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--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.
-- Compile order for the differential equation solver:
-- oderf.spad odealg.spad nlode.spad nlinsol.spad riccati.spad odeef.spad
)abbrev package ODEPRRIC PrimitiveRatRicDE
++ Author: Manuel Bronstein
++ Date Created: 22 October 1991
++ Date Last Updated: 2 February 1993
++ Description: In-field solution of Riccati equations, primitive case.
PrimitiveRatRicDE(F, UP, L, LQ): Exports == Implementation where
F : Join(Field, CharacteristicZero, RetractableTo Fraction Integer)
UP : UnivariatePolynomialCategory F
L : LinearOrdinaryDifferentialOperatorCategory UP
LQ : LinearOrdinaryDifferentialOperatorCategory Fraction UP
N ==> NonNegativeInteger
Z ==> Integer
RF ==> Fraction UP
UP2 ==> SparseUnivariatePolynomial UP
REC ==> Record(deg:N, eq:UP)
REC2 ==> Record(deg:N, eq:UP2)
POL ==> Record(poly:UP, eq:L)
FRC ==> Record(frac:RF, eq:L)
CNT ==> Record(constant:F, eq:L)
IJ ==> Record(ij: List Z, deg:N)
Exports ==> with
denomRicDE: L -> UP
++ denomRicDE(op) returns a polynomial \spad{d} such that any rational
++ solution of the associated Riccati equation of \spad{op y = 0} is
++ of the form \spad{p/d + q'/q + r} for some polynomials p and q
++ and a reduced r. Also, \spad{deg(p) < deg(d)} and {gcd(d,q) = 1}.
leadingCoefficientRicDE: L -> List REC
++ leadingCoefficientRicDE(op) returns
++ \spad{[[m1, p1], [m2, p2], ... , [mk, pk]]} such that the polynomial
++ part of any rational solution of the associated Riccati equation of
++ \spad{op y = 0} must have degree mj for some j, and its leading
++ coefficient is then a zero of pj. In addition,\spad{m1>m2> ... >mk}.
constantCoefficientRicDE: (L, UP -> List F) -> List CNT
++ constantCoefficientRicDE(op, ric) returns
++ \spad{[[a1, L1], [a2, L2], ... , [ak, Lk]]} such that any rational
++ solution with no polynomial part of the associated Riccati equation of
++ \spad{op y = 0} must be one of the ai's in which case the equation for
++ \spad{z = y e^{-int ai}} is \spad{Li z = 0}.
++ \spad{ric} is a Riccati equation solver over \spad{F}, whose input
++ is the associated linear equation.
polyRicDE: (L, UP -> List F) -> List POL
++ polyRicDE(op, zeros) returns
++ \spad{[[p1, L1], [p2, L2], ... , [pk, Lk]]} such that the polynomial
++ part of any rational solution of the associated Riccati equation of
++ \spad{op y=0} must be one of the pi's (up to the constant coefficient),
++ in which case the equation for \spad{z=y e^{-int p}} is \spad{Li z =0}.
++ \spad{zeros} is a zero finder in \spad{UP}.
singRicDE: (L, (UP, UP2) -> List UP, UP -> Factored UP) -> List FRC
++ singRicDE(op, zeros, ezfactor) returns
++ \spad{[[f1, L1], [f2, L2], ... , [fk, Lk]]} such that the singular
++ part of any rational solution of the associated Riccati equation of
++ \spad{op y=0} must be one of the fi's (up to the constant coefficient),
++ in which case the equation for \spad{z=y e^{-int p}} is \spad{Li z=0}.
++ \spad{zeros(C(x),H(x,y))} returns all the \spad{P_i(x)}'s such that
++ \spad{H(x,P_i(x)) = 0 modulo C(x)}.
++ Argument \spad{ezfactor} is a factorisation in \spad{UP},
++ not necessarily into irreducibles.
changeVar: (L, UP) -> L
++ changeVar(+/[ai D^i], a) returns the operator \spad{+/[ai (D+a)^i]}.
changeVar: (L, RF) -> L
++ changeVar(+/[ai D^i], a) returns the operator \spad{+/[ai (D+a)^i]}.
