/usr/share/axiom-20170501/src/algebra/GOSPER.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 | )abbrev package GOSPER GosperSummationMethod
++ Author: SMW
++ Date Last Updated: 19 August 1991
++ Description:
++ Gosper's summation algorithm.
GosperSummationMethod(E, V, R, P, Q) : SIG == CODE 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
SIG ==> 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},
++ \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 that \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)}.
CODE ==> 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(x+->x::P::Q, y+->y::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(x +-> x::PQ,
y +-> (retractIfCan(y::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).
(k := GosperDegBd(pn, qn, rn, n, newV)) < 0 => "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)
degree p ^= 1 => "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)
rt < 0 or denom rt ^= 1 => "failed"
numer rt
|