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\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}
<<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()$List(%)
for eq in leq repeat
(k := retractIfCan(lhs eq)@Union(K, "failed")) case "failed" =>
error "left hand side must be a single kernel"
if not member?(k::K, lk) then
lk := concat(k::K, lk)
lv := concat(rhs eq, lv)
[lk, lv]
if % has RetractableTo Integer then
intpred?: (%, Integer -> Boolean) -> Boolean
even? x == intpred?(x, even?)
odd? x == intpred?(x, odd?)
intpred?(x, pred?) ==
(u := retractIfCan(x)@Union(Integer, "failed")) case Integer
and pred?(u::Integer)
@
\section{package ES1 ExpressionSpaceFunctions1}
<<package ES1 ExpressionSpaceFunctions1>>=
)abbrev package ES1 ExpressionSpaceFunctions1
++ Lifting of maps from expression spaces to kernels over them
++ Author: Manuel Bronstein
++ Date Created: 23 March 1988
++ Date Last Updated: 19 April 1991
++ Description:
++ This package allows a map from any expression space into any object
++ to be lifted to a kernel over the expression set, using a given
++ property of the operator of the kernel.
-- should not be exposed
ExpressionSpaceFunctions1(F:ExpressionSpace, S:Type): with
map: (F -> S, String, Kernel F) -> S
++ map(f, p, k) uses the property p of the operator
++ of k, in order to lift f and apply it to k.
== add
-- prop contains an evaluation function List S -> S
map(F2S, prop, k) ==
args := [F2S x for x in argument k]$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}
<<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}
<<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)$PolynomialCategoryLifting(IndexedExponents SY, SY,
Q, Polynomial Q, %)
if R has RetractableTo Z then
coerce(x:Fraction Z):% == numer(x)::MP / denom(x)::MP
isMult x ==
(u := smpIsMult numer x) case "failed"
or (v := retractIfCan(denom x)@Union(R, "failed")) case "failed"
or (w := retractIfCan(v::R)@Union(Z, "failed")) case "failed"
=> "failed"
r := u::Record(coef:Z, var:K)
(q := r.coef exquo w::Z) case "failed" => "failed"
[q::Z, r.var]
if R has ConvertibleTo Pattern Z then
convert(x:%):Pattern(Z) == convert(numer x) / convert(denom x)
if R has ConvertibleTo Pattern Float then
convert(x:%):Pattern(Float) ==
convert(numer x) / convert(denom x)
@
\section{package FS2 FunctionSpaceFunctions2}
<<package FS2 FunctionSpaceFunctions2>>=
)abbrev package FS2 FunctionSpaceFunctions2
++ Lifting of maps to function spaces
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 3 May 1994
++ Description:
++ This package allows a mapping R -> S to be lifted to a mapping
++ from a function space over R to a function space over S;
FunctionSpaceFunctions2(R, A, S, B): Exports == Implementation where
R, S: Ring
A : FunctionSpace R
B : FunctionSpace S
K ==> Kernel A
P ==> SparseMultivariatePolynomial(R, K)
Exports ==> with
map: (R -> S, A) -> B
++ map(f, a) applies f to all the constants in R appearing in \spad{a}.
Implementation ==> add
smpmap: (R -> S, P) -> B
smpmap(fn, p) ==
map(map(map(fn, #1), #1)$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}
<<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.
@
<<*>>=
<<license>>
-- SPAD files for the functional world should be compiled in the
-- following order:
--
-- op kl FSPACE expr funcpkgs
<<category ES ExpressionSpace>>
<<package ES1 ExpressionSpaceFunctions1>>
<<package ES2 ExpressionSpaceFunctions2>>
<<category FS FunctionSpace>>
<<package FS2 FunctionSpaceFunctions2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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