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--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.
)abbrev domain PFR PartialFraction
++ Author: Robert S. Sutor
++ Date Created: 1986
++ Change History:
++ 05/20/91 BMT Converted to the new library
++ Basic Operations: (Field), (Algebra),
++ coerce, compactFraction, firstDenom, firstNumer,
++ nthFractionalTerm, numberOfFractionalTerms, padicallyExpand,
++ padicFraction, partialFraction, wholePart
++ Related Constructors:
++ Also See: ContinuedFraction
++ AMS Classifications:
++ Keywords: partial fraction, factorization, euclidean domain
++ References:
++ Description:
++ The domain \spadtype{PartialFraction} implements partial fractions
++ over a euclidean domain \spad{R}. This requirement on the
++ argument domain allows us to normalize the fractions. Of
++ particular interest are the 2 forms for these fractions. The
++ ``compact'' form has only one fractional term per prime in the
++ denominator, while the ``p-adic'' form expands each numerator
++ p-adically via the prime p in the denominator. For computational
++ efficiency, the compact form is used, though the p-adic form may
++ be gotten by calling the function \spadfunFrom{padicFraction}{PartialFraction}. For a
++ general euclidean domain, it is not known how to factor the
++ denominator. Thus the function \spadfunFrom{partialFraction}{PartialFraction} takes as its
++ second argument an element of \spadtype{Factored(R)}.
PartialFraction(R: EuclideanDomain): Cat == Capsule where
FRR ==> Factored R
SUPR ==> SparseUnivariatePolynomial R
Cat == Join(Field, Algebra R) with
coerce: % -> Fraction R
++ coerce(p) sums up the components of the partial fraction and
++ returns a single fraction.
coerce: Fraction FRR -> %
++ coerce(f) takes a fraction with numerator and denominator in
++ factored form and creates a partial fraction. It is
++ necessary for the parts to be factored because it is not
++ known in general how to factor elements of \spad{R} and
++ this is needed to decompose into partial fractions.
compactFraction: % -> %
++ compactFraction(p) normalizes the partial fraction \spad{p}
++ to the compact representation. In this form, the partial
++ fraction has only one fractional term per prime in the
++ denominator.
firstDenom: % -> FRR
++ firstDenom(p) extracts the denominator of the first fractional
++ term. This returns 1 if there is no fractional part (use
++ \spadfunFrom{wholePart}{PartialFraction} to get the whole part).
firstNumer: % -> R
++ firstNumer(p) extracts the numerator of the first fractional
++ term. This returns 0 if there is no fractional part (use
++ \spadfunFrom{wholePart}{PartialFraction} to get the whole part).
nthFractionalTerm: (%,Integer) -> %
++ nthFractionalTerm(p,n) extracts the nth fractional term from
++ the partial fraction \spad{p}. This returns 0 if the index
++ \spad{n} is out of range.
numberOfFractionalTerms: % -> Integer
++ numberOfFractionalTerms(p) computes the number of fractional
++ terms in \spad{p}. This returns 0 if there is no fractional
++ part.
padicallyExpand: (R,R) -> SUPR
++ padicallyExpand(p,x) is a utility function that expands
++ the second argument \spad{x} ``p-adically'' in
++ the first.
padicFraction: % -> %
++ padicFraction(q) expands the fraction p-adically in the primes
++ \spad{p} in the denominator of \spad{q}. For example,
++ \spad{padicFraction(3/(2**2)) = 1/2 + 1/(2**2)}.
++ Use \spadfunFrom{compactFraction}{PartialFraction} to return to compact form.
partialFraction: (R, FRR) -> %
++ partialFraction(numer,denom) is the main function for
++ constructing partial fractions. The second argument is the
++ denominator and should be factored.
wholePart: % -> R
++ wholePart(p) extracts the whole part of the partial fraction
++ \spad{p}.
