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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.
-- SPAD files for the integration world should be compiled in the
-- following order:
--
-- intaux rderf intrf curve curvepkg divisor pfo
-- intalg INTAF efstruc rdeef intef irexpand integrat
)abbrev package INTG0 GenusZeroIntegration
++ Rationalization of several types of genus 0 integrands;
++ Author: Manuel Bronstein
++ Date Created: 11 October 1988
++ Date Last Updated: 24 June 1994
++ Description:
++ This internal package rationalises integrands on curves of the form:
++ \spad{y\^2 = a x\^2 + b x + c}
++ \spad{y\^2 = (a x + b) / (c x + d)}
++ \spad{f(x, y) = 0} where f has degree 1 in x
++ The rationalization is done for integration, limited integration,
++ extended integration and the risch differential equation;
GenusZeroIntegration(R, F, L): Exports == Implementation where
R: Join(GcdDomain, RetractableTo Integer, CharacteristicZero,
LinearlyExplicitRingOver Integer)
F: Join(FunctionSpace R, AlgebraicallyClosedField,
TranscendentalFunctionCategory)
L: SetCategory
SY ==> Symbol
Q ==> Fraction Integer
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial F
RF ==> Fraction UP
UPUP ==> SparseUnivariatePolynomial RF
IR ==> IntegrationResult F
LOG ==> Record(coeff:F, logand:F)
U1 ==> Union(F, "failed")
U2 ==> Union(Record(ratpart:F, coeff:F),"failed")
U3 ==> Union(Record(mainpart:F, limitedlogs:List LOG), "failed")
REC ==> Record(coeff:F, var:List K, val:List F)
ODE ==> Record(particular: Union(F, "failed"), basis: List F)
LODO==> LinearOrdinaryDifferentialOperator1 RF
Exports ==> with
palgint0 : (F, K, K, F, UP) -> IR
++ palgint0(f, x, y, d, p) returns the integral of \spad{f(x,y)dx}
++ where y is an algebraic function of x satisfying
++ \spad{d(x)\^2 y(x)\^2 = P(x)}.
palgint0 : (F, K, K, K, F, RF) -> IR
++ palgint0(f, x, y, z, t, c) returns the integral of \spad{f(x,y)dx}
++ where y is an algebraic function of x satisfying
++ \spad{f(x,y)dx = c f(t,y) dy}; c and t are rational functions of y.
++ Argument z is a dummy variable not appearing in \spad{f(x,y)}.
palgextint0: (F, K, K, F, F, UP) -> U2
++ palgextint0(f, x, y, g, d, p) returns functions \spad{[h, c]} such
++ that \spad{dh/dx = f(x,y) - c g}, where y is an algebraic function
++ of x satisfying \spad{d(x)\^2 y(x)\^2 = P(x)},
++ or "failed" if no such functions exist.
palgextint0: (F, K, K, F, K, F, RF) -> U2
++ palgextint0(f, x, y, g, z, t, c) returns functions \spad{[h, d]} such
++ that \spad{dh/dx = f(x,y) - d g}, where y is an algebraic function
++ of x satisfying \spad{f(x,y)dx = c f(t,y) dy}, and c and t are rational
++ functions of y.
++ Argument z is a dummy variable not appearing in \spad{f(x,y)}.
++ The operation returns "failed" if no such functions exist.
palglimint0: (F, K, K, List F, F, UP) -> U3
++ palglimint0(f, x, y, [u1,...,un], d, p) returns functions
++ \spad{[h,[[ci, ui]]]} such that the ui's are among \spad{[u1,...,un]}
++ and \spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,
++ and "failed" otherwise.
++ Argument y is an algebraic function of x satisfying
++ \spad{d(x)\^2y(x)\^2 = P(x)}.
palglimint0: (F, K, K, List F, K, F, RF) -> U3
++ palglimint0(f, x, y, [u1,...,un], z, t, c) returns functions
++ \spad{[h,[[ci, ui]]]} such that the ui's are among \spad{[u1,...,un]}
++ and \spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,
++ and "failed" otherwise.
++ Argument y is an algebraic function of x satisfying
++ \spad{f(x,y)dx = c f(t,y) dy}; c and t are rational functions of y.
palgRDE0 : (F, F, K, K, (F, F, SY) -> U1, F, UP) -> U1
++ palgRDE0(f, g, x, y, foo, d, p) returns a function \spad{z(x,y)}
++ such that \spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a z exists,
++ and "failed" otherwise.
++ Argument y is an algebraic function of x satisfying
++ \spad{d(x)\^2y(x)\^2 = P(x)}.