Implementation ==> add
import PrimitiveRatDE(F, UP, L, LQ)
import BalancedFactorisation(F, UP)
bound : (UP, L) -> N
lambda : (UP, L) -> List IJ
infmax : (IJ, L) -> List Z
dmax : (IJ, UP, L) -> List Z
getPoly : (IJ, L, List Z) -> UP
getPol : (IJ, UP, L, List Z) -> UP2
innerlb : (L, UP -> Z) -> List IJ
innermax : (IJ, L, UP -> Z) -> List Z
tau0 : (UP, UP) -> UP
poly1 : (UP, UP, Z) -> UP2
getPol1 : (List Z, UP, L) -> UP2
getIndices : (N, List IJ) -> List Z
refine : (List UP, UP -> Factored UP) -> List UP
polysol : (L, N, Boolean, UP -> List F) -> List POL
fracsol : (L, (UP, UP2) -> List UP, List UP) -> List FRC
padicsol : (UP, L, N, Boolean, (UP, UP2) -> List UP) -> List FRC
leadingDenomRicDE : (UP, L) -> List REC2
factoredDenomRicDE: L -> List UP
constantCoefficientOperator: (L, N) -> UP
infLambda: L -> List IJ
-- infLambda(op) returns
-- \spad{[[[i,j], (\deg(a_i)-\deg(a_j))/(i-j) ]]} for all the pairs
-- of indices \spad{i,j} such that \spad{(\deg(a_i)-\deg(a_j))/(i-j)} is
-- an integer.
diff := D()$L
diffq := D()$LQ
lambda(c, l) == innerlb(l, order(#1, c)::Z)
infLambda l == innerlb(l, -(degree(#1)::Z))
infmax(rec, l) == innermax(rec, l, degree(#1)::Z)
dmax(rec, c, l) == innermax(rec, l, - order(#1, c)::Z)
tau0(p, q) == ((q exquo (p ** order(q, p)))::UP) rem p
poly1(c, cp, i) == */[monomial(1,1)$UP2 - (j * cp)::UP2 for j in 0..i-1]
getIndices(n, l) == removeDuplicates! concat [r.ij for r in l | r.deg=n]
denomRicDE l == */[c ** bound(c, l) for c in factoredDenomRicDE l]
polyRicDE(l, zeros) == concat([0, l], polysol(l, 0, false, zeros))
-- refine([p1,...,pn], foo) refines the list of factors using foo
refine(l, ezfactor) ==
concat [[r.factor for r in factors ezfactor p] for p in l]
-- returns [] if the solutions of l have no p-adic component at c
padicsol(c, op, b, finite?, zeros) ==
ans:List(FRC) := empty()
finite? and zero? b => ans
lc := leadingDenomRicDE(c, op)
if finite? then lc := select!(#1.deg <= b, lc)
for rec in lc repeat
for r in zeros(c, rec.eq) | r ~= 0 repeat
rcn := r /$RF (c ** rec.deg)
neweq := changeVar(op, rcn)
sols := padicsol(c, neweq, (rec.deg-1)::N, true, zeros)
ans :=
empty? sols => concat([rcn, neweq], ans)
concat!([[rcn + sol.frac, sol.eq] for sol in sols], ans)
ans
leadingDenomRicDE(c, l) ==
ind:List(Z) -- to cure the compiler... (won't compile without)
lb := lambda(c, l)
done:List(N) := empty()
ans:List(REC2) := empty()
for rec in lb | (not member?(rec.deg, done)) and
not(empty?(ind := dmax(rec, c, l))) repeat
ans := concat([rec.deg, getPol(rec, c, l, ind)], ans)
done := concat(rec.deg, done)
sort!(#1.deg > #2.deg, ans)
getPol(rec, c, l, ind) ==
one?(rec.deg) => getPol1(ind, c, l)
+/[monomial(tau0(c, coefficient(l, i::N)), i::N)$UP2 for i in ind]
getPol1(ind, c, l) ==
cp := diff c
+/[tau0(c, coefficient(l, i::N)) * poly1(c, cp, i) for i in ind]
constantCoefficientRicDE(op, ric) ==
m := "max"/[degree p for p in coefficients op]
[[a, changeVar(op,a::UP)] for a in ric constantCoefficientOperator(op,m)]
constantCoefficientOperator(op, m) ==
ans:UP := 0
while op ~= 0 repeat
if degree(p := leadingCoefficient op) = m then
ans := ans + monomial(leadingCoefficient p, degree op)
op := reductum op
