/usr/lib/open-axiom/src/algebra/rdeef.spad is in open-axiom-source 1.4.1+svn~2626-2ubuntu2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 | --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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 INTTOOLS IntegrationTools
++ Tools for the integrator
++ Author: Manuel Bronstein
++ Date Created: 25 April 1990
++ Date Last Updated: 9 June 1993
++ Keywords: elementary, function, integration.
IntegrationTools(R: SetCategory, F:FunctionSpace R): Exp == Impl where
K ==> Kernel F
SE ==> Symbol
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial F
IR ==> IntegrationResult F
ANS ==> Record(special:F, integrand:F)
U ==> Union(ANS, "failed")
Exp ==> with
varselect: (List K, SE) -> List K
++ varselect([k1,...,kn], x) returns the ki which involve x.
kmax : List K -> K
++ kmax([k1,...,kn]) returns the top-level ki for integration.
ksec : (K, List K, SE) -> K
++ ksec(k, [k1,...,kn], x) returns the second top-level ki
++ after k involving x.
union : (List K, List K) -> List K
++ union(l1, l2) returns set-theoretic union of l1 and l2.
vark : (List F, SE) -> List K
++ vark([f1,...,fn],x) returns the set-theoretic union of
++ \spad{(varselect(f1,x),...,varselect(fn,x))}.
if R has IntegralDomain then
removeConstantTerm: (F, SE) -> F
++ removeConstantTerm(f, x) returns f minus any additive constant
++ with respect to x.
if R has GcdDomain and F has ElementaryFunctionCategory then
mkPrim: (F, SE) -> F
++ mkPrim(f, x) makes the logs in f which are linear in x
++ primitive with respect to x.
if R has ConvertibleTo Pattern Integer and R has PatternMatchable Integer
and F has LiouvillianFunctionCategory and F has RetractableTo SE then
intPatternMatch: (F, SE, (F, SE) -> IR, (F, SE) -> U) -> IR
++ intPatternMatch(f, x, int, pmint) tries to integrate \spad{f}
++ first by using the integration function \spad{int}, and then
++ by using the pattern match intetgration function \spad{pmint}
++ on any remaining unintegrable part.
Impl ==> add
macro ALGOP == '%alg
better?: (K, K) -> Boolean
union(l1, l2) == setUnion(l1, l2)
varselect(l, x) == [k for k in l | member?(x, variables(k::F))]
ksec(k, l, x) == kmax setUnion(remove(k, l), vark(argument k, x))
vark(l, x) ==
varselect(reduce("setUnion",[kernels f for f in l],empty()$List(K)), x)
kmax l ==
ans := first l
for k in rest l repeat
if better?(k, ans) then ans := k
ans
-- true if x should be considered before y in the tower
better?(x, y) ==
height(y) ~= height(x) => height(y) < height(x)
has?(operator y, ALGOP) or
(is?(y,'exp) and not is?(x, 'exp)
and not has?(operator x, ALGOP))
if R has IntegralDomain then
removeConstantTerm(f, x) ==
not freeOf?((den := denom f)::F, x) => f
(u := isPlus(num := numer f)) case "failed" =>
freeOf?(num::F, x) => 0
f
ans:P := 0
for term in u::List(P) repeat
if not freeOf?(term::F, x) then ans := ans + term
ans / den
if R has GcdDomain and F has ElementaryFunctionCategory then
psimp : (P, SE) -> Record(coef:Integer, logand:F)
