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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 package ISUMP InnerPolySum
++ Summation of polynomials
++ Author: SMW
++ Date Created: ???
++ Date Last Updated: 19 April 1991
++ Description: tools for the summation packages.
InnerPolySum(E, V, R, P): Exports == Impl where
E: OrderedAbelianMonoidSup
V: OrderedSet
R: IntegralDomain
P: PolynomialCategory(R, E, V)
Z ==> Integer
Q ==> Fraction Z
SUP ==> SparseUnivariatePolynomial
Exports ==> with
sum: (P, V, Segment P) -> Record(num:P, den:Z)
++ sum(p(n), n = a..b) returns \spad{p(a) + p(a+1) + ... + p(b)}.
sum: (P, V) -> Record(num:P, den: Z)
++ sum(p(n), n) returns \spad{P(n)},
++ the indefinite sum of \spad{p(n)} with respect to
++ upward difference on n, i.e. \spad{P(n+1) - P(n) = a(n)};
Impl ==> add
import PolynomialNumberTheoryFunctions()
import UnivariatePolynomialCommonDenominator(Z, Q, SUP Q)
pmul: (P, SUP Q) -> Record(num:SUP P, den:Z)
pmul(c, p) ==
pn := (rec := splitDenominator p).num
[map(numer(#1) * c,
pn)$SparseUnivariatePolynomialFunctions2(Q, P), rec.den]
sum(p, v, s) ==
indef := sum(p, v)
[eval(indef.num, v, 1 + hi s) - eval(indef.num, v, lo s),
indef.den]
sum(p, v) ==
up := univariate(p, v)
lp := nil()$List(SUP P)
ld := nil()$List(Z)
while up ~= 0 repeat
ud := degree up; uc := leadingCoefficient up
up := reductum up
rec := pmul(uc, 1 / (ud+1) * bernoulli(ud+1))
lp := concat(rec.num, lp)
ld := concat(rec.den, ld)
d := lcm ld
vp := +/[(d exquo di)::Z * pi for di in ld for pi in lp]
[multivariate(vp, v), d]
)abbrev package GOSPER GosperSummationMethod
++ Gosper's summation algorithm
++ Author: SMW
++ Date Created: ???
++ Date Last Updated: 19 August 1991
++ Description: Gosper's summation algorithm.
GosperSummationMethod(E, V, R, P, Q): Exports == Impl where
E: OrderedAbelianMonoidSup
V: OrderedSet
R: IntegralDomain
P: PolynomialCategory(R, E, V)
Q: Join(RetractableTo Fraction Integer, Field with
(coerce: P -> %; numer : % -> P; denom : % -> P))
I ==> Integer
RN ==> Fraction I
PQ ==> SparseMultivariatePolynomial(RN, V)
RQ ==> Fraction PQ
Exports ==> with
GospersMethod: (Q, V, () -> V) -> Union(Q, "failed")
++ GospersMethod(b, n, new) returns a rational function
++ \spad{rf(n)} such that \spad{a(n) * rf(n)} is the indefinite
++ sum of \spad{a(n)}
++ with respect to upward difference on \spad{n}, i.e.
++ \spad{a(n+1) * rf(n+1) - a(n) * rf(n) = a(n)},
++ where \spad{b(n) = a(n)/a(n-1)} is a rational function.
++ Returns "failed" if no such rational function \spad{rf(n)}
++ exists.
++ Note: \spad{new} is a nullary function returning a new
++ V every time.
++ The condition on \spad{a(n)} is that \spad{a(n)/a(n-1)}
++ is a rational function of \spad{n}.
--++ \spad{sum(a(n), n) = rf(n) * a(n)}.
Impl ==> add
import PolynomialCategoryQuotientFunctions(E, V, R, P, Q)
import LinearSystemMatrixPackage(RQ,Vector RQ,Vector RQ,Matrix RQ)
InnerGospersMethod: (RQ, V, () -> V) -> Union(RQ, "failed")
GosperPQR: (PQ, PQ, V, () -> V) -> List PQ
GosperDegBd: (PQ, PQ, PQ, V, () -> V) -> I
GosperF: (I, PQ, PQ, PQ, V, () -> V) -> Union(RQ, "failed")
linearAndNNIntRoot: (PQ, V) -> Union(I, "failed")
deg0: (PQ, V) -> I -- degree with deg 0 = -1.
