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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra manip.spad}
\author{Manuel Bronstein, Robert Sutor}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package FACTFUNC FactoredFunctions}
<<package FACTFUNC FactoredFunctions>>=
)abbrev package FACTFUNC FactoredFunctions
++ Author: Manuel Bronstein
++ Date Created: 2 Feb 1988
++ Date Last Updated: 25 Jun 1990
++ Description: computes various functions on factored arguments.
-- not visible to the user
FactoredFunctions(M:IntegralDomain): Exports == Implementation where
N ==> NonNegativeInteger
Exports ==> with
nthRoot: (Factored M,N) -> Record(exponent:N,coef:M,radicand:List M)
++ nthRoot(f, n) returns \spad{(p, r, [r1,...,rm])} such that
++ the nth-root of f is equal to \spad{r * pth-root(r1 * ... * rm)},
++ where r1,...,rm are distinct factors of f,
++ each of which has an exponent smaller than p in f.
log : Factored M -> List Record(coef:N, logand:M)
++ log(f) returns \spad{[(a1,b1),...,(am,bm)]} such that
++ the logarithm of f is equal to \spad{a1*log(b1) + ... + am*log(bm)}.
Implementation ==> add
nthRoot(ff, n) ==
coeff:M := 1
radi:List(M) := (one? unit ff => empty(); [unit ff])
lf := factors ff
d:N :=
empty? radi => gcd(concat(n, [t.exponent::N for t in lf]))::N
1
n := n quo d
for term in lf repeat
qr := divide(term.exponent::N quo d, n)
coeff := coeff * term.factor ** qr.quotient
not zero?(qr.remainder) =>
radi := concat!(radi, term.factor ** qr.remainder)
[n, coeff, radi]
log ff ==
ans := unit ff
concat([1, unit ff],
[[term.exponent::N, term.factor] for term in factors ff])
@
\section{package POLYROOT PolynomialRoots}
<<package POLYROOT PolynomialRoots>>=
)abbrev package POLYROOT PolynomialRoots
++ Author: Manuel Bronstein
++ Date Created: 15 July 1988
++ Date Last Updated: 10 November 1993
++ Description: computes n-th roots of quotients of
++ multivariate polynomials
-- not visible to the user
PolynomialRoots(E, V, R, P, F):Exports == Implementation where
E: OrderedAbelianMonoidSup
V: OrderedSet
R: IntegralDomain
P: PolynomialCategory(R, E, V)
F: Field with
coerce: P -> %
numer : $ -> P
++ numer(x) \undocumented
denom : $ -> P
++ denom(x) \undocumented
N ==> NonNegativeInteger
Z ==> Integer
Q ==> Fraction Z
REC ==> Record(exponent:N, coef:F, radicand:F)
Exports ==> with
rroot: (R, N) -> REC
++ rroot(f, n) returns \spad{[m,c,r]} such
++ that \spad{f**(1/n) = c * r**(1/m)}.
qroot : (Q, N) -> REC
++ qroot(f, n) returns \spad{[m,c,r]} such
++ that \spad{f**(1/n) = c * r**(1/m)}.
if R has GcdDomain then froot: (F, N) -> REC
++ froot(f, n) returns \spad{[m,c,r]} such
++ that \spad{f**(1/n) = c * r**(1/m)}.