ans
getPoly(rec, l, ind) ==
+/[monomial(leadingCoefficient coefficient(l,i::N),i::N)$UP for i in ind]
-- returns empty() if rec is does not reach the max,
-- the list of indices (including rec) that reach the max otherwise
innermax(rec, l, nu) ==
n := degree l
i := first(rec.ij)
m := i * (d := rec.deg) + nu coefficient(l, i::N)
ans:List(Z) := empty()
for j in 0..n | (f := coefficient(l, j)) ~= 0 repeat
if ((k := (j * d + nu f)) > m) then return empty()
else if (k = m) then ans := concat(j, ans)
ans
leadingCoefficientRicDE l ==
ind:List(Z) -- to cure the compiler... (won't compile without)
lb := infLambda l
done:List(N) := empty()
ans:List(REC) := empty()
for rec in lb | (not member?(rec.deg, done)) and
not(empty?(ind := infmax(rec, l))) repeat
ans := concat([rec.deg, getPoly(rec, l, ind)], ans)
done := concat(rec.deg, done)
sort!(#1.deg > #2.deg, ans)
factoredDenomRicDE l ==
bd := factors balancedFactorisation(leadingCoefficient l, coefficients l)
[dd.factor for dd in bd]
changeVar(l:L, a:UP) ==
dpa := diff + a::L -- the operator (D + a)
dpan:L := 1 -- will accumulate the powers of (D + a)
op:L := 0
for i in 0..degree l repeat
op := op + coefficient(l, i) * dpan
dpan := dpa * dpan
primitivePart op
changeVar(l:L, a:RF) ==
dpa := diffq + a::LQ -- the operator (D + a)
dpan:LQ := 1 -- will accumulate the powers of (D + a)
op:LQ := 0
for i in 0..degree l repeat
op := op + coefficient(l, i)::RF * dpan
dpan := dpa * dpan
splitDenominator(op, empty()).eq
bound(c, l) ==
empty?(lb := lambda(c, l)) => 1
"max"/[rec.deg for rec in lb]
-- returns all the pairs [[i, j], n] such that
-- n = (nu(i) - nu(j)) / (i - j) is an integer
innerlb(l, nu) ==
lb:List(IJ) := empty()
n := degree l
for i in 0..n | (li := coefficient(l, i)) ~= 0 repeat
for j in i+1..n | (lj := coefficient(l, j)) ~= 0 repeat
u := (nu li - nu lj) exquo (i-j)
if (u case Z) and positive?(b := u::Z) then
lb := concat([[i, j], b::N], lb)
lb
singRicDE(l, zeros, ezfactor) ==
concat([0, l], fracsol(l, zeros, refine(factoredDenomRicDE l, ezfactor)))
-- returns [] if the solutions of l have no singular component
fracsol(l, zeros, lc) ==
ans:List(FRC) := empty()
empty? lc => ans
empty?(sols := padicsol(first lc, l, 0, false, zeros)) =>
fracsol(l, zeros, rest lc)
for rec in sols repeat
neweq := changeVar(l, rec.frac)
sols := fracsol(neweq, zeros, rest lc)
ans :=
empty? sols => concat(rec, ans)
concat!([[rec.frac + sol.frac, sol.eq] for sol in sols], ans)
ans
-- returns [] if the solutions of l have no polynomial component
polysol(l, b, finite?, zeros) ==
ans:List(POL) := empty()
finite? and zero? b => ans
lc := leadingCoefficientRicDE l
if finite? then lc := select!(#1.deg <= b, lc)
for rec in lc repeat
for a in zeros(rec.eq) | a ~= 0 repeat
atn:UP := monomial(a, rec.deg)
neweq := changeVar(l, atn)
sols := polysol(neweq, (rec.deg - 1)::N, true, zeros)
ans :=
empty? sols => concat([atn, neweq], ans)
concat!([[atn + sol.poly, sol.eq] for sol in sols], ans)
ans
)abbrev package ODERTRIC RationalRicDE
++ Author: Manuel Bronstein
++ Date Created: 22 October 1991
++ Date Last Updated: 11 April 1994
++ Description: In-field solution of Riccati equations, rational case.
