ variables(f) returns the list of all the variables of f. applyQuote: (SY, %) -> % ++ applyQuote(foo, x) returns \spad{'foo(x)}. applyQuote: (SY, %, %) -> % ++ applyQuote(foo, x, y) returns \spad{'foo(x,y)}. applyQuote: (SY, %, %, %) -> % ++ applyQuote(foo, x, y, z) returns \spad{'foo(x,y,z)}. applyQuote: (SY, %, %, %, %) -> % ++ applyQuote(foo, x, y, z, t) returns \spad{'foo(x,y,z,t)}. applyQuote: (SY, List %) -> % ++ applyQuote(foo, [x1,...,xn]) returns \spad{'foo(x1,...,xn)}. if R has ConvertibleTo InputForm then ConvertibleTo InputForm eval : (%, SY) -> % ++ eval(f, foo) unquotes all the foo's in f. eval : (%, List SY) -> % ++ eval(f, [foo1,...,foon]) unquotes all the \spad{fooi}'s in f. eval : % -> % ++ eval(f) unquotes all the quoted operators in f. eval : (%, OP, %, SY) -> % ++ eval(x, s, f, y) replaces every \spad{s(a)} in x by \spad{f(y)} ++ with \spad{y} replaced by \spad{a} for any \spad{a}. eval : (%, List OP, List %, SY) -> % ++ eval(x, [s1,...,sm], [f1,...,fm], y) replaces every ++ \spad{si(a)} in x by \spad{fi(y)} ++ with \spad{y} replaced by \spad{a} for any \spad{a}. if R has SemiGroup then Monoid -- the following line is necessary because of a compiler bug ** : (%, N) -> % ++ x**n returns x * x * x * ... * x (n times). isTimes: % -> Union(List %, "failed") ++ isTimes(p) returns \spad{[a1,...,an]} ++ if \spad{p = a1*...*an} and \spad{n > 1}. isExpt : % -> Union(Record(var:K,exponent:Z),"failed") ++ isExpt(p) returns \spad{[x, n]} if \spad{p = x**n} ++ and \spad{n <> 0}. if R has Group then Group if R has AbelianSemiGroup then AbelianMonoid isPlus: % -> Union(List %, "failed") ++ isPlus(p) returns \spad{[m1,...,mn]} ++ if \spad{p = m1 +...+ mn} and \spad{n > 1}. isMult: % -> Union(Record(coef:Z, var:K),"failed") ++ isMult(p) returns \spad{[n, x]} if \spad{p = n * x} ++ and \spad{n <> 0}. if R has AbelianGroup then AbelianGroup if R has Ring then Ring RetractableTo PR PartialDifferentialRing SY FullyLinearlyExplicitRingOver R coerce : MP -> % ++ coerce(p) returns p as an element of %. numer : % -> MP ++ numer(f) returns the ++ numerator of f viewed as a polynomial in the kernels over R ++ if R is an integral domain. If not, then numer(f) = f viewed ++ as a polynomial in the kernels over R. -- DO NOT change this meaning of numer! MB 1/90 numerator : % -> % ++ numerator(f) returns the numerator of \spad{f} converted to %. isExpt:(%,OP) -> Union(Record(var:K,exponent:Z),"failed") ++ isExpt(p,op) returns \spad{[x, n]} if \spad{p = x**n} ++ and \spad{n <> 0} and \spad{x = op(a)}. isExpt:(%,SY) -> Union(Record(var:K,exponent:Z),"failed") ++ isExpt(p,f) returns \spad{[x, n]} if \spad{p = x**n} ++ and \spad{n <> 0} and \spad{x = f(a)}. isPower : % -> Union(Record(val:%,exponent:Z),"failed") ++ isPower(p) returns \spad{[x, n]} if \spad{p = x**n} ++ and \spad{n <> 0}. eval: (%, List SY, List N, List(% -> %)) -> % ++ eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm]) replaces ++ every \spad{si(a)**ni} in x by \spad{fi(a)} for any \spad{a}. eval: (%, List SY, List N, List(List % -> %)) -> % ++ eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm]) replaces ++ every \spad{si(a1,...,an)**ni} in x by \spad{fi(a1,...,an)} ++ for any a1,...,am. eval: (%, SY, N, List % -> %) -> % ++ eval(x, s, n, f) replaces every \spad{s(a1,...,am)**n} in x ++ by \spad{f(a1,...,am)} for any a1,...,am. eval: (%, SY, N, % -> %) -> % ++ eval(x, s, n, f) replaces every \spad{s(a)**n} in x ++ by \spad{f(a)} for any \spad{a}. if R has CharacteristicZero then CharacteristicZero if R has CharacteristicNonZero then CharacteristicNonZero if R has CommutativeRing then Algebra R if R has IntegralDomain then Field RetractableTo Fraction PR convert : Factored % -> % ++ convert(f1\^e1 ... fm\^em) returns \spad{(f1)\^e1 ... (fm)\^em} ++ as an element of %, using formal kernels ++ created using a \spadfunFrom{paren}{ExpressionSpace}. denom : % -> MP ++ denom(f) returns the denominator of f viewed as a ++ polynomial in the kernels over R. denominator : % -> % ++ denominator(f) returns the denominator of \spad{f} converted to %. / : (MP, MP) -> % ++ p1/p2 returns the quotient of p1 and p2 as an element of %. coerce : Q -> % ++ coerce(q) returns q as an element of %. coerce : Polynomial Q -> % ++ coerce(p) returns p as an element of %. coerce : Fraction Polynomial Q -> % ++ coerce(f) returns f as an element of %. univariate: (%, K) -> Fraction SparseUnivariatePolynomial % ++ univariate(f, k) returns f viewed as a univariate fraction in k. if R has RetractableTo Z then RetractableTo Fraction Z add macro ODD == 'odd macro EVEN == 'even macro SPECIALDIFF == '%specialDiff macro SPECIALDISP == '%specialDisp macro SPECIALEQUAL == '%specialEqual macro SPECIALINPUT == '%specialInput import BasicOperatorFunctions1(%) -- these are needed in Ring only, but need to be declared here -- because of compiler bug: if they are declared inside the Ring -- case, then they are not visible inside the IntegralDomain case. smpIsMult : MP -> Union(Record(coef:Z, var:K),"failed") smpret : MP -> Union(PR, "failed") smpeval : (MP, List K, List %) -> % smpsubst : (MP, List K, List %) -> % smpderiv : (MP, SY) -> % smpunq : (MP, List SY, Boolean) -> % kerderiv : (K, SY) -> % kderiv : K -> List % opderiv : (OP, N) -> List(List % -> %) smp2O : MP -> O bestKernel: List K -> K worse? : (K, K) -> Boolean diffArg : (List %, OP, N) -> List % substArg : (OP, List %, Z, %) -> % dispdiff : List % -> Record(name:O, sub:O, arg:List O, level:N) ddiff : List % -> O diffEval : List % -> % dfeval : (List %, K) -> % smprep : (List SY, List N, List(List % -> %), MP) -> % diffdiff : (List %, SY) -> % diffdiff0 : (List %, SY, %, K, List %) -> % subs : (% -> %, K) -> % symsub : (SY, Z) -> SY kunq : (K, List SY, Boolean) -> % pushunq : (List SY, List %) -> List % notfound : (K -> %, List K, K) -> % equaldiff : (K,K)->Boolean debugA: (List % ,List %,Boolean) -> Boolean opdiff := operator('%diff)$CommonOperators() opquote := operator('applyQuote)$CommonOperators ground? x == retractIfCan(x)@Union(R,"failed") case R ground x == retract x coerce(x:SY):% == kernel(x)@K :: % retract(x:%):SY == symbolIfCan(retract(x)@K)::SY applyQuote(s:SY, x:%) == applyQuote(s, [x]) applyQuote(s, x, y) == applyQuote(s, [x, y]) applyQuote(s, x, y, z) == applyQuote(s, [x, y, z]) applyQuote(s, x, y, z, t) == applyQuote(s, [x, y, z, t]) applyQuote(s:SY, l:List %) == opquote concat(s::%, l) belong? op == op = opdiff or op = opquote subs(fn, k) == kernel(operator k,[fn x for x in argument k]$List(%)) operator op == is?(op, '%diff) => opdiff is?