++ Argument foo, called by \spad{foo(a, b, x)}, is a function that solves
++ \spad{du/dx + n * da/dx u(x) = u(x)}
++ for an unknown \spad{u(x)} not involving y.
palgRDE0 : (F, F, K, K, (F, F, SY) -> U1, K, F, RF) -> U1
++ palgRDE0(f, g, x, y, foo, t, c) returns a function \spad{z(x,y)}
++ such that \spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a z exists,
++ and "failed" otherwise.
++ Argument y is an algebraic function of x satisfying
++ \spad{f(x,y)dx = c f(t,y) dy}; c and t are rational functions of y.
++ Argument \spad{foo}, called by \spad{foo(a, b, x)}, is a function that
++ solves \spad{du/dx + n * da/dx u(x) = u(x)}
++ for an unknown \spad{u(x)} not involving y.
univariate: (F, K, K, UP) -> UPUP
++ univariate(f,k,k,p) \undocumented
multivariate: (UPUP, K, F) -> F
++ multivariate(u,k,f) \undocumented
lift: (UP, K) -> UPUP
++ lift(u,k) \undocumented
if L has LinearOrdinaryDifferentialOperatorCategory F then
palgLODE0 : (L, F, K, K, F, UP) -> ODE
++ palgLODE0(op, g, x, y, d, p) returns the solution of \spad{op f = g}.
++ Argument y is an algebraic function of x satisfying
++ \spad{d(x)\^2y(x)\^2 = P(x)}.
palgLODE0 : (L, F, K, K, K, F, RF) -> ODE
++ palgLODE0(op,g,x,y,z,t,c) returns the solution of \spad{op f = g}
++ Argument y is an algebraic function of x satisfying
++ \spad{f(x,y)dx = c f(t,y) dy}; c and t are rational functions of y.
Implementation ==> add
import RationalIntegration(F, UP)
import AlgebraicManipulations(R, F)
import IntegrationResultFunctions2(RF, F)
import ElementaryFunctionStructurePackage(R, F)
import SparseUnivariatePolynomialFunctions2(F, RF)
import PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
mkRat : (F, REC, List K) -> RF
mkRatlx : (F, K, K, F, K, RF) -> RF
quadsubst: (K, K, F, UP) -> Record(diff:F, subs:REC, newk:List K)
kerdiff : (F, F) -> List K
checkroot: (F, List K) -> F
univ : (F, List K, K) -> RF
dummy := kernel(new()$SY)@K
kerdiff(sa, a) == setDifference(kernels sa, kernels a)
checkroot(f, l) == (empty? l => f; rootNormalize(f, first l))
univ(c, l, x) == univariate(checkroot(c, l), x)
univariate(f, x, y, p) == lift(univariate(f, y, p), x)
lift(p, k) == map(univariate(#1, k), p)
palgint0(f, x, y, den, radi) ==
-- y is a square root so write f as f1 y + f0 and integrate separately
ff := univariate(f, x, y, minPoly y)
f0 := reductum ff
pr := quadsubst(x, y, den, radi)
map(#1(x::F), integrate(retract(f0)@RF)) +
map(#1(pr.diff),
integrate
mkRat(multivariate(leadingMonomial ff,x,y::F), pr.subs, pr.newk))
-- the algebraic relation is (den * y)**2 = p where p is a * x**2 + b * x + c
-- if p is squarefree, then parametrize in the following form:
-- u = y - x \sqrt{a}
-- x = (u^2 - c) / (b - 2 u \sqrt{a}) = h(u)
-- dx = h'(u) du
-- y = (u + a h(u)) / den = g(u)
-- if a is a perfect square,
-- u = (y - \sqrt{c}) / x
-- x = (b - 2 u \sqrt{c}) / (u^2 - a) = h(u)
-- dx = h'(u) du
-- y = (u h(u) + \sqrt{c}) / den = g(u)
-- otherwise.
-- if p is a square p = a t^2, then we choose only one branch for now:
-- u = x
-- x = u = h(u)
-- dx = du
-- y = t \sqrt{a} / den = g(u)
-- returns [u(x,y), [h'(u), [x,y], [h(u), g(u)], l] in both cases,
-- where l is empty if no new square root was needed,
-- l := [k] if k is the new square root kernel that was created.