pCoef: (PQ, PQ) -> PQ -- pCoef(p, a*b**2)
RF2QIfCan: Q -> Union(RQ, "failed")
UP2QIfCan: P -> Union(PQ,"failed")
RFQ2R : RQ -> Q
PQ2R : PQ -> Q
rat? : R -> Boolean
deg0(p, v) == (zero? p => -1; degree(p, v))
rat? x == retractIfCan(x::P::Q)@Union(RN, "failed") case RN
RFQ2R f == PQ2R(numer f) / PQ2R(denom f)
PQ2R p ==
map(#1::P::Q, #1::Q, p)$PolynomialCategoryLifting(
IndexedExponents V, V, RN, PQ, Q)
GospersMethod(aquo, n, newV) ==
((q := RF2QIfCan aquo) case "failed") or
((u := InnerGospersMethod(q::RQ, n, newV)) case "failed") =>
"failed"
RFQ2R(u::RQ)
RF2QIfCan f ==
(n := UP2QIfCan numer f) case "failed" => "failed"
(d := UP2QIfCan denom f) case "failed" => "failed"
n::PQ / d::PQ
UP2QIfCan p ==
every?(rat?, coefficients p) =>
map(#1::PQ, (retractIfCan(#1::P::Q)@Union(RN, "failed"))::RN::PQ,
p)$PolynomialCategoryLifting(E, V, R, P, PQ)
"failed"
InnerGospersMethod(aquo, n, newV) ==
-- 1. Define coprime polys an,anm1 such that
-- an/anm1=a(n)/a(n-1)
an := numer aquo
anm1 := denom aquo
-- 2. Define p,q,r such that
-- a(n)/a(n-1) = (p(n)/p(n-1)) * (q(n)/r(n))
-- and
-- gcd(q(n), r(n+j)) = 1, for all j: NNI.
pqr:= GosperPQR(an, anm1, n, newV)
pn := first pqr; qn := second pqr; rn := third pqr
-- 3. If the sum is a rational fn, there is a poly f with
-- sum(a(n), n) = q(n+1)/p(n) * a(n) * f(n).
-- 4. Bound the degree of f(n).
negative?(k := GosperDegBd(pn, qn, rn, n, newV)) => "failed"
-- 5. Find a polynomial f of degree at most k, satisfying
-- p(n) = q(n+1)*f(n) - r(n)*f(n-1)
(ufn := GosperF(k, pn, qn, rn, n, newV)) case "failed" =>
"failed"
fn := ufn::RQ
-- 6. The sum is q(n-1)/p(n)*f(n) * a(n). We leave out a(n).
--qnm1 := eval(qn,n,n::PQ - 1)
--qnm1/pn * fn
qn1 := eval(qn,n,n::PQ + 1)
qn1/pn * fn
GosperF(k, pn, qn, rn, n, newV) ==
mv := newV(); mp := mv::PQ; np := n::PQ
fn: PQ := +/[mp**(i+1) * np**i for i in 0..k]
fnminus1: PQ := eval(fn, n, np-1)
qnplus1 := eval(qn, n, np+1)
zro := qnplus1 * fn - rn * fnminus1 - pn
zron := univariate(zro, n)
dz := degree zron
mat: Matrix RQ := zero(dz+1, (k+1)::NonNegativeInteger)
vec: Vector RQ := new(dz+1, 0)
while zron ~= 0 repeat
cz := leadingCoefficient zron
dz := degree zron
zron := reductum zron
mz := univariate(cz, mv)
while mz ~= 0 repeat
cmz := leadingCoefficient(mz)::RQ
dmz := degree mz
mz := reductum mz
dmz = 0 => vec(dz + minIndex vec) := -cmz
qsetelt!(mat, dz + minRowIndex mat,
dmz + minColIndex(mat) - 1, cmz)
(soln := particularSolution(mat, vec)) case "failed" => "failed"
vec := soln::Vector RQ
(+/[np**i * vec(i + minIndex vec) for i in 0..k])@RQ
GosperPQR(an, anm1, n, newV) ==
np := n::PQ -- polynomial version of n
-- Initial guess.
pn: PQ := 1
qn: PQ := an
rn: PQ := anm1
-- Find all j: NNI giving common factors to q(n) and r(n+j).
j := newV()
rnj := eval(rn, n, np + j::PQ)
res := resultant(qn, rnj, n)
fres := factor(res)$MRationalFactorize(IndexedExponents V,
V, I, PQ)
js := [rt::I for fe in factors fres
| (rt := linearAndNNIntRoot(fe.factor,j)) case I]
-- For each such j, change variables to remove the gcd.
for rt in js repeat
rtp:= rt::PQ -- polynomial version of rt
gn := gcd(qn, eval(rn,n,np+rtp))
qn := (qn exquo gn)::PQ
rn := (rn exquo eval(gn, n, np-rtp))::PQ
pn := pn * */[eval(gn, n, np-i::PQ) for i in 0..rt-1]
[pn, qn, rn]
-- Find a degree bound for the polynomial f(n) which satisfies
-- p(n) = q(n+1)*f(n) - r(n)*f(n-1).