nthr: (P, N) -> Record(exponent:N,coef:P,radicand:List P)
++ nthr(p,n) should be local but conditional
Implementation ==> add
import FactoredFunctions Z
import FactoredFunctions P
rsplit: List P -> Record(coef:R, poly:P)
zroot : (Z, N) -> Record(exponent:N, coef:Z, radicand:Z)
zroot(x, n) ==
zero? x or one? x => [1, x, 1]
s := nthRoot(squareFree x, n)
[s.exponent, s.coef, */s.radicand]
if R has imaginary: () -> R then
czroot: (Z, N) -> REC
czroot(x, n) ==
rec := zroot(x, n)
rec.exponent = 2 and negative? rec.radicand =>
[rec.exponent, rec.coef * imaginary()::P::F, (-rec.radicand)::F]
[rec.exponent, rec.coef::F, rec.radicand::F]
qroot(x, n) ==
sn := czroot(numer x, n)
sd := czroot(denom x, n)
m := lcm(sn.exponent, sd.exponent)::N
[m, sn.coef / sd.coef,
(sn.radicand ** (m quo sn.exponent)) /
(sd.radicand ** (m quo sd.exponent))]
else
qroot(x, n) ==
sn := zroot(numer x, n)
sd := zroot(denom x, n)
m := lcm(sn.exponent, sd.exponent)::N
[m, sn.coef::F / sd.coef::F,
(sn.radicand ** (m quo sn.exponent))::F /
(sd.radicand ** (m quo sd.exponent))::F]
if R has RetractableTo Fraction Z then
rroot(x, n) ==
(r := retractIfCan(x)@Union(Fraction Z,"failed")) case "failed"
=> [n, 1, x::P::F]
qroot(r::Q, n)
else
if R has RetractableTo Z then
rroot(x, n) ==
(r := retractIfCan(x)@Union(Z,"failed")) case "failed"
=> [n, 1, x::P::F]
qroot(r::Z::Q, n)
else
rroot(x, n) == [n, 1, x::P::F]
rsplit l ==
r := 1$R
p := 1$P
for q in l repeat
if (u := retractIfCan(q)@Union(R, "failed")) case "failed"
then p := p * q
else r := r * u::R
[r, p]
if R has GcdDomain then
if R has RetractableTo Z then
nthr(x, n) ==
(r := retractIfCan(x)@Union(Z,"failed")) case "failed"
=> nthRoot(squareFree x, n)
rec := zroot(r::Z, n)
[rec.exponent, rec.coef::P, [rec.radicand::P]]
else nthr(x, n) == nthRoot(squareFree x, n)
froot(x, n) ==
zero? x or one? x => [1, x, 1]
sn := nthr(numer x, n)
sd := nthr(denom x, n)
pn := rsplit(sn.radicand)
pd := rsplit(sd.radicand)
rn := rroot(pn.coef, sn.exponent)
rd := rroot(pd.coef, sd.exponent)
m := lcm([rn.exponent, rd.exponent, sn.exponent, sd.exponent])::N
[m, (sn.coef::F / sd.coef::F) * (rn.coef / rd.coef),
((rn.radicand ** (m quo rn.exponent)) /
(rd.radicand ** (m quo rd.exponent))) *
(pn.poly ** (m quo sn.exponent))::F /
(pd.poly ** (m quo sd.exponent))::F]
@
\section{package ALGMANIP AlgebraicManipulations}
<<package ALGMANIP AlgebraicManipulations>>=
)abbrev package ALGMANIP AlgebraicManipulations
++ Author: Manuel Bronstein
++ Date Created: 28 Mar 1988
++ Date Last Updated: 5 August 1993
++ Description:
++ AlgebraicManipulations provides functions to simplify and expand
++ expressions involving algebraic operators.
++ Keywords: algebraic, manipulation.
AlgebraicManipulations(R, F): Exports == Implementation where
R : IntegralDomain
F : Join(Field, ExpressionSpace) with
numer : $ -> SparseMultivariatePolynomial(R, Kernel $)
++ numer(x) \undocumented
denom : $ -> SparseMultivariatePolynomial(R, Kernel $)
++ denom(x) \undocumented
coerce : SparseMultivariatePolynomial(R, Kernel $) -> $
++ coerce(x) \undocumented
N ==> NonNegativeInteger
Z ==> Integer
OP ==> BasicOperator
SY ==> Symbol
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
RF ==> Fraction P
REC ==> Record(ker:List K, exponent: List Z)
Exports ==> with
rootSplit: F -> F
++ rootSplit(f) transforms every radical of the form
++ \spad{(a/b)**(1/n)} appearing in f into \spad{a**(1/n) / b**(1/n)}.