cont : (P, List K) -> P
logsimp : (F, SE) -> F
linearLog?: (K, F, SE) -> Boolean
logsimp(f, x) ==
r1 := psimp(numer f, x)
r2 := psimp(denom f, x)
g := gcd(r1.coef, r2.coef)
g * log(r1.logand ** (r1.coef quo g) / r2.logand ** (r2.coef quo g))
cont(p, l) ==
empty? l => p
q := univariate(p, first l)
cont(unitNormal(leadingCoefficient q).unit * content q, rest l)
linearLog?(k, f, x) ==
is?(k, 'log) and
((u := retractIfCan(univariate(f,k))@Union(UP,"failed")) case UP)
and one?(degree(u::UP))
and not member?(x, variables leadingCoefficient(u::UP))
mkPrim(f, x) ==
lg := [k for k in kernels f | linearLog?(k, f, x)]
eval(f, lg, [logsimp(first argument k, x) for k in lg])
psimp(p, x) ==
(u := isExpt(p := ((p exquo cont(p, varselect(variables p, x)))::P)))
case "failed" => [1, p::F]
[u.exponent, u.var::F]
if R has Join(ConvertibleTo Pattern Integer, PatternMatchable Integer)
and F has Join(LiouvillianFunctionCategory, RetractableTo SE) then
intPatternMatch(f, x, int, pmint) ==
ir := int(f, x)
empty?(l := notelem ir) => ir
ans := ratpart ir
nl:List(Record(integrand:F, intvar:F)) := empty()
lg := logpart ir
for rec in l repeat
u := pmint(rec.integrand, retract(rec.intvar))
if u case ANS then
rc := u::ANS
ans := ans + rc.special
if rc.integrand ~= 0 then
ir0 := intPatternMatch(rc.integrand, x, int, pmint)
ans := ans + ratpart ir0
lg := concat(logpart ir0, lg)
nl := concat(notelem ir0, nl)
else nl := concat(rec, nl)
mkAnswer(ans, lg, nl)
)abbrev package RDEEF ElementaryRischDE
++ Risch differential equation, elementary case.
++ Author: Manuel Bronstein
++ Date Created: 1 February 1988
++ Date Last Updated: 2 November 1995
++ Keywords: elementary, function, integration.
ElementaryRischDE(R, F): Exports == Implementation where
R : Join(GcdDomain, CharacteristicZero,
RetractableTo Integer, LinearlyExplicitRingOver Integer)
F : Join(TranscendentalFunctionCategory, AlgebraicallyClosedField,
FunctionSpace R)
N ==> NonNegativeInteger
Z ==> Integer
SE ==> Symbol
LF ==> List F
K ==> Kernel F
LK ==> List K
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial F
RF ==> Fraction UP
GP ==> LaurentPolynomial(F, UP)
Data ==> List Record(coeff:Z, argument:P)
RRF ==> Record(mainpart:F,limitedlogs:List NL)
NL ==> Record(coeff:F,logand:F)
U ==> Union(RRF, "failed")
UF ==> Union(F, "failed")
UUP ==> Union(UP, "failed")
UGP ==> Union(GP, "failed")
URF ==> Union(RF, "failed")
UEX ==> Union(Record(ratpart:F, coeff:F), "failed")
PSOL==> Record(ans:F, right:F, sol?:Boolean)
FAIL==> error("Function not supported by Risch d.e.")
Exports ==> with
rischDE: (Z, F, F, SE, (F, LF) -> U, (F, F) -> UEX) -> PSOL
++ rischDE(n, f, g, x, lim, ext) returns \spad{[y, h, b]} such that
++ \spad{dy/dx + n df/dx y = h} and \spad{b := h = g}.
++ The equation \spad{dy/dx + n df/dx y = g} has no solution
++ if \spad{h \~~= g} (y is a partial solution in that case).
++ Notes: lim is a limited integration function, and
++ ext is an extended integration function.