quadsubst(x, y, den, p) ==
u := dummy::F
b := coefficient(p, 1)
c := coefficient(p, 0)
sa := rootSimp sqrt(a := coefficient(p, 2))
zero?(b * b - 4 * a * c) => -- case where p = a (x + b/(2a))^2
[x::F, [1, [x, y], [u, sa * (u + b / (2*a)) / eval(den,x,u)]], empty()]
empty? kerdiff(sa, a) =>
bm2u := b - 2 * u * sa
q := eval(den, x, xx := (u**2 - c) / bm2u)
yy := (ua := u + xx * sa) / q
[y::F - x::F * sa, [2 * ua / bm2u, [x, y], [xx, yy]], empty()]
u2ma:= u**2 - a
sc := rootSimp sqrt c
q := eval(den, x, xx := (b - 2 * u * sc) / u2ma)
yy := (ux := xx * u + sc) / q
[(y::F - sc) / x::F, [- 2 * ux / u2ma, [x ,y], [xx, yy]], kerdiff(sc, c)]
mkRatlx(f,x,y,t,z,dx) ==
rat := univariate(eval(f, [x, y], [t, z::F]), z) * dx
numer(rat) / denom(rat)
mkRat(f, rec, l) ==
rat:=univariate(checkroot(rec.coeff * eval(f,rec.var,rec.val), l), dummy)
numer(rat) / denom(rat)
palgint0(f, x, y, z, xx, dx) ==
map(multivariate(#1, y), integrate mkRatlx(f, x, y, xx, z, dx))
palgextint0(f, x, y, g, z, xx, dx) ==
map(multivariate(#1, y),
extendedint(mkRatlx(f,x,y,xx,z,dx), mkRatlx(g,x,y,xx,z,dx)))
palglimint0(f, x, y, lu, z, xx, dx) ==
map(multivariate(#1, y), limitedint(mkRatlx(f, x, y, xx, z, dx),
[mkRatlx(u, x, y, xx, z, dx) for u in lu]))
palgRDE0(f, g, x, y, rischde, z, xx, dx) ==
(u := rischde(eval(f, [x, y], [xx, z::F]),
multivariate(dx, z) * eval(g, [x, y], [xx, z::F]),
symbolIfCan(z)::SY)) case "failed" => "failed"
eval(u::F, z, y::F)
-- given p = sum_i a_i(X) Y^i, returns sum_i a_i(x) y^i
multivariate(p, x, y) ==
(map(multivariate(#1, x),
p)$SparseUnivariatePolynomialFunctions2(RF, F))
(y)
palgextint0(f, x, y, g, den, radi) ==
pr := quadsubst(x, y, den, radi)
map(#1(pr.diff),
extendedint(mkRat(f, pr.subs, pr.newk), mkRat(g, pr.subs, pr.newk)))
palglimint0(f, x, y, lu, den, radi) ==
pr := quadsubst(x, y, den, radi)
map(#1(pr.diff),
limitedint(mkRat(f, pr.subs, pr.newk),
[mkRat(u, pr.subs, pr.newk) for u in lu]))
palgRDE0(f, g, x, y, rischde, den, radi) ==
pr := quadsubst(x, y, den, radi)
(u := rischde(checkroot(eval(f, pr.subs.var, pr.subs.val), pr.newk),
checkroot(pr.subs.coeff * eval(g, pr.subs.var, pr.subs.val),
pr.newk), symbolIfCan(dummy)::SY)) case "failed"
=> "failed"
eval(u::F, dummy, pr.diff)
if L has LinearOrdinaryDifferentialOperatorCategory F then
import RationalLODE(F, UP)
palgLODE0(eq, g, x, y, den, radi) ==
pr := quadsubst(x, y, den, radi)
d := monomial(univ(inv(pr.subs.coeff), pr.newk, dummy), 1)$LODO
di:LODO := 1 -- will accumulate the powers of d
op:LODO := 0 -- will accumulate the new LODO
for i in 0..degree eq repeat
op := op + univ(eval(coefficient(eq, i), pr.subs.var, pr.subs.val),
pr.newk, dummy) * di
di := d * di
rec := ratDsolve(op,univ(eval(g,pr.subs.var,pr.subs.val),pr.newk,dummy))
bas:List(F) := [b(pr.diff) for b in rec.basis]
rec.particular case "failed" => ["failed", bas]
[((rec.particular)::RF) (pr.diff), bas]
palgLODE0(eq, g, x, y, kz, xx, dx) ==
d := monomial(univariate(inv multivariate(dx, kz), kz), 1)$LODO
di:LODO := 1 -- will accumulate the powers of d
op:LODO := 0 -- will accumulate the new LODO
lk:List(K) := [x, y]
lv:List(F) := [xx, kz::F]
for i in 0..degree eq repeat
op := op + univariate(eval(coefficient(eq, i), lk, lv), kz) * di
di := d * di
rec := ratDsolve(op, univariate(eval(g, lk, lv), kz))
bas:List(F) := [multivariate(b, y) for b in rec.basis]
rec.particular case "failed" => ["failed", bas]