++ This transformation is not in general valid for all
++ complex numbers \spad{a} and b.
ratDenom : F -> F
++ ratDenom(f) rationalizes the denominators appearing in f
++ by moving all the algebraic quantities into the numerators.
ratDenom : (F, F) -> F
++ ratDenom(f, a) removes \spad{a} from the denominators in f
++ if \spad{a} is an algebraic kernel.
ratDenom : (F, List F) -> F
++ ratDenom(f, [a1,...,an]) removes the ai's which are
++ algebraic kernels from the denominators in f.
ratDenom : (F, List K) -> F
++ ratDenom(f, [a1,...,an]) removes the ai's which are
++ algebraic from the denominators in f.
ratPoly : F -> SparseUnivariatePolynomial F
++ ratPoly(f) returns a polynomial p such that p has no
++ algebraic coefficients, and \spad{p(f) = 0}.
if R has Join(GcdDomain, RetractableTo Integer)
and F has FunctionSpace(R) then
rootPower : F -> F
++ 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.
rootProduct: F -> F
++ 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.
rootSimp : F -> F
++ rootSimp(f) transforms every radical of the form
++ \spad{(a * b**(q*n+r))**(1/n)} appearing in f into
++ \spad{b**q * (a * b**r)**(1/n)}.
++ This transformation is not in general valid for all
++ complex numbers b.
rootKerSimp: (OP, F, N) -> F
++ rootKerSimp(op,f,n) should be local but conditional.
Implementation ==> add
macro ALGOP == '%alg
macro NTHR == 'nthRoot
import PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,P,F)
innerRF : (F, List K) -> F
rootExpand : K -> F
algkernels : List K -> List K
rootkernels: List K -> List K
dummy := kernel(new()$SY)$K
ratDenom x == innerRF(x, algkernels tower x)
ratDenom(x:F, l:List K):F == innerRF(x, algkernels l)
ratDenom(x:F, y:F) == ratDenom(x, [y])
ratDenom(x:F, l:List F) == ratDenom(x, [retract(y)@K for y in l]$List(K))
algkernels l == select!(has?(operator #1, ALGOP), l)
rootkernels l == select!(is?(operator #1, NTHR::SY), l)
ratPoly x ==
numer univariate(denom(ratDenom inv(dummy::P::F - x))::F, dummy)
rootSplit x ==
lk := rootkernels tower x
eval(x, lk, [rootExpand k for k in lk])
rootExpand k ==
x := first argument k
n := second argument k
op := operator k
op(numer(x)::F, n) / op(denom(x)::F, n)
-- all the kernels in ll must be algebraic
innerRF(x, ll) ==
empty?(l := sort!(#1 > #2, kernels x)$List(K)) or
empty? setIntersection(ll, tower x) => x
lk := empty()$List(K)
k : K
while not member?(k := first l, ll) repeat
lk := concat(k, lk)
empty?(l := rest l) =>
return eval(x, lk, [map(innerRF(#1, ll), kk) for kk in lk])
q := univariate(eval(x, lk,
[map(innerRF(#1, ll), kk) for kk in lk]), k, minPoly k)
map(innerRF(#1, ll), q) (map(innerRF(#1, ll), k))
if R has Join(GcdDomain, RetractableTo Integer)
and F has FunctionSpace(R) then
import PolynomialRoots(IndexedExponents K, K, R, P, F)
sroot : K -> F
inroot : (OP, F, N) -> F
radeval: (P, K) -> F
breakup: List K -> List REC
if R has RadicalCategory then
rootKerSimp(op, x, n) ==
(r := retractIfCan(x)@Union(R, "failed")) case R =>
nthRoot(r::R, n)::F
inroot(op, x, n)
else
rootKerSimp(op, x, n) == inroot(op, x, n)
-- l is a list of nth-roots, returns a list of records of the form
-- [a**(1/n1),a**(1/n2),...], [n1,n2,...]]