Implementation ==> add
macro ALGOP == '%alg
import IntegrationTools(R, F)
import TranscendentalRischDE(F, UP)
import TranscendentalIntegration(F, UP)
import PureAlgebraicIntegration(R, F, F)
import FunctionSpacePrimitiveElement(R, F)
import ElementaryFunctionStructurePackage(R, F)
import PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
RF2GP: RF -> GP
makeData : (F, SE, K) -> Data
normal0 : (Z, F, F, SE) -> UF
normalise0: (Z, F, F, SE) -> PSOL
normalise : (Z, F, F, F, SE, K, (F, LF) -> U, (F, F) -> UEX) -> PSOL
rischDEalg: (Z, F, F, F, K, LK, SE, (F, LF) -> U, (F, F) -> UEX) -> PSOL
rischDElog: (LK, RF, RF, SE, K, UP->UP,(F,LF)->U,(F,F)->UEX) -> URF
rischDEexp: (LK, RF, RF, SE, K, UP->UP,(F,LF)->U,(F,F)->UEX) -> URF
polyDElog : (LK, UP, UP,UP,SE,K,UP->UP,(F,LF)->U,(F,F)->UEX) -> UUP
polyDEexp : (LK, UP, UP,UP,SE,K,UP->UP,(F,LF)->U,(F,F)->UEX) -> UUP
gpolDEexp : (LK, UP, GP,GP,SE,K,UP->UP,(F,LF)->U,(F,F)->UEX) -> UGP
boundAt0 : (LK, F, Z, Z, SE, K, (F, LF) -> U) -> Z
boundInf : (LK, F, Z, Z, Z, SE, K, (F, LF) -> U) -> Z
logdegrad : (LK, F, UP, Z, SE, K,(F,LF)->U, (F,F) -> UEX) -> UUP
expdegrad : (LK, F, UP, Z, SE, K,(F,LF)->U, (F,F) -> UEX) -> UUP
logdeg : (UP, F, Z, SE, F, (F, LF) -> U, (F, F) -> UEX) -> UUP
expdeg : (UP, F, Z, SE, F, (F, LF) -> U, (F, F) -> UEX) -> UUP
exppolyint: (UP, (Z, F) -> PSOL) -> UUP
RRF2F : RRF -> F
logdiff : (List K, List K) -> List K
tab:AssociationList(F, Data) := table()
RF2GP f == (numer(f)::GP exquo denom(f)::GP)::GP
logdiff(twr, bad) ==
[u for u in twr | is?(u, 'log) and not member?(u, bad)]
rischDEalg(n, nfp, f, g, k, l, x, limint, extint) ==
symbolIfCan(kx := ksec(k, l, x)) case SE =>
(u := palgRDE(nfp, f, g, kx, k, normal0(n, #1, #2, #3))) case "failed"
=> [0, 0, false]
[u::F, g, true]
has?(operator kx, ALGOP) =>
rec := primitiveElement(kx::F, k::F)
y := rootOf(rec.prim)
lk:LK := [kx, k]
lv:LF := [(rec.pol1) y, (rec.pol2) y]
rc := rischDE(n, eval(f, lk, lv), eval(g, lk, lv), x, limint, extint)
rc.sol? => [eval(rc.ans, retract(y)@K, rec.primelt), rc.right, true]
[0, 0, false]
FAIL
-- solve y' + n f'y = g for a rational function y
rischDE(n, f, g, x, limitedint, extendedint) ==
zero? g => [0, g, true]
zero?(nfp := n * differentiate(f, x)) =>
(u := limitedint(g, empty())) case "failed" => [0, 0, false]
[u.mainpart, g, true]
freeOf?(y := g / nfp, x) => [y, g, true]
vl := varselect(union(kernels nfp, kernels g), x)
symbolIfCan(k := kmax vl) case SE => normalise0(n, f, g, x)
is?(k, 'log) or is?(k, 'exp) =>
normalise(n, nfp, f, g, x, k, limitedint, extendedint)
has?(operator k, ALGOP) =>
rischDEalg(n, nfp, f, g, k, vl, x, limitedint, extendedint)
FAIL
normal0(n, f, g, x) ==
rec := normalise0(n, f, g, x)
rec.sol? => rec.ans
"failed"
-- solve y' + n f' y = g
-- when f' and g are rational functions over a constant field
normalise0(n, f, g, x) ==
k := kernel(x)@K
if (data1 := search(f, tab)) case "failed" then
tab.f := data := makeData(f, x, k)