RationalRicDE(F, UP): Exports == Implementation where
F : Join(Field, CharacteristicZero, RetractableTo Integer,
RetractableTo Fraction Integer)
UP : UnivariatePolynomialCategory F
N ==> NonNegativeInteger
Z ==> Integer
SY ==> Symbol
P ==> Polynomial F
RF ==> Fraction P
EQ ==> Equation RF
QF ==> Fraction UP
UP2 ==> SparseUnivariatePolynomial UP
SUP ==> SparseUnivariatePolynomial P
REC ==> Record(poly:SUP, vars:List SY)
SOL ==> Record(var:List SY, val:List F)
POL ==> Record(poly:UP, eq:L)
FRC ==> Record(frac:QF, eq:L)
CNT ==> Record(constant:F, eq:L)
UTS ==> UnivariateTaylorSeries(F, dummy, 0)
UPS ==> SparseUnivariatePolynomial UTS
L ==> LinearOrdinaryDifferentialOperator2(UP, QF)
LQ ==> LinearOrdinaryDifferentialOperator1 QF
Exports ==> with
ricDsolve: (LQ, UP -> List F) -> List QF
++ ricDsolve(op, zeros) returns the rational solutions of the associated
++ Riccati equation of \spad{op y = 0}.
++ \spad{zeros} is a zero finder in \spad{UP}.
ricDsolve: (LQ, UP -> List F, UP -> Factored UP) -> List QF
++ ricDsolve(op, zeros, ezfactor) returns the rational
++ solutions of the associated Riccati equation of \spad{op y = 0}.
++ \spad{zeros} is a zero finder in \spad{UP}.
++ Argument \spad{ezfactor} is a factorisation in \spad{UP},
++ not necessarily into irreducibles.
ricDsolve: (L, UP -> List F) -> List QF
++ ricDsolve(op, zeros) returns the rational solutions of the associated
++ Riccati equation of \spad{op y = 0}.
++ \spad{zeros} is a zero finder in \spad{UP}.
ricDsolve: (L, UP -> List F, UP -> Factored UP) -> List QF
++ ricDsolve(op, zeros, ezfactor) returns the rational
++ solutions of the associated Riccati equation of \spad{op y = 0}.
++ \spad{zeros} is a zero finder in \spad{UP}.
++ Argument \spad{ezfactor} is a factorisation in \spad{UP},
++ not necessarily into irreducibles.
singRicDE: (L, UP -> Factored UP) -> List FRC
++ singRicDE(op, ezfactor) returns \spad{[[f1,L1], [f2,L2],..., [fk,Lk]]}
++ such that the singular ++ part of any rational solution of the
++ associated Riccati equation of \spad{op y = 0} must be one of the fi's
++ (up to the constant coefficient), in which case the equation for
++ \spad{z = y e^{-int ai}} is \spad{Li z = 0}.
++ Argument \spad{ezfactor} is a factorisation in \spad{UP},
++ not necessarily into irreducibles.
polyRicDE: (L, UP -> List F) -> List POL
++ polyRicDE(op, zeros) returns \spad{[[p1, L1], [p2, L2], ... , [pk,Lk]]}
++ such that the polynomial part of any rational solution of the
++ associated Riccati equation of \spad{op y = 0} must be one of the pi's
++ (up to the constant coefficient), in which case the equation for
++ \spad{z = y e^{-int p}} is \spad{Li z = 0}.
++ \spad{zeros} is a zero finder in \spad{UP}.
if F has AlgebraicallyClosedField then
ricDsolve: LQ -> List QF
++ ricDsolve(op) returns the rational solutions of the associated
++ Riccati equation of \spad{op y = 0}.
ricDsolve: (LQ, UP -> Factored UP) -> List QF
++ ricDsolve(op, ezfactor) returns the rational solutions of the
++ associated Riccati equation of \spad{op y = 0}.
++ Argument \spad{ezfactor} is a factorisation in \spad{UP},
++ not necessarily into irreducibles.
ricDsolve: L -> List QF
++ ricDsolve(op) returns the rational solutions of the associated
++ Riccati equation of \spad{op y = 0}.
ricDsolve: (L, UP -> Factored UP) -> List QF
++ ricDsolve(op, ezfactor) returns the rational solutions of the
++ associated Riccati equation of \spad{op y = 0}.