-- such that the whole list covers l exactly
breakup l ==
empty? l => empty()
k := first l
a := first(arg := argument(k := first l))
n := retract(second arg)@Z
expo := empty()$List(Z)
others := same := empty()$List(K)
for kk in rest l repeat
if (a = first(arg := argument kk)) then
same := concat(kk, same)
expo := concat(retract(second arg)@Z, expo)
else others := concat(kk, others)
ll := breakup others
concat([concat(k, same), concat(n, expo)], ll)
rootProduct x ==
for rec in breakup rootkernels tower x repeat
k0 := first(l := rec.ker)
nx := numer x; dx := denom x
if empty? rest l then x := radeval(nx, k0) / radeval(dx, k0)
else
n := lcm(rec.exponent)
k := kernel(operator k0, [first argument k0, n::F], height k0)$K
lv := [monomial(1, k, (n quo m)::N) for m in rec.exponent]$List(P)
x := radeval(eval(nx, l, lv), k) / radeval(eval(dx, l, lv), k)
x
rootPower x ==
for k in rootkernels tower x repeat
x := radeval(numer x, k) / radeval(denom x, k)
x
-- replaces (a**(1/n))**m in p by a power of a simpler radical of a if
-- n and m have a common factor
radeval(p, k) ==
a := first(arg := argument k)
n := (retract(second arg)@Integer)::NonNegativeInteger
ans:F := 0
q := univariate(p, k)
while (d := degree q) > 0 repeat
term :=
one?(g := gcd(d, n)) => monomial(1, k, d)
monomial(1, kernel(operator k, [a,(n quo g)::F], height k), d quo g)
ans := ans + leadingCoefficient(q)::F * term::F
q := reductum q
leadingCoefficient(q)::F + ans
inroot(op, x, n) ==
one? x => x
(x ~= -1) and (one?(num := numer x) or (num = -1)) =>
inv inroot(op, (num * denom x)::F, n)
(u := isExpt(x, op)) case "failed" => kernel(op, [x, n::F])
pr := u::Record(var:K, exponent:Integer)
q := pr.exponent /$Fraction(Z)
(n * retract(second argument(pr.var))@Z)
qr := divide(numer q, denom q)
x := first argument(pr.var)
x ** qr.quotient * rootKerSimp(op,x,denom(q)::N) ** qr.remainder
sroot k ==
pr := froot(first(arg := argument k),(retract(second arg)@Z)::N)
pr.coef * rootKerSimp(operator k, pr.radicand, pr.exponent)
rootSimp x ==
lk := rootkernels tower x
eval(x, lk, [sroot k for k in lk])
@
\section{package SIMPAN SimplifyAlgebraicNumberConvertPackage}
<<package SIMPAN SimplifyAlgebraicNumberConvertPackage>>=
)abbrev package SIMPAN SimplifyAlgebraicNumberConvertPackage
++ Package to allow simplify to be called on AlgebraicNumbers
++ by converting to EXPR(INT)
SimplifyAlgebraicNumberConvertPackage(): with
simplify: AlgebraicNumber -> Expression(Integer)
++ simplify(an) applies simplifications to an
== add
simplify(a:AlgebraicNumber) ==
simplify(a::Expression(Integer))$TranscendentalManipulations(Integer, Expression Integer)
@
\section{package TRMANIP TranscendentalManipulations}
<<package TRMANIP TranscendentalManipulations>>=
)abbrev package TRMANIP TranscendentalManipulations
++ Transformations on transcendental objects
++ Author: Bob Sutor, Manuel Bronstein
++ Date Created: Way back
++ Date Last Updated: 22 January 1996, added simplifyLog MCD.
++ Description:
++ TranscendentalManipulations provides functions to simplify and
++ expand expressions involving transcendental operators.
++ Keywords: transcendental, manipulation.