[multivariate((rec.particular)::RF, y), bas]
)abbrev package INTPAF PureAlgebraicIntegration
++ Integration of pure algebraic functions;
++ Author: Manuel Bronstein
++ Date Created: 27 May 1988
++ Date Last Updated: 24 June 1994
++ Description:
++ This package provides functions for integration, limited integration,
++ extended integration and the risch differential equation for
++ pure algebraic integrands;
PureAlgebraicIntegration(R, F, L): Exports == Implementation where
R: Join(GcdDomain,RetractableTo Integer, CharacteristicZero,
LinearlyExplicitRingOver Integer)
F: Join(FunctionSpace R, AlgebraicallyClosedField,
TranscendentalFunctionCategory)
L: SetCategory
SY ==> Symbol
N ==> NonNegativeInteger
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial F
RF ==> Fraction UP
UPUP==> SparseUnivariatePolynomial RF
IR ==> IntegrationResult F
IR2 ==> IntegrationResultFunctions2(curve, F)
ALG ==> AlgebraicIntegrate(R, F, UP, UPUP, curve)
LDALG ==> LinearOrdinaryDifferentialOperator1 curve
RDALG ==> PureAlgebraicLODE(F, UP, UPUP, curve)
LOG ==> Record(coeff:F, logand:F)
REC ==> Record(particular:U1, basis:List F)
CND ==> Record(left:UP, right:UP)
CHV ==> Record(int:UPUP, left:UP, right:UP, den:RF, deg:N)
U1 ==> Union(F, "failed")
U2 ==> Union(Record(ratpart:F, coeff:F),"failed")
U3 ==> Union(Record(mainpart:F, limitedlogs:List LOG), "failed")
FAIL==> error "failed - cannot handle that integrand"
Exports ==> with
palgint : (F, K, K) -> IR
++ palgint(f, x, y) returns the integral of \spad{f(x,y)dx}
++ where y is an algebraic function of x.
palgextint: (F, K, K, F) -> U2
++ palgextint(f, x, y, g) returns functions \spad{[h, c]} such that
++ \spad{dh/dx = f(x,y) - c g}, where y is an algebraic function of x;
++ returns "failed" if no such functions exist.
palglimint: (F, K, K, List F) -> U3
++ palglimint(f, x, y, [u1,...,un]) returns functions
++ \spad{[h,[[ci, ui]]]} such that the ui's are among \spad{[u1,...,un]}
++ and \spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,
++ "failed" otherwise;
++ y is an algebraic function of x.
palgRDE : (F, F, F, K, K, (F, F, SY) -> U1) -> U1
++ palgRDE(nfp, f, g, x, y, foo) returns a function \spad{z(x,y)}
++ such that \spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a z exists,
++ "failed" otherwise;
++ y is an algebraic function of x;
++ \spad{foo(a, b, x)} is a function that solves
++ \spad{du/dx + n * da/dx u(x) = u(x)}
++ for an unknown \spad{u(x)} not involving y.
++ \spad{nfp} is \spad{n * df/dx}.
if L has LinearOrdinaryDifferentialOperatorCategory F then
palgLODE: (L, F, K, K, SY) -> REC
++ palgLODE(op, g, kx, y, x) returns the solution of \spad{op f = g}.
++ y is an algebraic function of x.
Implementation ==> add
import IntegrationTools(R, F)
import RationalIntegration(F, UP)
import GenusZeroIntegration(R, F, L)
import ChangeOfVariable(F, UP, UPUP)
import IntegrationResultFunctions2(F, F)
import IntegrationResultFunctions2(RF, F)
import SparseUnivariatePolynomialFunctions2(F, RF)
import UnivariatePolynomialCommonDenominator(UP, RF, UPUP)
import PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
quadIfCan : (K, K) -> Union(Record(coef:F, poly:UP), "failed")
linearInXIfCan : (K, K) -> Union(Record(xsub:F, dxsub:RF), "failed")
prootintegrate : (F, K, K) -> IR
prootintegrate1: (UPUP, K, K, UPUP) -> IR
prootextint : (F, K, K, F) -> U2