++ Argument \spad{ezfactor} is a factorisation in \spad{UP},
++ not necessarily into irreducibles.
Implementation ==> add
import RationalLODE(F, UP)
import NonLinearSolvePackage F
import PrimitiveRatDE(F, UP, L, LQ)
import PrimitiveRatRicDE(F, UP, L, LQ)
FifCan : RF -> Union(F, "failed")
UP2SUP : UP -> SUP
innersol : (List UP, Boolean) -> List QF
mapeval : (SUP, List SY, List F) -> UP
ratsol : List List EQ -> List SOL
ratsln : List EQ -> Union(SOL, "failed")
solveModulo : (UP, UP2) -> List UP
logDerOnly : L -> List QF
nonSingSolve : (N, L, UP -> List F) -> List QF
constantRic : (UP, UP -> List F) -> List F
nopoly : (N, UP, L, UP -> List F) -> List QF
reverseUP : UP -> UTS
reverseUTS : (UTS, N) -> UP
newtonSolution : (L, F, N, UP -> List F) -> UP
newtonSolve : (UPS, F, N) -> Union(UTS, "failed")
genericPolynomial: (SY, Z) -> Record(poly:SUP, vars:List SY)
-- genericPolynomial(s, n) returns
-- \spad{[[s0 + s1 X +...+ sn X^n],[s0,...,sn]]}.
dummy := new()$SY
UP2SUP p == map(#1::P,p)$UnivariatePolynomialCategoryFunctions2(F,UP,P,SUP)
logDerOnly l == [differentiate(s) / s for s in ratDsolve(l, 0).basis]
ricDsolve(l:LQ, zeros:UP -> List F) == ricDsolve(l, zeros, squareFree)
ricDsolve(l:L, zeros:UP -> List F) == ricDsolve(l, zeros, squareFree)
singRicDE(l, ezfactor) == singRicDE(l, solveModulo, ezfactor)
ricDsolve(l:LQ, zeros:UP -> List F, ezfactor:UP -> Factored UP) ==
ricDsolve(splitDenominator(l, empty()).eq, zeros, ezfactor)
mapeval(p, ls, lv) ==
map(ground eval(#1, ls, lv),
p)$UnivariatePolynomialCategoryFunctions2(P, SUP, F, UP)
FifCan f ==
((n := retractIfCan(numer f))@Union(F, "failed") case F) and
((d := retractIfCan(denom f))@Union(F, "failed") case F) =>
(n::F) / (d::F)
"failed"
-- returns [0, []] if n < 0
genericPolynomial(s, n) ==
ans:SUP := 0
l:List(SY) := empty()
for i in 0..n repeat
ans := ans + monomial((sy := new s)::P, i::N)
l := concat(sy, l)
[ans, reverse! l]
ratsln l ==
ls:List(SY) := empty()
lv:List(F) := empty()
for eq in l repeat
((u := FifCan rhs eq) case "failed") or
((v := retractIfCan(lhs eq)@Union(SY, "failed")) case "failed")
=> return "failed"
lv := concat(u::F, lv)
ls := concat(v::SY, ls)
[ls, lv]
ratsol l ==
ans:List(SOL) := empty()
for sol in l repeat
if ((u := ratsln sol) case SOL) then ans := concat(u::SOL, ans)
ans
-- returns [] if the solutions of l have no polynomial component
polyRicDE(l, zeros) ==
ans:List(POL) := [[0, l]]
empty?(lc := leadingCoefficientRicDE l) => ans
rec := first lc -- one with highest degree
for a in zeros(rec.eq) | a ~= 0 repeat
if (p := newtonSolution(l, a, rec.deg, zeros)) ~= 0 then
ans := concat([p, changeVar(l, p)], ans)
ans
-- reverseUP(a_0 + a_1 x + ... + an x^n) = a_n + ... + a_0 x^n
reverseUP p ==
ans:UTS := 0
n := degree(p)::Z
while p ~= 0 repeat
ans := ans + monomial(leadingCoefficient p, (n - degree p)::N)
p := reductum p
ans
-- reverseUTS(a_0 + a_1 x + ..., n) = a_n + ... + a_0 x^n
reverseUTS(s, n) ==
+/[monomial(coefficient(s, i), (n - i)::N)$UP for i in 0..n]
-- returns a potential polynomial solution p with leading coefficient a*?**n
newtonSolution(l, a, n, zeros) ==
i:N
m:Z := 0
aeq:UPS := 0
op := l
while op ~= 0 repeat