TranscendentalManipulations(R, F): Exports == Implementation where
R : GcdDomain
F : Join(FunctionSpace R, TranscendentalFunctionCategory)
Z ==> Integer
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial P
POWER ==> '%power
POW ==> Record(val: F,exponent: Z)
PRODUCT ==> Record(coef : Z, var : K)
FPR ==> Fraction Polynomial R
Exports ==> with
expand : F -> F
++ expand(f) performs the following expansions on 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}
simplify : F -> F
++ simplify(f) performs the following simplifications on 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}
htrigs : F -> F
++ htrigs(f) converts all the exponentials in f into
++ hyperbolic sines and cosines.
simplifyExp: F -> F
++ simplifyExp(f) converts every product \spad{exp(a)*exp(b)}
++ appearing in f into \spad{exp(a+b)}.
simplifyLog : F -> F
++ simplifyLog(f) converts every \spad{log(a) - log(b)} appearing in 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)}.
expandPower: F -> F
++ expandPower(f) converts every power \spad{(a/b)**c} appearing
++ in f into \spad{a**c * b**(-c)}.
expandLog : F -> F
++ expandLog(f) converts every \spad{log(a/b)} appearing in f into
++ \spad{log(a) - log(b)}, and every \spad{log(a*b)} into
++ \spad{log(a) + log(b)}..
cos2sec : F -> F
++ cos2sec(f) converts every \spad{cos(u)} appearing in f into
++ \spad{1/sec(u)}.
cosh2sech : F -> F
++ cosh2sech(f) converts every \spad{cosh(u)} appearing in f into
++ \spad{1/sech(u)}.
cot2trig : F -> F
++ cot2trig(f) converts every \spad{cot(u)} appearing in f into
++ \spad{cos(u)/sin(u)}.
coth2trigh : F -> F
++ coth2trigh(f) converts every \spad{coth(u)} appearing in f into
++ \spad{cosh(u)/sinh(u)}.
csc2sin : F -> F
++ csc2sin(f) converts every \spad{csc(u)} appearing in f into
++ \spad{1/sin(u)}.
csch2sinh : F -> F
++ csch2sinh(f) converts every \spad{csch(u)} appearing in f into
++ \spad{1/sinh(u)}.
sec2cos : F -> F
++ sec2cos(f) converts every \spad{sec(u)} appearing in f into
++ \spad{1/cos(u)}.
sech2cosh : F -> F
++ sech2cosh(f) converts every \spad{sech(u)} appearing in f into
++ \spad{1/cosh(u)}.
sin2csc : F -> F
++ sin2csc(f) converts every \spad{sin(u)} appearing in f into
++ \spad{1/csc(u)}.
sinh2csch : F -> F
++ sinh2csch(f) converts every \spad{sinh(u)} appearing in f into
++ \spad{1/csch(u)}.
tan2trig : F -> F
++ tan2trig(f) converts every \spad{tan(u)} appearing in f into
++ \spad{sin(u)/cos(u)}.
tanh2trigh : F -> F
++ tanh2trigh(f) converts every \spad{tanh(u)} appearing in f into
++ \spad{sinh(u)/cosh(u)}.
tan2cot : F -> F
++ tan2cot(f) converts every \spad{tan(u)} appearing in f into
++ \spad{1/cot(u)}.
tanh2coth : F -> F
++ tanh2coth(f) converts every \spad{tanh(u)} appearing in f into
++ \spad{1/coth(u)}.
cot2tan : F -> F
++ cot2tan(f) converts every \spad{cot(u)} appearing in f into
++ \spad{1/tan(u)}.
coth2tanh : F -> F
++ coth2tanh(f) converts every \spad{coth(u)} appearing in f into
++ \spad{1/tanh(u)}.
removeCosSq: F -> F
++ removeCosSq(f) converts every \spad{cos(u)**2} appearing in f into
++ \spad{1 - sin(x)**2}, and also reduces higher
++ powers of \spad{cos(u)} with that formula.
removeSinSq: F -> F
++ removeSinSq(f) converts every \spad{sin(u)**2} appearing in f into
++ \spad{1 - cos(x)**2}, and also reduces higher powers of
++ \spad{sin(u)} with that formula.
removeCoshSq:F -> F
++ removeCoshSq(f) converts every \spad{cosh(u)**2} appearing in f into
++ \spad{1 - sinh(x)**2}, and also reduces higher powers of
++ \spad{cosh(u)} with that formula.
removeSinhSq:F -> F
++ removeSinhSq(f) converts every \spad{sinh(u)**2} appearing in f into
++ \spad{1 - cosh(x)**2}, and also reduces higher powers
++ of \spad{sinh(u)} with that formula.
if R has PatternMatchable(R) and R has ConvertibleTo(Pattern(R)) and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then
expandTrigProducts : F -> F
++ expandTrigProducts(e) replaces \axiom{sin(x)*sin(y)} by
++ \spad{(cos(x-y)-cos(x+y))/2}, \axiom{cos(x)*cos(y)} by
++ \spad{(cos(x-y)+cos(x+y))/2}, and \axiom{sin(x)*cos(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.