prootlimint : (F, K, K, List F) -> U3
prootRDE : (F, F, F, K, K, (F, F, SY) -> U1) -> U1
palgRDE1 : (F, F, K, K) -> U1
palgLODE1 : (List F, F, K, K, SY) -> REC
palgintegrate : (F, K, K) -> IR
palgext : (F, K, K, F) -> U2
palglim : (F, K, K, List F) -> U3
UPUP2F1 : (UPUP, RF, RF, K, K) -> F
UPUP2F0 : (UPUP, K, K) -> F
RF2UPUP : (RF, UPUP) -> UPUP
algaddx : (IR, F) -> IR
chvarIfCan : (UPUP, RF, UP, RF) -> Union(UPUP, "failed")
changeVarIfCan : (UPUP, RF, N) -> Union(CHV, "failed")
rationalInt : (UPUP, N, UP) -> IntegrationResult RF
chv : (UPUP, N, F, F) -> RF
chv0 : (UPUP, N, F, F) -> F
candidates : UP -> List CND
dummy := new()$SY
dumk := kernel(dummy)@K
UPUP2F1(p, t, cf, kx, k) == UPUP2F0(eval(p, t, cf), kx, k)
UPUP2F0(p, kx, k) == multivariate(p, kx, k::F)
chv(f, n, a, b) == univariate(chv0(f, n, a, b), dumk)
RF2UPUP(f, modulus) ==
bc := extendedEuclidean(map(#1::UP::RF, denom f), modulus,
1)::Record(coef1:UPUP, coef2:UPUP)
(map(#1::UP::RF, numer f) * bc.coef1) rem modulus
-- returns "failed", or (xx, c) such that f(x, y)dx = f(xx, y) c dy
-- if p(x, y) = 0 is linear in x
linearInXIfCan(x, y) ==
a := b := 0$UP
p := clearDenominator lift(minPoly y, x)
while p ~= 0 repeat
degree(q := numer leadingCoefficient p) > 1 => return "failed"
a := a + monomial(coefficient(q, 1), d := degree p)
b := b - monomial(coefficient(q, 0), d)
p := reductum p
xx:RF := b / a
[xx(dumk::F), differentiate(xx, differentiate)]
-- return Int(f(x,y)dx) where y is an n^th root of a rational function in x
prootintegrate(f, x, y) ==
modulus := lift(p := minPoly y, x)
rf := reductum(ff := univariate(f, x, y, p))
((r := retractIfCan(rf)@Union(RF,"failed")) case RF) and rf ~= 0 =>
-- in this case, ff := lc(ff) y^i + r so we integrate both terms
-- separately to gain time
map(#1(x::F), integrate(r::RF)) +
prootintegrate1(leadingMonomial ff, x, y, modulus)
prootintegrate1(ff, x, y, modulus)
prootintegrate1(ff, x, y, modulus) ==
chv:CHV
r := radPoly(modulus)::Record(radicand:RF, deg:N)
(uu := changeVarIfCan(ff, r.radicand, r.deg)) case CHV =>
chv := uu::CHV
newalg := nthRoot((chv.left)(dumk::F), chv.deg)
kz := retract(numer newalg)@K
newf := multivariate(chv.int, ku := dumk, newalg)
vu := (chv.right)(x::F)
vz := (chv.den)(x::F) * (y::F) * denom(newalg)::F
map(eval(#1, [ku, kz], [vu, vz]), palgint(newf, ku, kz))
cv := chvar(ff, modulus)
r := radPoly(cv.poly)::Record(radicand:RF, deg:N)
qprime := differentiate(q := retract(r.radicand)@UP)::RF
not zero? qprime and
((u := chvarIfCan(cv.func, 1, q, inv qprime)) case UPUP) =>
m := monomial(1, r.deg)$UPUP - q::RF::UPUP
map(UPUP2F1(RF2UPUP(#1, m), cv.c1, cv.c2, x, y),
rationalInt(u::UPUP, r.deg, monomial(1, 1)))
curve := RadicalFunctionField(F, UP, UPUP, q::RF, r.deg)
algaddx(map(UPUP2F1(lift #1, cv.c1, cv.c2, x, y),
palgintegrate(reduce(cv.func), differentiate$UP)$ALG)$IR2, x::F)
-- Do the rationalizing change of variable
-- Int(f(x, y) dx) --> Int(n u^(n-1) f((u^n - b)/a, u) / a du) where
-- u^n = y^n = g(x) = a x + b
-- returns the integral as an integral of a rational function in u
rationalInt(f, n, g) ==
not one? degree g => error "rationalInt: radicand must be linear"
a := leadingCoefficient g
integrate(n * monomial(inv a, (n-1)::N)$UP
* chv(f, n, a, leadingCoefficient reductum g))
-- Do the rationalizing change of variable f(x,y) --> f((u^n - b)/a, u) where
-- u = y = (a x + b)^(1/n).