Capsule == add
-- some constructor assignments and macros
Ex ==> OutputForm
fTerm ==> Record(num: R, den: FRR) -- den should have
-- unit = 1 and only
-- 1 factor
LfTerm ==> List Record(num: R, den: FRR)
QR ==> Record(quotient: R, remainder: R)
Rep := Record(whole:R, fract: LfTerm)
import %before?: (FRR,FRR) -> Boolean from Foreign Builtin
-- private function signatures
copypf: % -> %
multiplyFracTerms: (fTerm, fTerm) -> %
normalizeFracTerm: fTerm -> %
partialFractionNormalized: (R, FRR) -> %
-- declarations
a,b: %
n: Integer
r: R
-- private function definitions
copypf(a: %): % == [a.whole,copy a.fract]$%
LessThan(s: fTerm, t: fTerm): Boolean ==
-- have to wait until FR has < operation
%before?(s.den,t.den)
multiplyFracTerms(s : fTerm, t : fTerm) ==
nthFactor(s.den,1) = nthFactor(t.den,1) =>
normalizeFracTerm([s.num * t.num, s.den * t.den]$fTerm) : Rep
i : Union(Record(coef1: R, coef2: R),"failed")
coefs : Record(coef1: R, coef2: R)
i := extendedEuclidean(expand t.den, expand s.den,s.num * t.num)
i case "failed" => error "PartialFraction: not in ideal"
coefs := (i :: Record(coef1: R, coef2: R))
c : % := copypf 0$%
d : %
if coefs.coef2 ~= 0$R then
c := normalizeFracTerm ([coefs.coef2, t.den]$fTerm)
if coefs.coef1 ~= 0$R then
d := normalizeFracTerm ([coefs.coef1, s.den]$fTerm)
c.whole := c.whole + d.whole
not (null d.fract) => c.fract := append(d.fract,c.fract)
c
normalizeFracTerm(s : fTerm) ==
-- makes sure num is "less than" den, whole may be non-zero
qr : QR := divide(s.num, (expand s.den))
qr.remainder = 0$R => [qr.quotient, nil()$LfTerm]
-- now verify num and den are coprime
f : R := nthFactor(s.den,1)
nexpon : Integer := nthExponent(s.den,1)
expon : Integer := 0
q : QR := divide(qr.remainder, f)
while (q.remainder = 0$R) and (expon < nexpon) repeat
expon := expon + 1
qr.remainder := q.quotient
q := divide(qr.remainder,f)
expon = 0 => [qr.quotient,[[qr.remainder, s.den]$fTerm]$LfTerm]
expon = nexpon => (qr.quotient + qr.remainder) :: %
[qr.quotient,[[qr.remainder, nilFactor(f,nexpon-expon)]$fTerm]$LfTerm]
partialFractionNormalized(nm: R, dn : FRR) ==
-- assume unit dn = 1
nm = 0$R => 0$%
dn = 1$FRR => nm :: %
qr : QR := divide(nm, expand dn)
c : % := [0$R,[[qr.remainder,
nilFactor(nthFactor(dn,1), nthExponent(dn,1))]$fTerm]$LfTerm]
d : %
for i in 2..numberOfFactors(dn) repeat
d :=
[0$R,[[1$R,nilFactor(nthFactor(dn,i), nthExponent(dn,i))]$fTerm]$LfTerm]
c := c * d
(qr.quotient :: %) + c
-- public function definitions
padicFraction(a : %) ==
b: % := compactFraction a
null b.fract => b
l : LfTerm := nil
f : R
e,d: Integer
for s in b.fract repeat
e := nthExponent(s.den,1)
e = 1 => l := cons(s,l)
f := nthFactor(s.den,1)
d := degree(sp := padicallyExpand(f,s.num))
while (sp ~= 0$SUPR) repeat
l := cons([leadingCoefficient sp,nilFactor(f,e-d)]$fTerm, l)
d := degree(sp := reductum sp)
[b.whole, sort(LessThan,l)]$%
compactFraction(a : %) ==
-- only one power for each distinct denom will remain
2 > # a.fract => a
af : LfTerm := reverse a.fract
bf : LfTerm := nil
bw : R := a.whole
b : %
s : fTerm := [(first af).num,(first af).den]$fTerm
f : R := nthFactor(s.den,1)
e : Integer := nthExponent(s.den,1)
for t in rest af repeat
f = nthFactor(t.den,1) =>
s.num := s.num + (t.num *
(f **$R ((e - nthExponent(t.den,1)) : NonNegativeInteger)))
b := normalizeFracTerm s
bw := bw + b.whole
if not (null b.fract) then bf := cons(first b.fract,bf)
s := [t.num, t.den]$fTerm
f := nthFactor(s.den,1)
e := nthExponent(s.den,1)
b := normalizeFracTerm s
[bw + b.whole,append(b.fract,bf)]$%
0 == [0$R, nil()$LfTerm]
1 == [1$R, nil()$LfTerm]
characteristic == characteristic$R
coerce(r): % == [r, nil()$LfTerm]
coerce(n): % == [(n :: R), nil()$LfTerm]
coerce(a): Fraction R ==
q : Fraction R := (a.whole :: Fraction R)
for s in a.fract repeat
q := q + (s.num / (expand s.den))
q
coerce(q: Fraction FRR): % ==
u : R := (recip unit denom q):: R
r1 : R := u * expand numer q
partialFractionNormalized(r1, u * denom q)
a exquo b ==
b = 0$% => "failed"
b = 1$% => a
br : Fraction R := inv (b :: Fraction R)
a * partialFraction(numer br,(denom br) :: FRR)