Implementation ==> add
import FactoredFunctions(P)
import PolynomialCategoryLifting(IndexedExponents K, K, R, P, F)
import
PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,P,F)
smpexp : P -> F
termexp : P -> F
exlog : P -> F
smplog : P -> F
smpexpand : P -> F
smp2htrigs: P -> F
kerexpand : K -> F
expandpow : K -> F
logexpand : K -> F
sup2htrigs: (UP, F) -> F
supexp : (UP, F, F, Z) -> F
ueval : (F, String, F -> F) -> F
ueval2 : (F, String, F -> F) -> F
powersimp : (P, List K) -> F
t2t : F -> F
c2t : F -> F
c2s : F -> F
s2c : F -> F
s2c2 : F -> F
th2th : F -> F
ch2th : F -> F
ch2sh : F -> F
sh2ch : F -> F
sh2ch2 : F -> F
simplify0 : F -> F
simplifyLog1 : F -> F
logArgs : List F -> F
import F
import List F
if R has PatternMatchable R and R has ConvertibleTo Pattern R and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then
XX : F := coerce new()$Symbol
YY : F := coerce new()$Symbol
sinCosRule : RewriteRule(R,R,F) :=
rule(cos(XX)*sin(YY),(sin(XX+YY)-sin(XX-YY))/2::F)
sinSinRule : RewriteRule(R,R,F) :=
rule(sin(XX)*sin(YY),(cos(XX-YY)-cos(XX+YY))/2::F)
cosCosRule : RewriteRule(R,R,F) :=
rule(cos(XX)*cos(YY),(cos(XX-YY)+cos(XX+YY))/2::F)
expandTrigProducts(e:F):F ==
applyRules([sinCosRule,sinSinRule,cosCosRule],e,10)$ApplyRules(R,R,F)
logArgs(l:List F):F ==
-- This function will take a list of Expressions (implicitly a sum) and
-- add them up, combining log terms. It also replaces n*log(x) by
-- log(x^n).
import K
sum : F := 0
arg : F := 1
for term in l repeat
is?(term,'log) =>
arg := arg * simplifyLog(first(argument(first(kernels(term)))))
-- Now look for multiples, including negative ones.
prod : Union(PRODUCT, "failed") := isMult(term)
(prod case PRODUCT) and is?(prod.var,'log) =>
arg := arg * simplifyLog ((first argument(prod.var))**(prod.coef))
sum := sum+term
sum+log(arg)
simplifyLog(e:F):F ==
simplifyLog1(numerator e)/simplifyLog1(denominator e)
simplifyLog1(e:F):F ==
freeOf?(e,'log) => e
-- Check for n*log(u)
prod : Union(PRODUCT, "failed") := isMult(e)
(prod case PRODUCT) and is?(prod.var,'log) =>
log simplifyLog ((first argument(prod.var))**(prod.coef))
termList : Union(List(F),"failed") := isTimes(e)
-- I'm using two variables, termList and terms, to work round a
-- bug in the old compiler.
not (termList case "failed") =>
-- We want to simplify each log term in the product and then multiply
-- them together. However, if there is a constant or arithmetic
-- expression (i.e. somwthing which looks like a Polynomial) we would
-- like to combine it with a log term.
terms :List F := [simplifyLog(term) for term in termList::List(F)]
exprs :List F := []
nterms :List F := []
for term in terms repeat
if retractIfCan(term)@Union(FPR,"failed") case FPR then
exprs := cons(term, exprs)
else
nterms := cons(term, nterms)
terms := nterms
if not empty? exprs then
foundLog := false
i : NonNegativeInteger := 0
while (not(foundLog) and (i < #terms)) repeat
i := i+1
if is?(terms.i,'log) then
args : List F := argument(retract(terms.i)@K)
setelt(terms,i, log simplifyLog1(first(args)**(*/exprs)))
foundLog := true
-- The next line deals with a situation which shouldn't occur,
-- since we have checked whether we are freeOf log already.