-- Returns f((u^n - b)/a,u) as an element of F
chv0(f, n, a, b) ==
d := dumk::F
(f (d::UP::RF)) ((d ** n - b) / a)
-- candidates(p) returns a list of pairs [g, u] such that p(x) = g(u(x)),
-- those u's are candidates for change of variables
-- currently uses a dumb heuristic where the candidates u's are p itself
-- and all the powers x^2, x^3, ..., x^{deg(p)},
-- will use polynomial decomposition in smarter days MB 8/93
candidates p ==
l:List(CND) := empty()
ground? p => l
for i in 2..degree p repeat
if (u := composite(p, xi := monomial(1, i))) case UP then
l := concat([u::UP, xi], l)
concat([monomial(1, 1), p], l)
-- checks whether Int(p(x, y) dx) can be rewritten as
-- Int(r(u, z) du) where u is some polynomial of x,
-- z = d y for some polynomial d, and z^m = g(u)
-- returns either [r(u, z), g, u, d, m] or "failed"
-- we have y^n = radi
changeVarIfCan(p, radi, n) ==
rec := rootPoly(radi, n)
for cnd in candidates(rec.radicand) repeat
(u := chvarIfCan(p, rec.coef, cnd.right,
inv(differentiate(cnd.right)::RF))) case UPUP =>
return [u::UPUP, cnd.left, cnd.right, rec.coef, rec.exponent]
"failed"
-- checks whether Int(p(x, y) dx) can be rewritten as
-- Int(r(u, z) du) where u is some polynomial of x and z = d y
-- we have y^n = a(x)/d(x)
-- returns either "failed" or r(u, z)
chvarIfCan(p, d, u, u1) ==
ans:UPUP := 0
while p ~= 0 repeat
(v := composite(u1 * leadingCoefficient(p) / d ** degree(p), u))
case "failed" => return "failed"
ans := ans + monomial(v::RF, degree p)
p := reductum p
ans
algaddx(i, xx) ==
elem? i => i
mkAnswer(ratpart i, logpart i,
[[- ne.integrand / (xx**2), xx] for ne in notelem i])
prootRDE(nfp, f, g, x, k, rde) ==
modulus := lift(p := minPoly k, x)
r := radPoly(modulus)::Record(radicand:RF, deg:N)
rec := rootPoly(r.radicand, r.deg)
dqdx := inv(differentiate(q := rec.radicand)::RF)
((uf := chvarIfCan(ff := univariate(f,x,k,p),rec.coef,q,1)) case UPUP) and
((ug:=chvarIfCan(gg:=univariate(g,x,k,p),rec.coef,q,dqdx)) case UPUP) =>
(u := rde(chv0(uf::UPUP, rec.exponent, 1, 0), rec.exponent *
(dumk::F) ** (rec.exponent * (rec.exponent - 1))
* chv0(ug::UPUP, rec.exponent, 1, 0),
symbolIfCan(dumk)::SY)) case "failed" => "failed"
eval(u::F, dumk, k::F)
one?(rec.coef) =>
curve := RadicalFunctionField(F, UP, UPUP, q::RF, rec.exponent)
rc := algDsolve(D()$LDALG + reduce(univariate(nfp, x, k, p))::LDALG,
reduce univariate(g, x, k, p))$RDALG
rc.particular case "failed" => "failed"
UPUP2F0(lift((rc.particular)::curve), x, k)
palgRDE1(nfp, g, x, k)
prootlimint(f, x, k, lu) ==
modulus := lift(p := minPoly k, x)
r := radPoly(modulus)::Record(radicand:RF, deg:N)
rec := rootPoly(r.radicand, r.deg)
dqdx := inv(differentiate(q := rec.radicand)::RF)
(uf := chvarIfCan(ff := univariate(f,x,k,p),rec.coef,q,dqdx)) case UPUP =>
l := empty()$List(RF)
n := rec.exponent * monomial(1, (rec.exponent - 1)::N)$UP
for u in lu repeat
if ((v:=chvarIfCan(uu:=univariate(u,x,k,p),rec.coef,q,dqdx))case UPUP)
then l := concat(n * chv(v::UPUP,rec.exponent, 1, 0), l) else FAIL
m := monomial(1, rec.exponent)$UPUP - q::RF::UPUP
map(UPUP2F0(RF2UPUP(#1,m), x, k),
limitedint(n * chv(uf::UPUP, rec.exponent, 1, 0), reverse! l))
cv := chvar(ff, modulus)
r := radPoly(cv.poly)::Record(radicand:RF, deg:N)
dqdx := inv(differentiate(q := retract(r.radicand)@UP)::RF)
curve := RadicalFunctionField(F, UP, UPUP, q::RF, r.deg)
(ui := palginfieldint(reduce(cv.func), differentiate$UP)$ALG)