else data := data1::Data
f' := nfprime := n * differentiate(f, x)
p:P := 1
for v in data | positive?(m := n * v.coeff) repeat
p := p * v.argument ** (m::N)
f' := f' - m * differentiate(v.argument::F, x) / (v.argument::F)
rec := baseRDE(univariate(f', k), univariate(p::F * g, k))
y := multivariate(rec.ans, k) / p::F
rec.nosol => [y, differentiate(y, x) + nfprime * y, false]
[y, g, true]
-- make f weakly normalized, and solve y' + n f' y = g
normalise(n, nfp, f, g, x, k, limitedint, extendedint) ==
if (data1:= search(f, tab)) case "failed" then
tab.f := data := makeData(f, x, k)
else data := data1::Data
p:P := 1
for v in data | positive?(m := n * v.coeff) repeat
p := p * v.argument ** (m::N)
f := f - v.coeff * log(v.argument::F)
nfp := nfp - m * differentiate(v.argument::F, x) / (v.argument::F)
newf := univariate(nfp, k)
newg := univariate(p::F * g, k)
twr := union(logdiff(tower f, empty()), logdiff(tower g, empty()))
ans1 :=
is?(k, 'log) =>
rischDElog(twr, newf, newg, x, k,
differentiate(#1, differentiate(#1, x),
differentiate(k::F, x)::UP),
limitedint, extendedint)
is?(k, 'exp) =>
rischDEexp(twr, newf, newg, x, k,
differentiate(#1, differentiate(#1, x),
monomial(differentiate(first argument k, x), 1)),
limitedint, extendedint)
ans1 case "failed" => [0, 0, false]
[multivariate(ans1::RF, k) / p::F, g, true]
-- find the n * log(P) appearing in f, where P is in P, n in Z
makeData(f, x, k) ==
disasters := empty()$Data
fnum := numer f
fden := denom f
for u in varselect(kernels f, x) | is?(u, 'log) repeat
logand := first argument u
if zero?(degree univariate(fden, u)) and
one?(degree(num := univariate(fnum, u))) then
cf := (leadingCoefficient num) / fden
if (n := retractIfCan(cf)@Union(Z, "failed")) case Z then
if positive? degree(numer logand, k) then
disasters := concat([n::Z, numer logand], disasters)
if positive? degree(denom logand, k) then
disasters := concat([-(n::Z), denom logand], disasters)
disasters
rischDElog(twr, f, g, x, theta, driv, limint, extint) ==
(u := monomRDE(f, g, driv)) case "failed" => "failed"
(v := polyDElog(twr, u.a, retract(u.b), retract(u.c), x, theta, driv,
limint, extint)) case "failed" => "failed"
v::UP / u.t
rischDEexp(twr, f, g, x, theta, driv, limint, extint) ==
(u := monomRDE(f, g, driv)) case "failed" => "failed"
(v := gpolDEexp(twr, u.a, RF2GP(u.b), RF2GP(u.c), x, theta, driv,
limint, extint)) case "failed" => "failed"
convert(v::GP)@RF / u.t::RF
polyDElog(twr, aa, bb, cc, x, t, driv, limint, extint) ==
zero? cc => 0
t' := differentiate(t::F, x)
zero? bb =>
(u := cc exquo aa) case "failed" => "failed"
primintfldpoly(u::UP, extint(#1, t'), t')
n := degree(cc)::Z - (db := degree(bb)::Z)
if ((da := degree(aa)::Z) = db) and positive? da then
lk0 := tower(f0 :=
- (leadingCoefficient bb) / (leadingCoefficient aa))
lk1 := logdiff(twr, lk0)
(if0 := limint(f0, [first argument u for u in lk1]))
case "failed" => error "Risch's theorem violated"