recip a == (1$% exquo a)
firstDenom a == -- denominator of 1st fractional term
null a.fract => 1$FRR
(first a.fract).den
firstNumer a == -- numerator of 1st fractional term
null a.fract => 0$R
(first a.fract).num
numberOfFractionalTerms a == # a.fract
nthFractionalTerm(a,n) ==
l : LfTerm := a.fract
(n < 1) or (n > # l) => 0$%
[0$R,[l.n]$LfTerm]$%
wholePart a == a.whole
partialFraction(nm: R, dn : FRR) ==
nm = 0$R => 0$%
-- move inv unit of den to numerator
u : R := unit dn
u := (recip u) :: R
partialFractionNormalized(u * nm,u * dn)
padicallyExpand(p : R, r : R) ==
-- expands r as a sum of powers of p, with coefficients
-- r = HornerEval(padicallyExpand(p,r),p)
qr : QR := divide(r, p)
qr.quotient = 0$R => qr.remainder :: SUPR
(qr.remainder :: SUPR) + monomial(1$R,1$NonNegativeInteger)$SUPR *
padicallyExpand(p,qr.quotient)
a = b ==
a.whole ~= b.whole => false -- must verify this
(null a.fract) =>
null b.fract => a.whole = b.whole
false
null b.fract => false
-- oh, no! following is temporary
(a :: Fraction R) = (b :: Fraction R)
- a ==
l: LfTerm := nil
for s in reverse a.fract repeat l := cons([- s.num,s.den]$fTerm,l)
[- a.whole,l]
r * a ==
r = 0$R => 0$%
r = 1$R => a
b : % := (r * a.whole) :: %
c : %
for s in reverse a.fract repeat
c := normalizeFracTerm [r * s.num, s.den]$fTerm
b.whole := b.whole + c.whole
not (null c.fract) => b.fract := append(c.fract, b.fract)
b
n * a == (n :: R) * a
a + b ==
compactFraction
[a.whole + b.whole,
sort(LessThan,append(a.fract,copy b.fract))]$%
a * b ==
null a.fract => a.whole * b
null b.fract => b.whole * a
af : % := [0$R, a.fract]$% -- a - a.whole
c: % := (a.whole * b) + (b.whole * af)
for s in a.fract repeat
for t in b.fract repeat
c := c + multiplyFracTerms(s,t)
c
coerce(a): Ex ==
null a.fract => a.whole :: Ex
l : List Ex
if a.whole = 0 then l := nil else l := [a.whole :: Ex]
for s in a.fract repeat
s.den = 1$FRR => l := cons(s.num :: Ex, l)
l := cons(s.num :: Ex / s.den :: Ex, l)
# l = 1 => first l
reduce("+", reverse l)
)abbrev package PFRPAC PartialFractionPackage
++ Author: Barry M. Trager
++ Date Created: 1992
++ BasicOperations:
++ Related Constructors: PartialFraction
++ Also See:
++ AMS Classifications:
++ Keywords: partial fraction, factorization, euclidean domain
++ References:
++ Description:
++ The package \spadtype{PartialFractionPackage} gives an easier
++ to use interfact the domain \spadtype{PartialFraction}.
++ The user gives a fraction of polynomials, and a variable and
++ the package converts it to the proper datatype for the
++ \spadtype{PartialFraction} domain.
PartialFractionPackage(R): Cat == Capsule where
-- R : UniqueFactorizationDomain -- not yet supported
R : Join(EuclideanDomain, CharacteristicZero)
FPR ==> Fraction Polynomial R
INDE ==> IndexedExponents Symbol
PR ==> Polynomial R
SUP ==> SparseUnivariatePolynomial
Cat == with
partialFraction: (FPR, Symbol) -> Any
++ partialFraction(rf, var) returns the partial fraction decomposition
++ of the rational function rf with respect to the variable var.
partialFraction: (PR, Factored PR, Symbol) -> Any
++ partialFraction(num, facdenom, var) returns the partial fraction
++ decomposition of the rational function whose numerator is num and
++ whose factored denominator is facdenom with respect to the variable var.
Capsule == add
partialFraction(rf, v) ==
df := factor(denom rf)$MultivariateFactorize(Symbol, INDE,R,PR)
partialFraction(numer rf, df, v)
makeSup(p:Polynomial R, v:Symbol) : SparseUnivariatePolynomial FPR ==
up := univariate(p,v)
map(#1::FPR,up)$UnivariatePolynomialCategoryFunctions2(PR, SUP PR, FPR, SUP FPR)
partialFraction(p, facq, v) ==
up := UnivariatePolynomial(v, Fraction Polynomial R)
fup := Factored up
ffact : List(Record(irr:up,pow:Integer))
ffact:=[[makeSup(u.factor,v) pretend up,u.exponent]
for u in factors facq]
fcont:=makeSup(unit facq,v) pretend up
nflist:fup := fcont*(*/[primeFactor(ff.irr,ff.pow) for ff in ffact])
pfup:=partialFraction(makeSup(p,v) pretend up, nflist)$PartialFraction(up)
coerce(pfup)$AnyFunctions1(PartialFraction up)
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