case "failed" => FAIL
[UPUP2F1(lift(ui::curve), cv.c1, cv.c2, x, k), empty()]
prootextint(f, x, k, g) ==
modulus := lift(p := minPoly k, x)
r := radPoly(modulus)::Record(radicand:RF, deg:N)
rec := rootPoly(r.radicand, r.deg)
dqdx := inv(differentiate(q := rec.radicand)::RF)
((uf:=chvarIfCan(ff:=univariate(f,x,k,p),rec.coef,q,dqdx)) case UPUP) and
((ug:=chvarIfCan(gg:=univariate(g,x,k,p),rec.coef,q,dqdx)) case UPUP) =>
m := monomial(1, rec.exponent)$UPUP - q::RF::UPUP
n := rec.exponent * monomial(1, (rec.exponent - 1)::N)$UP
map(UPUP2F0(RF2UPUP(#1,m), x, k),
extendedint(n * chv(uf::UPUP, rec.exponent, 1, 0),
n * chv(ug::UPUP, rec.exponent, 1, 0)))
cv := chvar(ff, modulus)
r := radPoly(cv.poly)::Record(radicand:RF, deg:N)
dqdx := inv(differentiate(q := retract(r.radicand)@UP)::RF)
curve := RadicalFunctionField(F, UP, UPUP, q::RF, r.deg)
(u := palginfieldint(reduce(cv.func), differentiate$UP)$ALG)
case "failed" => FAIL
[UPUP2F1(lift(u::curve), cv.c1, cv.c2, x, k), 0]
palgRDE1(nfp, g, x, y) ==
palgLODE1([nfp, 1], g, x, y, symbolIfCan(x)::SY).particular
palgLODE1(eq, g, kx, y, x) ==
modulus:= lift(p := minPoly y, kx)
curve := AlgebraicFunctionField(F, UP, UPUP, modulus)
neq:LDALG := 0
for f in eq for i in 0.. repeat
neq := neq + monomial(reduce univariate(f, kx, y, p), i)
empty? remove!(y, remove!(kx, varselect(kernels g, x))) =>
rec := algDsolve(neq, reduce univariate(g, kx, y, p))$RDALG
bas:List(F) := [UPUP2F0(lift h, kx, y) for h in rec.basis]
rec.particular case "failed" => ["failed", bas]
[UPUP2F0(lift((rec.particular)::curve), kx, y), bas]
rec := algDsolve(neq, 0)
["failed", [UPUP2F0(lift h, kx, y) for h in rec.basis]]
palgintegrate(f, x, k) ==
modulus:= lift(p := minPoly k, x)
cv := chvar(univariate(f, x, k, p), modulus)
curve := AlgebraicFunctionField(F, UP, UPUP, cv.poly)
knownInfBasis(cv.deg)
algaddx(map(UPUP2F1(lift #1, cv.c1, cv.c2, x, k),
palgintegrate(reduce(cv.func), differentiate$UP)$ALG)$IR2, x::F)
palglim(f, x, k, lu) ==
modulus:= lift(p := minPoly k, x)
cv := chvar(univariate(f, x, k, p), modulus)
curve := AlgebraicFunctionField(F, UP, UPUP, cv.poly)
knownInfBasis(cv.deg)
(u := palginfieldint(reduce(cv.func), differentiate$UP)$ALG)
case "failed" => FAIL
[UPUP2F1(lift(u::curve), cv.c1, cv.c2, x, k), empty()]
palgext(f, x, k, g) ==
modulus:= lift(p := minPoly k, x)
cv := chvar(univariate(f, x, k, p), modulus)
curve := AlgebraicFunctionField(F, UP, UPUP, cv.poly)
knownInfBasis(cv.deg)
(u := palginfieldint(reduce(cv.func), differentiate$UP)$ALG)
case "failed" => FAIL
[UPUP2F1(lift(u::curve), cv.c1, cv.c2, x, k), 0]
palgint(f, x, y) ==
(v := linearInXIfCan(x, y)) case "failed" =>
(u := quadIfCan(x, y)) case "failed" =>
is?(y, 'nthRoot) => prootintegrate(f, x, y)
is?(y, 'rootOf) => palgintegrate(f, x, y)
FAIL
palgint0(f, x, y, u.coef, u.poly)
palgint0(f, x, y, dumk, v.xsub, v.dxsub)
palgextint(f, x, y, g) ==
(v := linearInXIfCan(x, y)) case "failed" =>
(u := quadIfCan(x, y)) case "failed" =>
is?(y, 'nthRoot) => prootextint(f, x, y, g)
is?(y, 'rootOf) => palgext(f, x, y, g)
FAIL
palgextint0(f, x, y, g, u.coef, u.poly)
palgextint0(f, x, y, g, dumk, v.xsub, v.dxsub)
palglimint(f, x, y, lu) ==
(v := linearInXIfCan(x, y)) case "failed" =>
(u := quadIfCan(x, y)) case "failed" =>
is?(y, 'nthRoot) => prootlimint(f, x, y, lu)
is?(y, 'rootOf) => palglim(f, x, y, lu)
FAIL