(alph := validExponential(lk0, RRF2F(if0::RRF), x)) case F =>
return
(ans := polyDElog(twr, alph::F * aa,
differentiate(alph::F, x) * aa + alph::F * bb,
cc, x, t, driv, limint, extint)) case "failed" => "failed"
alph::F * ans::UP
if (da > db + 1) then n := max(0, degree(cc)::Z - da + 1)
if (da = db + 1) then
i := limint(- (leadingCoefficient bb) / (leadingCoefficient aa),
[first argument t])
if not(i case "failed") then
r :=
null(i.limitedlogs) => 0$F
i.limitedlogs.first.coeff
if (nn := retractIfCan(r)@Union(Z, "failed")) case Z then
n := max(nn::Z, n)
(v := polyRDE(aa, bb, cc, n, driv)) case ans =>
v.ans.nosol => "failed"
v.ans.ans
w := v.eq
zero?(w.b) =>
degree(w.c) > w.m => "failed"
(u := primintfldpoly(w.c, extint(#1,t'), t')) case "failed" => "failed"
degree(u::UP) > w.m => "failed"
w.alpha * u::UP + w.beta
(u := logdegrad(twr, retract(w.b), w.c, w.m, x, t, limint, extint))
case "failed" => "failed"
w.alpha * u::UP + w.beta
gpolDEexp(twr, a, b, c, x, t, driv, limint, extint) ==
zero? c => 0
zero? b =>
(u := c exquo (a::GP)) case "failed" => "failed"
expintfldpoly(u::GP,
rischDE(#1, first argument t, #2, x, limint, extint))
lb := boundAt0(twr, - coefficient(b, 0) / coefficient(a, 0),
nb := order b, nc := order c, x, t, limint)
tm := monomial(1, (m := max(0, max(-nb, lb - nc)))::N)$UP
(v := polyDEexp(twr,a * tm,lb * differentiate(first argument t, x)
* a * tm + retract(b * tm::GP)@UP,
retract(c * monomial(1, m - lb))@UP,
x, t, driv, limint, extint)) case "failed" => "failed"
v::UP::GP * monomial(1, lb)
polyDEexp(twr, aa, bb, cc, x, t, driv, limint, extint) ==
zero? cc => 0
zero? bb =>
(u := cc exquo aa) case "failed" => "failed"
exppolyint(u::UP, rischDE(#1, first argument t, #2, x, limint, extint))
n := boundInf(twr,-leadingCoefficient(bb) / (leadingCoefficient aa),
degree(aa)::Z, degree(bb)::Z, degree(cc)::Z, x, t, limint)
(v := polyRDE(aa, bb, cc, n, driv)) case ans =>
v.ans.nosol => "failed"
v.ans.ans
w := v.eq
zero?(w.b) =>
degree(w.c) > w.m => "failed"
(u := exppolyint(w.c,
rischDE(#1, first argument t, #2, x, limint, extint)))
case "failed" => "failed"
w.alpha * u::UP + w.beta
(u := expdegrad(twr, retract(w.b), w.c, w.m, x, t, limint, extint))
case "failed" => "failed"
w.alpha * u::UP + w.beta
exppolyint(p, rischdiffeq) ==
(u := expintfldpoly(p::GP, rischdiffeq)) case "failed" => "failed"
retractIfCan(u::GP)@Union(UP, "failed")
boundInf(twr, f0, da, db, dc, x, t, limitedint) ==
da < db => dc - db
da > db => max(0, dc - da)
l1 := logdiff(twr, l0 := tower f0)
(if0 := limitedint(f0, [first argument u for u in l1]))
case "failed" => error "Risch's theorem violated"
(alpha := validExponential(concat(t, l0), RRF2F(if0::RRF), x))
case F =>
al := separate(univariate(alpha::F, t))$GP
zero?(al.fracPart) and monomial?(al.polyPart) =>
max(0, max(degree(al.polyPart), dc - db))
dc - db
dc - db
boundAt0(twr, f0, nb, nc, x, t, limitedint) ==
nb ~= 0 => min(0, nc - min(0, nb))