mu := degree(op) * n + degree leadingCoefficient op
op := reductum op
if mu > m then m := mu
while l ~= 0 repeat
c := leadingCoefficient l
d := degree l
s:UTS := monomial(1, (m - d * n - degree c)::N)$UTS * reverseUP c
aeq := aeq + monomial(s, d)
l := reductum l
(u := newtonSolve(aeq, a, n)) case UTS => reverseUTS(u::UTS, n)
-- newton lifting failed, so revert to traditional method
atn := monomial(a, n)$UP
neq := changeVar(l, atn)
sols := [sol.poly for sol in polyRicDE(neq, zeros) | degree(sol.poly) < n]
empty? sols => atn
atn + first sols
-- solves the algebraic equation eq for y, returns a solution of degree n with
-- initial term a
-- uses naive newton approximation for now
-- an example where this fails is y^2 + 2 x y + 1 + x^2 = 0
-- which arises from the differential operator D^2 + 2 x D + 1 + x^2
newtonSolve(eq, a, n) ==
deq := differentiate eq
sol := a::UTS
for i in 1..n repeat
(xquo := eq(sol) exquo deq(sol)) case "failed" => return "failed"
sol := truncate(sol - xquo::UTS, i)
sol
-- there could be the same solutions coming in different ways, so we
-- stop when the number of solutions reaches the order of the equation
ricDsolve(l:L, zeros:UP -> List F, ezfactor:UP -> Factored UP) ==
n := degree l
ans:List(QF) := empty()
for rec in singRicDE(l, ezfactor) repeat
ans := removeDuplicates! concat!(ans,
[rec.frac + f for f in nonSingSolve(n, rec.eq, zeros)])
#ans = n => return ans
ans
-- there could be the same solutions coming in different ways, so we
-- stop when the number of solutions reaches the order of the equation
nonSingSolve(n, l, zeros) ==
ans:List(QF) := empty()
for rec in polyRicDE(l, zeros) repeat
ans := removeDuplicates! concat!(ans, nopoly(n,rec.poly,rec.eq,zeros))
#ans = n => return ans
ans
constantRic(p, zeros) ==
zero? degree p => empty()
zeros squareFreePart p
-- there could be the same solutions coming in different ways, so we
-- stop when the number of solutions reaches the order of the equation
nopoly(n, p, l, zeros) ==
ans:List(QF) := empty()
for rec in constantCoefficientRicDE(l, constantRic(#1, zeros)) repeat
ans := removeDuplicates! concat!(ans,
[(rec.constant::UP + p)::QF + f for f in logDerOnly(rec.eq)])
#ans = n => return ans
ans
-- returns [p1,...,pn] s.t. h(x,pi(x)) = 0 mod c(x)
solveModulo(c, h) ==
rec := genericPolynomial(dummy, degree(c)::Z - 1)
unk:SUP := 0
while not zero? h repeat
unk := unk + UP2SUP(leadingCoefficient h) * (rec.poly ** degree h)
h := reductum h
sol := ratsol solve(coefficients(monicDivide(unk,UP2SUP c).remainder),
rec.vars)
[mapeval(rec.poly, s.var, s.val) for s in sol]
if F has AlgebraicallyClosedField then
zro1: UP -> List F
zro : (UP, UP -> Factored UP) -> List F
ricDsolve(l:L) == ricDsolve(l, squareFree)
ricDsolve(l:LQ) == ricDsolve(l, squareFree)
ricDsolve(l:L, ezfactor:UP -> Factored UP) ==
ricDsolve(l, zro(#1, ezfactor), ezfactor)
ricDsolve(l:LQ, ezfactor:UP -> Factored UP) ==
ricDsolve(l, zro(#1, ezfactor), ezfactor)
zro(p, ezfactor) ==
concat [zro1(r.factor) for r in factors ezfactor p]
zro1 p ==
[zeroOf(map(#1, p)$UnivariatePolynomialCategoryFunctions2(F, UP,
F, SparseUnivariatePolynomial F))]
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