(op, '%quote) => opquote error "Unknown operator" if R has ConvertibleTo InputForm then INP==>InputForm import MakeUnaryCompiledFunction(%, %, %) indiff: List % -> INP pint : List INP-> INP differentiand: List % -> % differentiand l == eval(first l, retract(second l)@K, third l) pint l == convert concat(convert("D"::SY)@INP, l) indiff l == r2:= convert([convert("::"::SY)@INP,convert(third l)@INP,convert("Symbol"::SY)@INP]@List INP)@INP pint [convert(differentiand l)@INP, r2] eval(f:%, s:SY) == eval(f, [s]) eval(f:%, s:OP, g:%, x:SY) == eval(f, [s], [g], x) eval(f:%, ls:List OP, lg:List %, x:SY) == eval(f, ls, [compiledFunction(g, x) for g in lg]) setProperty(opdiff,SPECIALINPUT,indiff@(List % -> InputForm) pretend None) variables x == l := empty()$List(SY) for k in tower x repeat if ((s := symbolIfCan k) case SY) then l := concat(s::SY, l) reverse_! l retractIfCan(x:%):Union(SY, "failed") == (k := retractIfCan(x)@Union(K,"failed")) case "failed" => "failed" symbolIfCan(k::K) if R has Ring then import UserDefinedPartialOrdering(SY) -- cannot use new()$Symbol because of possible re-instantiation gendiff := "%%0"::SY characteristic == characteristic$R coerce(k:K):% == k::MP::% symsub(sy, i) == concat(string sy, convert(i)@String)::SY numerator x == numer(x)::% eval(x:%, s:SY, n:N, f:% -> %) == eval(x,[s],[n],[f first #1]) eval(x:%, s:SY, n:N, f:List % -> %) == eval(x, [s], [n], [f]) eval(x:%, l:List SY, f:List(List % -> %)) == eval(x, l, new(#l, 1), f) elt(op:OP, args:List %) == unary? op and ((od? := has?(op, ODD)) or has?(op, EVEN)) and before?(leadingCoefficient(numer first args),0) => x := op(- first args) od? => -x x elt(op, args)$ExpressionSpace_&(%) eval(x:%, s:List SY, n:List N, l:List(% -> %)) == eval(x, s, n, [f first #1 for f in l]$List(List % -> %)) -- op(arg)**m ==> func(arg)**(m quo n) * op(arg)**(m rem n) smprep(lop, lexp, lfunc, p) == (v := mainVariable p) case "failed" => p::% k := v::K g := (op := operator k) (arg := [eval(a,lop,lexp,lfunc) for a in argument k]$List(%)) q := map(eval(#1::%, lop, lexp, lfunc), univariate(p, k))$SparseUnivariatePolynomialFunctions2(MP, %) (n := position(name op, lop)) < minIndex lop => q g a:% := 0 f := eval((lfunc.n) arg, lop, lexp, lfunc) e := lexp.n while q ~= 0 repeat m := degree q qr := divide(m, e) t1 := f ** (qr.quotient)::N t2 := g ** (qr.remainder)::N a := a + leadingCoefficient(q) * t1 * t2 q := reductum q a dispdiff l == s := second(l)::O t := third(l)::O a := argument(k := retract(first l)@K) is?(k, opdiff) => rec := dispdiff a i := position(s, rec.arg) rec.arg.i := t [rec.name, hconcat(rec.sub, hconcat(","::SY::O, (i+1-minIndex a)::O)), rec.arg, (zero?(rec.level) => 0; rec.level + 1)] i := position(second l, a) m := [x::O for x in a]$List(O) m.i := t [name(operator k)::O, hconcat(","::SY::O, (i+1-minIndex a)::O), m, (empty? rest a => 1; 0)] ddiff l == rec := dispdiff l opname := zero?(rec.level) => sub(rec.name, rec.sub) differentiate(rec.name, rec.level) prefix(opname, rec.arg) substArg(op, l, i, g) == z := copy l z.i := g kernel(op, z) diffdiff(l, x) == f := kernel(opdiff, l) diffdiff0(l, x, f, retract(f)@K, empty()) diffdiff0(l, x, expr, kd, done) == op := operator(k := retract(first l)@K) gg := second l u := third l arg := argument k ans:% := 0 if (not member?(u,done)) and (ans := differentiate(u,x))~=0 then ans := ans * kernel(opdiff, [subst(expr, [kd], [kernel(opdiff, [first l, gg, gg])]), gg, u]) done := concat(gg, done) is?(k, opdiff) => ans + diffdiff0(arg, x, expr, k, done) for i in minIndex arg .. maxIndex arg for b in arg repeat if (not member?