if not foundLog then terms := append(exprs,terms)
*/terms
terms : Union(List(F),"failed") := isPlus(e)
not (terms case "failed") => logArgs(terms)
expt : Union(POW, "failed") := isPower(e)
(expt case POW) and not one? expt.exponent =>
simplifyLog(expt.val)**(expt.exponent)
kers : List K := kernels e
not(one?(#kers)) => e -- Have a constant
kernel(operator first kers,[simplifyLog(u) for u in argument first kers])
if R has RetractableTo Integer then
simplify x == rootProduct(simplify0 x)$AlgebraicManipulations(R,F)
else simplify x == simplify0 x
expandpow k ==
a := expandPower first(arg := argument k)
b := expandPower second arg
ne:F := (one? numer a => 1; numer(a)::F ** b)
de:F := (one? denom a => 1; denom(a)::F ** (-b))
ne * de
termexp p ==
exponent:F := 0
coef := (leadingCoefficient p)::P
lpow := select(is?(#1, POWER)$K, lk := variables p)$List(K)
for k in lk repeat
d := degree(p, k)
if is?(k, 'exp) then
exponent := exponent + d * first argument k
else if not is?(k, POWER) then
-- Expand arguments to functions as well ... MCD 23/1/97
--coef := coef * monomial(1, k, d)
coef := coef * monomial(1, kernel(operator k,[simplifyExp u for u in argument k], height k), d)
coef::F * exp exponent * powersimp(p, lpow)
expandPower f ==
l := select(is?(#1, POWER)$K, kernels f)$List(K)
eval(f, l, [expandpow k for k in l])
-- l is a list of pure powers appearing as kernels in p
powersimp(p, l) ==
empty? l => 1
k := first l -- k = a**b
a := first(arg := argument k)
exponent := degree(p, k) * second arg
empty?(lk := select(a = first argument #1, rest l)) =>
(a ** exponent) * powersimp(p, rest l)
for k0 in lk repeat
exponent := exponent + degree(p, k0) * second argument k0
(a ** exponent) * powersimp(p, setDifference(rest l, lk))
t2t x == sin(x) / cos(x)
c2t x == cos(x) / sin(x)
c2s x == inv sin x
s2c x == inv cos x
s2c2 x == 1 - cos(x)**2
th2th x == sinh(x) / cosh(x)
ch2th x == cosh(x) / sinh(x)
ch2sh x == inv sinh x
sh2ch x == inv cosh x
sh2ch2 x == cosh(x)**2 - 1
ueval(x, s,f) == eval(x, s::Symbol, f)
ueval2(x,s,f) == eval(x, s::Symbol, 2, f)
cos2sec x == ueval(x, "cos", inv sec #1)
sin2csc x == ueval(x, "sin", inv csc #1)
csc2sin x == ueval(x, "csc", c2s)
sec2cos x == ueval(x, "sec", s2c)
tan2cot x == ueval(x, "tan", inv cot #1)
cot2tan x == ueval(x, "cot", inv tan #1)
tan2trig x == ueval(x, "tan", t2t)
cot2trig x == ueval(x, "cot", c2t)
cosh2sech x == ueval(x, "cosh", inv sech #1)
sinh2csch x == ueval(x, "sinh", inv csch #1)
csch2sinh x == ueval(x, "csch", ch2sh)
sech2cosh x == ueval(x, "sech", sh2ch)
tanh2coth x == ueval(x, "tanh", inv coth #1)
coth2tanh x == ueval(x, "coth", inv tanh #1)
tanh2trigh x == ueval(x, "tanh", th2th)
coth2trigh x == ueval(x, "coth", ch2th)
removeCosSq x == ueval2(x, "cos", 1 - (sin #1)**2)
removeSinSq x == ueval2(x, "sin", s2c2)
removeCoshSq x== ueval2(x, "cosh", 1 + (sinh #1)**2)
removeSinhSq x== ueval2(x, "sinh", sh2ch2)