l1 := logdiff(twr, l0 := tower f0)
(if0 := limitedint(f0, [first argument u for u in l1]))
case "failed" => error "Risch's theorem violated"
(alpha := validExponential(concat(t, l0), RRF2F(if0::RRF), x))
case F =>
al := separate(univariate(alpha::F, t))$GP
zero?(al.fracPart) and monomial?(al.polyPart) =>
min(0, min(degree(al.polyPart), nc))
min(0, nc)
min(0, nc)
-- case a = 1, deg(B) = 0, B <> 0
-- cancellation at infinity is possible
logdegrad(twr, b, c, n, x, t, limitedint, extint) ==
t' := differentiate(t::F, x)
lk1 := logdiff(twr, lk0 := tower(f0 := - b))
(if0 := limitedint(f0, [first argument u for u in lk1]))
case "failed" => error "Risch's theorem violated"
(alpha := validExponential(lk0, RRF2F(if0::RRF), x)) case F =>
(u1 := primintfldpoly(inv(alpha::F) * c, extint(#1, t'), t'))
case "failed" => "failed"
degree(u1::UP)::Z > n => "failed"
alpha::F * u1::UP
logdeg(c, - if0.mainpart -
+/[v.coeff * log(v.logand) for v in if0.limitedlogs],
n, x, t', limitedint, extint)
-- case a = 1, degree(b) = 0, and (exp integrate b) is not in F
-- this implies no cancellation at infinity
logdeg(c, f, n, x, t', limitedint, extint) ==
answr:UP := 0
repeat
zero? c => return answr
negative? n or ((m := degree c)::Z > n) => return "failed"
u := rischDE(1, f, leadingCoefficient c, x, limitedint, extint)
~u.sol? => return "failed"
zero? m => return(answr + u.ans::UP)
n := m::Z - 1
c := (reductum c) - monomial(m::Z * t' * u.ans, (m - 1)::N)
answr := answr + monomial(u.ans, m)
-- case a = 1, deg(B) = 0, B <> 0
-- cancellation at infinity is possible
expdegrad(twr, b, c, n, x, t, limint, extint) ==
lk1 := logdiff(twr, lk0 := tower(f0 := - b))
(if0 := limint(f0, [first argument u for u in lk1]))
case "failed" => error "Risch's theorem violated"
intf0 := - if0.mainpart -
+/[v.coeff * log(v.logand) for v in if0.limitedlogs]
(alpha := validExponential(concat(t, lk0), RRF2F(if0::RRF), x))
case F =>
al := separate(univariate(alpha::F, t))$GP
zero?(al.fracPart) and monomial?(al.polyPart) and
(degree(al.polyPart) >= 0) =>
(u1 := expintfldpoly(c::GP * recip(al.polyPart)::GP,
rischDE(#1, first argument t, #2, x, limint, extint)))
case "failed" => "failed"
degree(u1::GP) > n => "failed"
retractIfCan(al.polyPart * u1::GP)@Union(UP, "failed")
expdeg(c, intf0, n, x, first argument t, limint,extint)
expdeg(c, intf0, n, x, first argument t, limint, extint)
-- case a = 1, degree(b) = 0, and (exp integrate b) is not a monomial
-- this implies no cancellation at infinity
expdeg(c, f, n, x, eta, limitedint, extint) ==
answr:UP := 0
repeat
zero? c => return answr
negative? n or ((m := degree c)::Z > n) => return "failed"
u := rischDE(1, f + m * eta, leadingCoefficient c, x,limitedint,extint)
~u.sol? => return "failed"
zero? m => return(answr + u.ans::UP)
n := m::Z - 1
c := reductum c
answr := answr + monomial(u.ans, m)
RRF2F rrf ==
rrf.mainpart + +/[v.coeff*log(v.logand) for v in rrf.limitedlogs]
|