(b,done)) and (bp:=differentiate(b,x))~=0 then g := symsub(gendiff, i)::% ans := ans + bp * kernel(opdiff, [subst(expr, [kd], [kernel(opdiff, [substArg(op, arg, i, g), gg, u])]), g, b]) ans dfeval(l, g) == eval(differentiate(first l, symbolIfCan(g)::SY), g, third l) diffEval l == k:K g := retract(second l)@K ((u := retractIfCan(first l)@Union(K, "failed")) case "failed") or (u case K and symbolIfCan(k := u::K) case SY) => dfeval(l, g) op := operator k (ud := derivative op) case "failed" => -- possible trouble -- make sure it is a dummy var dumm:%:=symsub(gendiff,1)::% ss:=subst(l.1,l.2=dumm) -- output(nl::OutputForm)$OutputPackage -- output("fixed"::OutputForm)$OutputPackage nl:=[ss,dumm,l.3] kernel(opdiff, nl) (n := position(second l,argument k)) < minIndex l => dfeval(l,g) d := ud::List(List % -> %) eval((d.n)(argument k), g, third l) diffArg(l, op, i) == n := i - 1 + minIndex l z := copy l z.n := g := symsub(gendiff, n)::% [kernel(op, z), g, l.n] opderiv(op, n) == one? n => g := symsub(gendiff, n)::% [kernel(opdiff,[kernel(op, g), g, first #1])] [kernel(opdiff, diffArg(#1, op, i)) for i in 1..n] kderiv k == zero?(n := #(args := argument k)) => empty() op := operator k grad := (u := derivative op) case "failed" => opderiv(op, n) u::List(List % -> %) if #grad ~= n then grad := opderiv(op, n) [g args for g in grad] -- SPECIALDIFF contains a map (List %, Symbol) -> % -- it is used when the usual chain rule does not apply, -- for instance with implicit algebraics. kerderiv(k, x) == (v := symbolIfCan(k)) case SY => v::SY = x => 1 0 (fn := property(operator k, SPECIALDIFF)) case None => ((fn@None) pretend ((List %, SY) -> %)) (argument k, x) +/[g * differentiate(y,x) for g in kderiv k for y in argument k] smpderiv(p, x) == map(retract differentiate(#1::PR, x), p)::% + +/[differentiate(p,k)::% * kerderiv(k, x) for k in variables p] coerce(p:PR):% == map(#1::%, #1::%, p)$PolynomialCategoryLifting( IndexedExponents SY, SY, R, PR, %) worse?(k1, k2) == (u := less?(name operator k1,name operator k2)) case "failed" => k1 < k2 u::Boolean bestKernel l == empty? rest l => first l a := bestKernel rest l worse?(first l, a) => a first l smp2O p == (r:=retractIfCan(p)@Union(R,"failed")) case R =>r::R::OutputForm a := userOrdered?() => bestKernel variables p mainVariable(p)::K outputForm(map(#1::%, univariate(p, a))$SparseUnivariatePolynomialFunctions2(MP, %), a::OutputForm) smpsubst(p, lk, lv) == map(match(lk, lv, #1, notfound(subs(subst(#1, lk, lv), #1), lk, #1))$ListToMap(K,%), #1::%,p)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%) smpeval(p, lk, lv) == map(match(lk, lv, #1, notfound(map(eval(#1, lk, lv), #1), lk, #1))$ListToMap(K,%), #1::%,p)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%) -- this is called on k when k is not a member of lk notfound(fn, lk, k) == empty? setIntersection(tower(f := k::%), lk) => f fn k if R has ConvertibleTo InputForm then pushunq(l, arg) == empty? l => [eval a for a in arg] [eval(a, l) for a in arg] kunq(k, l, givenlist?) == givenlist? and empty? l => k::% is?(k, opquote) and (member?(s:=retract(first argument k)@SY, l) or empty? l) => interpret(convert(concat(convert(s)@InputForm, [convert a for a in pushunq(l, rest argument k) ]@List(InputForm)))@InputForm)$InputFormFunctions1(%) (operator k) pushunq(l, argument k) smpunq(p, l, givenlist?) == givenlist? and empty? l => p::% map(kunq(#1, l, givenlist?), #1::%, p)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%) smpret p == "or"/[symbolIfCan(k) case "failed" for k in variables p] => "failed" map(symbolIfCan(#1)::SY::PR, #1::PR, p)$PolynomialCategoryLifting(IndexedExponents K, K, R, MP, PR) isExpt(x:%, op:OP) == (u := isExpt x) case "failed" => "failed" v := (u::Record(var:K, exponent:Z)).var is?