GosperDegBd(pn, qn, rn, n, newV) ==
np := n::PQ
qnplus1 := eval(qn, n, np+1)
lplus := deg0(qnplus1 + rn, n)
lminus := deg0(qnplus1 - rn, n)
degp := deg0(pn, n)
k := degp - max(lplus-1, lminus)
lplus <= lminus => k
-- Find L(k), such that
-- p(n) = L(k)*c[k]*n**(k + lplus - 1) + ...
-- To do this, write f(n) and f(n-1) symbolically.
-- f(n) = c[k]*n**k + c[k-1]*n**(k-1) +O(n**(k-2))
-- f(n-1)=c[k]*n**k + (c[k-1]-k*c[k])*n**(k-1)+O(n**(k-2))
kk := newV()::PQ
ck := newV()::PQ
ckm1 := newV()::PQ
nkm1:= newV()::PQ
nk := np*nkm1
headfn := ck*nk + ckm1*nkm1
headfnm1 := ck*nk + (ckm1-kk*ck)*nkm1
-- Then p(n) = q(n+1)*f(n) - r(n)*f(n-1) gives L(k).
pk := qnplus1 * headfn - rn * headfnm1
lcpk := pCoef(pk, ck*np*nkm1)
-- The degree bd is now given by k, and the root of L.
k0 := linearAndNNIntRoot(lcpk, mainVariable(kk)::V)
k0 case "failed" => k
max(k0::I, k)
pCoef(p, nom) ==
not monomial? nom =>
error "pCoef requires a monomial 2nd arg"
vlist := variables nom
for v in vlist while p ~= 0 repeat
unom:= univariate(nom,v)
pow:=degree unom
nom:=leadingCoefficient unom
up := univariate(p, v)
p := coefficient(up, pow)
p
linearAndNNIntRoot(mp, v) ==
p := univariate(mp, v)
not one? degree p => "failed"
(p1 := retractIfCan(coefficient(p, 1))@Union(RN,"failed"))
case "failed" or
(p0 := retractIfCan(coefficient(p, 0))@Union(RN,"failed"))
case "failed" => "failed"
rt := -(p0::RN)/(p1::RN)
negative? rt or not one? denom rt => "failed"
numer rt
)abbrev package SUMRF RationalFunctionSum
++ Summation of rational functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 19 April 1991
++ Description: Computes sums of rational functions;
RationalFunctionSum(R): Exports == Impl where
R: Join(IntegralDomain, RetractableTo Integer)
P ==> Polynomial R
RF ==> Fraction P
FE ==> Expression R
SE ==> Symbol
Exports ==> with
sum: (P, SE) -> RF
++ sum(a(n), n) returns \spad{A} which
++ is the indefinite sum of \spad{a} with respect to
++ upward difference on \spad{n}, i.e. \spad{A(n+1) - A(n) = a(n)}.
sum: (RF, SE) -> Union(RF, FE)
++ sum(a(n), n) returns \spad{A} which
++ is the indefinite sum of \spad{a} with respect to
++ upward difference on \spad{n}, i.e. \spad{A(n+1) - A(n) = a(n)}.
sum: (P, SegmentBinding P) -> RF
++ sum(f(n), n = a..b) returns \spad{f(a) + f(a+1) + ... f(b)}.
sum: (RF, SegmentBinding RF) -> Union(RF, FE)
++ sum(f(n), n = a..b) returns \spad{f(a) + f(a+1) + ... f(b)}.
Impl ==> add
import RationalFunction R
import GosperSummationMethod(IndexedExponents SE, SE, R, P, RF)
innersum : (RF, SE) -> Union(RF, "failed")
innerpolysum: (P, SE) -> RF
sum(f:RF, s:SegmentBinding RF) ==
(indef := innersum(f, v := variable s)) case "failed" =>
summation(f::FE,map(#1::FE,s)$SegmentBindingFunctions2(RF,FE))
eval(indef::RF, v, 1 + hi segment s)
- eval(indef::RF, v,lo segment s)
sum(an:RF, n:SE) ==
(u := innersum(an, n)) case "failed" => summation(an::FE, n)
u::RF
sum(p:P, s:SegmentBinding P) ==
f := sum(p, v := variable s)
eval(f, v, (1 + hi segment s)::RF) - eval(f,v,lo(segment s)::RF)
innersum(an, n) ==
(r := retractIfCan(an)@Union(P, "failed")) case "failed" =>
an1 := eval(an, n, -1 + n::RF)
(u := GospersMethod(an/an1, n, new$SE)) case "failed" =>
"failed"
an1 * eval(u::RF, n, -1 + n::RF)
sum(r::P, n)
sum(p:P, n:SE) ==
rec := sum(p, n)$InnerPolySum(IndexedExponents SE, SE, R, P)
rec.num / (rec.den :: P)
|