palglimint0(f, x, y, lu, u.coef, u.poly)
palglimint0(f, x, y, lu, dumk, v.xsub, v.dxsub)
palgRDE(nfp, f, g, x, y, rde) ==
(v := linearInXIfCan(x, y)) case "failed" =>
(u := quadIfCan(x, y)) case "failed" =>
is?(y, 'nthRoot) => prootRDE(nfp, f, g, x, y, rde)
palgRDE1(nfp, g, x, y)
palgRDE0(f, g, x, y, rde, u.coef, u.poly)
palgRDE0(f, g, x, y, rde, dumk, v.xsub, v.dxsub)
-- returns "failed", or (d, P) such that (dy)**2 = P(x)
-- and degree(P) = 2
quadIfCan(x, y) ==
(degree(p := minPoly y) = 2) and zero?(coefficient(p, 1)) =>
d := denom(ff :=
univariate(- coefficient(p, 0) / coefficient(p, 2), x))
degree(radi := d * numer ff) = 2 => [d(x::F), radi]
"failed"
"failed"
if L has LinearOrdinaryDifferentialOperatorCategory F then
palgLODE(eq, g, kx, y, x) ==
(v := linearInXIfCan(kx, y)) case "failed" =>
(u := quadIfCan(kx, y)) case "failed" =>
palgLODE1([coefficient(eq, i) for i in 0..degree eq], g, kx, y, x)
palgLODE0(eq, g, kx, y, u.coef, u.poly)
palgLODE0(eq, g, kx, y, dumk, v.xsub, v.dxsub)
)abbrev package INTAF AlgebraicIntegration
++ Mixed algebraic integration;
++ Author: Manuel Bronstein
++ Date Created: 12 October 1988
++ Date Last Updated: 4 June 1988
++ Description:
++ This package provides functions for the integration of
++ algebraic integrands over transcendental functions;
AlgebraicIntegration(R, F): Exports == Implementation where
R : IntegralDomain
F : Join(AlgebraicallyClosedField, FunctionSpace R)
SY ==> Symbol
N ==> NonNegativeInteger
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial F
RF ==> Fraction UP
UPUP==> SparseUnivariatePolynomial RF
IR ==> IntegrationResult F
IR2 ==> IntegrationResultFunctions2(curve, F)
ALG ==> AlgebraicIntegrate(R, F, UP, UPUP, curve)
FAIL==> error "failed - cannot handle that integrand"
Exports ==> with
algint: (F, K, K, UP -> UP) -> IR
++ algint(f, x, y, d) returns the integral of \spad{f(x,y)dx}
++ where y is an algebraic function of x;
++ d is the derivation to use on \spad{k[x]}.
Implementation ==> add
import ChangeOfVariable(F, UP, UPUP)
import PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
rootintegrate: (F, K, K, UP -> UP) -> IR
algintegrate : (F, K, K, UP -> UP) -> IR
UPUP2F : (UPUP, RF, K, K) -> F
F2UPUP : (F, K, K, UP) -> UPUP
UP2UPUP : (UP, K) -> UPUP
F2UPUP(f, kx, k, p) == UP2UPUP(univariate(f, k, p), kx)
rootintegrate(f, t, k, derivation) ==
r1 := mkIntegral(modulus := UP2UPUP(p := minPoly k, t))
f1 := F2UPUP(f, t, k, p) monomial(inv(r1.coef), 1)
r := radPoly(r1.poly)::Record(radicand:RF, deg:N)
q := retract(r.radicand)
curve := RadicalFunctionField(F, UP, UPUP, q::RF, r.deg)
map(UPUP2F(lift #1, r1.coef, t, k),
algintegrate(reduce f1, derivation)$ALG)$IR2
algintegrate(f, t, k, derivation) ==
r1 := mkIntegral(modulus := UP2UPUP(p := minPoly k, t))
f1 := F2UPUP(f, t, k, p) monomial(inv(r1.coef), 1)
modulus:= UP2UPUP(p := minPoly k, t)
curve := AlgebraicFunctionField(F, UP, UPUP, r1.poly)
map(UPUP2F(lift #1, r1.coef, t, k),
algintegrate(reduce f1, derivation)$ALG)$IR2
UP2UPUP(p, k) ==
map(univariate(#1,k),p)$SparseUnivariatePolynomialFunctions2(F,RF)
UPUP2F(p, cf, t, k) ==
map(multivariate(#1, t),
p)$SparseUnivariatePolynomialFunctions2(RF, F)
(multivariate(cf, t) * k::F)
algint(f, t, y, derivation) ==
is?(y, 'nthRoot) => rootintegrate(f, t, y, derivation)
is?(y, 'rootOf) => algintegrate(f, t, y, derivation)
FAIL
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