(v,op) and #argument(v) = 1 => u "failed" isExpt(x:%, sy:SY) == (u := isExpt x) case "failed" => "failed" v := (u::Record(var:K, exponent:Z)).var is?(v, sy) and #argument(v) = 1 => u "failed" if R has RetractableTo Z then smpIsMult p == (u := mainVariable p) case K and one? degree(q:=univariate(p,u::K)) and zero?(leadingCoefficient reductum q) and ((r:=retractIfCan(leadingCoefficient q)@Union(R,"failed")) case R) and (n := retractIfCan(r::R)@Union(Z, "failed")) case Z => [n::Z, u::K] "failed" evaluate(opdiff, diffEval) debugA(a1,a2,t) == -- uncomment for debugging -- output(hconcat [a1::OutputForm,a2::OutputForm,t::OutputForm])$OutputPackage t equaldiff(k1,k2) == a1:=argument k1 a2:=argument k2 -- check the operator res:=operator k1 = operator k2 not res => debugA(a1,a2,res) -- check the evaluation point res:= (a1.3 = a2.3) not res => debugA(a1,a2,res) -- check all the arguments res:= (a1.1 = a2.1) and (a1.2 = a2.2) res => debugA(a1,a2,res) -- check the substituted arguments (subst(a1.1,[retract(a1.2)@K],[a2.2]) = a2.1) => debugA(a1,a2,true) debugA(a1,a2,false) setProperty(opdiff,SPECIALEQUAL, equaldiff@((K,K) -> Boolean) pretend None) setProperty(opdiff, SPECIALDIFF, diffdiff@((List %, SY) -> %) pretend None) setProperty(opdiff, SPECIALDISP, ddiff@(List % -> OutputForm) pretend None) if not(R has IntegralDomain) then mainKernel x == mainVariable numer x kernels x == variables numer x retract(x:%):R == retract numer x retract(x:%):PR == smpret(numer x)::PR retractIfCan(x:%):Union(R, "failed") == retract numer x retractIfCan(x:%):Union(PR, "failed") == smpret numer x eval(x:%, lk:List K, lv:List %) == smpeval(numer x, lk, lv) subst(x:%, lk:List K, lv:List %) == smpsubst(numer x, lk, lv) differentiate(x:%, s:SY) == smpderiv(numer x, s) coerce(x:%):OutputForm == smp2O numer x if R has ConvertibleTo InputForm then eval(f:%, l:List SY) == smpunq(numer f, l, true) eval f == smpunq(numer f, empty(), false) eval(x:%, s:List SY, n:List N, f:List(List % -> %)) == smprep(s, n, f, numer x) isPlus x == (u := isPlus numer x) case "failed" => "failed" [p::% for p in u::List(MP)] isTimes x == (u := isTimes numer x) case "failed" => "failed" [p::% for p in u::List(MP)] isExpt x == (u := isExpt numer x) case "failed" => "failed" r := u::Record(var:K, exponent:NonNegativeInteger) [r.var, r.exponent::Z] isPower x == (u := isExpt numer x) case "failed" => "failed" r := u::Record(var:K, exponent:NonNegativeInteger) [r.var::%, r.exponent::Z] if R has ConvertibleTo Pattern Z then convert(x:%):Pattern(Z) == convert numer x if R has ConvertibleTo Pattern Float then convert(x:%):Pattern(Float) == convert numer x if R has RetractableTo Z then isMult x == smpIsMult numer x if R has CommutativeRing then r:R * x:% == r::MP::% * x if R has IntegralDomain then par : % -> % mainKernel x == mainVariable(x)$QF kernels x == variables(x)$QF univariate(x:%, k:K) == univariate(x, k)$QF isPlus x == isPlus(x)$QF isTimes x == isTimes(x)$QF isExpt x == isExpt(x)$QF isPower x == isPower(x)$QF denominator x == denom(x)::% coerce(q:Q):% == (numer q)::MP / (denom q)::MP coerce(q:Fraction PR):% == (numer q)::% / (denom q)::% coerce(q:Fraction Polynomial Q) == (numer q)::% / (denom q)::% retract(x:%):PR == retract(retract(x)@Fraction(PR)) retract(x:%):Fraction(PR) == smpret(numer x)::PR / smpret(denom x)::PR retract(x:%):R == (retract(numer x)@R exquo retract(denom x)@R)::R coerce(x:%):OutputForm == one?