expandLog x == smplog(numer x) / smplog(denom x)
simplifyExp x == (smpexp numer x) / (smpexp denom x)
expand x == (smpexpand numer x) / (smpexpand denom x)
smpexpand p == map(kerexpand, #1::F, p)
smplog p == map(logexpand, #1::F, p)
smp2htrigs p == map(htrigs(#1::F), #1::F, p)
htrigs f ==
(m := mainKernel f) case "failed" => f
op := operator(k := m::K)
arg := [htrigs x for x in argument k]$List(F)
num := univariate(numer f, k)
den := univariate(denom f, k)
is?(op,'exp) =>
g1 := cosh(a := first arg) + sinh(a)
g2 := cosh(a) - sinh(a)
supexp(num,g1,g2,b:= (degree num)::Z quo 2)/supexp(den,g1,g2,b)
sup2htrigs(num, g1:= op arg) / sup2htrigs(den, g1)
supexp(p, f1, f2, bse) ==
ans:F := 0
while p ~= 0 repeat
g := htrigs(leadingCoefficient(p)::F)
if ((d := degree(p)::Z - bse) >= 0) then
ans := ans + g * f1 ** d
else ans := ans + g * f2 ** (-d)
p := reductum p
ans
sup2htrigs(p, f) ==
(map(smp2htrigs, p)$SparseUnivariatePolynomialFunctions2(P, F)) f
exlog p == +/[r.coef * log(r.logand::F) for r in log squareFree p]
logexpand k ==
nullary?(op := operator k) => k::F
is?(op,'log) =>
exlog(numer(x := expandLog first argument k)) - exlog denom x
op [expandLog x for x in argument k]$List(F)
kerexpand k ==
nullary?(op := operator k) => k::F
is?(op, POWER) => expandpow k
arg := first argument k
is?(op,'sec) => inv expand cos arg
is?(op,'csc) => inv expand sin arg
is?(op,'log) => exlog(numer(x := expand arg)) - exlog denom x
num := numer arg
den := denom arg
(b := (reductum num) / den) ~= 0 =>
a := (leadingMonomial num) / den
is?(op,'exp) => exp(expand a) * expand(exp b)
is?(op,'sin) =>
sin(expand a) * expand(cos b) + cos(expand a) * expand(sin b)
is?(op,'cos) =>
cos(expand a) * expand(cos b) - sin(expand a) * expand(sin b)
is?(op,'tan) =>
ta := tan expand a
tb := expand tan b
(ta + tb) / (1 - ta * tb)
is?(op,'cot) =>
cta := cot expand a
ctb := expand cot b
(cta * ctb - 1) / (ctb + cta)
op [expand x for x in argument k]$List(F)
op [expand x for x in argument k]$List(F)
smpexp p ==
ans:F := 0
while p ~= 0 repeat
ans := ans + termexp leadingMonomial p
p := reductum p
ans
-- this now works in 3 passes over the expression:
-- pass1 rewrites trigs and htrigs in terms of sin,cos,sinh,cosh
-- pass2 rewrites sin**2 and sinh**2 in terms of cos and cosh.
-- pass3 groups exponentials together
simplify0 x ==
simplifyExp eval(eval(x,
['tan,'cot,'sec,'csc,
'tanh,'coth,'sech,'csch],
[t2t,c2t,s2c,c2s,th2th,ch2th,sh2ch,ch2sh]),
['sin, 'sinh], [2, 2], [s2c2, sh2ch2])
@
\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 function funcpkgs MANIP combfunc
-- algfunc elemntry constant funceval fe
<<package FACTFUNC FactoredFunctions>>
<<package POLYROOT PolynomialRoots>>
<<package ALGMANIP AlgebraicManipulations>>
<<package SIMPAN SimplifyAlgebraicNumberConvertPackage>>
<<package TRMANIP TranscendentalManipulations>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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