(denom x) => smp2O numer x smp2O(numer x) / smp2O(denom x) retractIfCan(x:%):Union(R, "failed") == (n := retractIfCan(numer x)@Union(R, "failed")) case "failed" or (d := retractIfCan(denom x)@Union(R, "failed")) case "failed" or (r := n::R exquo d::R) case "failed" => "failed" r::R eval(f:%, l:List SY) == smpunq(numer f, l, true) / smpunq(denom f, l, true) if R has ConvertibleTo InputForm then eval f == smpunq(numer f, empty(), false) / smpunq(denom f, empty(), false) eval(x:%, s:List SY, n:List N, f:List(List % -> %)) == smprep(s, n, f, numer x) / smprep(s, n, f, denom x) differentiate(f:%, x:SY) == (smpderiv(numer f, x) * denom(f)::% - numer(f)::% * smpderiv(denom f, x)) / (denom(f)::% ** 2) eval(x:%, lk:List K, lv:List %) == smpeval(numer x, lk, lv) / smpeval(denom x, lk, lv) subst(x:%, lk:List K, lv:List %) == smpsubst(numer x, lk, lv) / smpsubst(denom x, lk, lv) par x == (r := retractIfCan(x)@Union(R, "failed")) case R => x paren x convert(x:Factored %):% == par(unit x) * */[par(f.factor) ** f.exponent for f in factors x] retractIfCan(x:%):Union(PR, "failed") == (u := retractIfCan(x)@Union(Fraction PR,"failed")) case "failed" => "failed" retractIfCan(u::Fraction(PR)) retractIfCan(x:%):Union(Fraction PR, "failed") == (n := smpret numer x) case "failed" => "failed" (d := smpret denom x) case "failed" => "failed" n::PR / d::PR coerce(p:Polynomial Q):% == map(#1::%, #1::%, p)$PolynomialCategoryLifting(IndexedExponents SY, SY, Q, Polynomial Q, %) if R has RetractableTo Z then coerce(x:Fraction Z):% == numer(x)::MP / denom(x)::MP isMult x == (u := smpIsMult numer x) case "failed" or (v := retractIfCan(denom x)@Union(R, "failed")) case "failed" or (w := retractIfCan(v::R)@Union(Z, "failed")) case "failed" => "failed" r := u::Record(coef:Z, var:K) (q := r.coef exquo w::Z) case "failed" => "failed" [q::Z, r.var] if R has ConvertibleTo Pattern Z then convert(x:%):Pattern(Z) == convert(numer x) / convert(denom x) if R has ConvertibleTo Pattern Float then convert(x:%):Pattern(Float) == convert(numer x) / convert(denom x) @ \section{package FS2 FunctionSpaceFunctions2} <>= )abbrev package FS2 FunctionSpaceFunctions2 ++ Lifting of maps to function spaces ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 3 May 1994 ++ Description: ++ This package allows a mapping R -> S to be lifted to a mapping ++ from a function space over R to a function space over S; FunctionSpaceFunctions2(R, A, S, B): Exports == Implementation where R, S: Ring A : FunctionSpace R B : FunctionSpace S K ==> Kernel A P ==> SparseMultivariatePolynomial(R, K) Exports ==> with map: (R -> S, A) -> B ++ map(f, a) applies f to all the constants in R appearing in \spad{a}. Implementation ==> add smpmap: (R -> S, P) -> B smpmap(fn, p) == map(map(map(fn, #1), #1)$ExpressionSpaceFunctions2(A,B),fn(#1)::B, p)$PolynomialCategoryLifting(IndexedExponents K, K, R, P, B) if R has IntegralDomain then if S has IntegralDomain then map(f, x) == smpmap(f, numer x) / smpmap(f, denom x) else map(f, x) == smpmap(f, numer x) * (recip(smpmap(f, denom x))::B) else map(f, x) == smpmap(f, numer x) @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2009, Gabriel Dos Reis. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. @ <<*>>= <> -- SPAD files for the functional world should be compiled in the -- following order: -- -- op kl FSPACE expr funcpkgs <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}