forked from fricas/fricas
-
Notifications
You must be signed in to change notification settings - Fork 0
/
intalg.spad
766 lines (699 loc) · 31.2 KB
/
intalg.spad
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
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
)abbrev package DBLRESP DoubleResultantPackage
++ Residue resultant
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Description:
++ This package provides functions for computing the residues
++ of a function on an algebraic curve.
DoubleResultantPackage(F, UP, UPUP, R) : Exports == Implementation where
F : Field
UP : UnivariatePolynomialCategory F
UPUP : UnivariatePolynomialCategory Fraction UP
R : FunctionFieldCategory(F, UP, UPUP)
RF ==> Fraction UP
UP2 ==> SparseUnivariatePolynomial UP
UP3 ==> SparseUnivariatePolynomial UP2
Exports ==> with
doubleResultant : (R, UP -> UP) -> UP
++ doubleResultant(f, ') returns p(x) whose roots are
++ rational multiples of the residues of f at all its
++ finite poles. Argument ' is the derivation to use.
Implementation ==> add
import from CommuteUnivariatePolynomialCategory(F, UP, UP2)
import from UnivariatePolynomialCommonDenominator(UP, RF, UPUP)
UP22 : UP -> UP2
UP23 : UPUP -> UP3
remove0 : UP -> UP -- removes the power of x dividing p
remove0 p ==
primitivePart((p exquo monomial(1, minimumDegree p))::UP)
UP22 p ==
map(x+->x::UP, p)$UnivariatePolynomialCategoryFunctions2(F, UP, UP, UP2)
UP23 p ==
map(x1+->UP22(retract(x1)@UP),
p)$UnivariatePolynomialCategoryFunctions2(RF, UPUP, UP2, UP3)
doubleResultant(h, derivation) ==
cd := splitDenominator lift h
d := (cd.den exquo (g := gcd(cd.den, derivation(cd.den))))::UP
r := swap primitivePart swap resultant(UP23(cd.num)
- ((monomial(1, 1)$UP :: UP2) * UP22(g * derivation d))::UP3,
UP23 definingPolynomial())
remove0 resultant(r, UP22 d)
)abbrev package INTHERAL AlgebraicHermiteIntegration
++ Hermite integration, algebraic case
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Description: algebraic Hermite redution.
AlgebraicHermiteIntegration(F, UP, UPUP, R) : Exports == Implementation where
F : Field
UP : UnivariatePolynomialCategory F
UPUP : UnivariatePolynomialCategory Fraction UP
R : FunctionFieldCategory(F, UP, UPUP)
N ==> NonNegativeInteger
RF ==> Fraction UP
Exports ==> with
HermiteIntegrate : (R, UP -> UP) -> Record(answer : R, logpart : R)
++ HermiteIntegrate(f, ') returns \spad{[g, h]} such that
++ \spad{f = g' + h} and h has a only simple finite normal poles.
HermiteIntegrate : (R, UP -> UP, R) -> Record(answer : R, logpart : R)
++ HermiteIntegrate(f, ', d0) returns \spad{[g, h]} such that
++ \spad{f = g' + d0*g + h} and h has a only simple finite normal poles.
++ Note: d0 must be integral.
Implementation ==> add
localsolve : (Matrix UP, Vector UP, UP) -> Vector UP
-- the denominator of f should have no prime factor P s.t. P | P'
-- (which happens only for P = t in the exponential case)
HermiteIntegrate(f, derivation) == HermiteIntegrate(f, derivation, 0)
HermiteIntegrate(f, derivation, d0) ==
ratform : R := 0
n := rank()
m := transpose((mat := integralDerivationMatrix derivation).num)
integralCoordinates(d0).den ~= 1 => error "nonintegral d0"
inum := (cform := integralCoordinates f).num
if ((iden := cform.den) exquo (e := mat.den)) case "failed" then
iden := (coef := (e exquo gcd(e, iden))::UP) * iden
inum := coef * inum
for trm in factorList squareFree iden | (j : N := trm.exponent) > 1 repeat
u' := (u := (iden exquo (v := trm.factor)^j)::UP) * derivation v
sys := ((u * v) exquo e)::UP * m
nn := minRowIndex sys - minIndex inum
while j > 1 repeat
j := (j - 1)::N
p := - j * u'
sol := localsolve(sys + scalarMatrix(n, p), inum, v)
s0 := integralRepresents(sol, 1)
corr := integralCoordinates(d0*s0)
corr.den ~= 1 => error "impossible"
cnum := corr.num
ratform := ratform + integralRepresents(sol, v ^ j)
inum := [((qelt(inum, i) - p * qelt(sol, i) -
dot(row(sys, i - nn), sol))
exquo v)::UP - u*(derivation qelt(sol, i) +
qelt(cnum, i))
for i in minIndex inum .. maxIndex inum]
iden := u * v
[ratform, integralRepresents(inum, iden)]
localsolve(mat, vec, modulus) ==
ans : Vector(UP) := new(nrows mat, 0)
diagonal? mat =>
for i in minIndex ans .. maxIndex ans
for j in minRowIndex mat .. maxRowIndex mat
for k in minColIndex mat .. maxColIndex mat repeat
(bc := extendedEuclidean(qelt(mat, j, k), modulus,
qelt(vec, i))) case "failed" => return new(0, 0)
qsetelt!(ans, i, bc.coef1)
ans
sol := particularSolution(map(x+->x::RF, mat)$MatrixCategoryFunctions2(UP,
Vector UP, Vector UP, Matrix UP, RF,
Vector RF, Vector RF, Matrix RF),
map(x+->x::RF, vec)$VectorFunctions2(UP,
RF))$LinearSystemMatrixPackage(RF,
Vector RF, Vector RF, Matrix RF)
sol case "failed" => new(0, 0)
for i in minIndex ans .. maxIndex ans repeat
(bc := extendedEuclidean(denom qelt(sol, i), modulus, 1))
case "failed" => return new(0, 0)
qsetelt!(ans, i, (numer qelt(sol, i) * bc.coef1) rem modulus)
ans
)abbrev package INTALG AlgebraicIntegrate
)boot $tryRecompileArguments := nil
++ Integration of an algebraic function
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Description:
++ This package provides functions for integrating a function
++ on an algebraic curve.
AlgebraicIntegrate(R0, F, UP, UPUP, R) : Exports == Implementation where
R0 : Join(GcdDomain, Comparable, RetractableTo Integer)
F : Join(AlgebraicallyClosedField, FunctionSpace R0)
UP : UnivariatePolynomialCategory F
UPUP : UnivariatePolynomialCategory Fraction UP
R : FunctionFieldCategory(F, UP, UPUP)
SY ==> Symbol
Z ==> Integer
Q ==> Fraction Z
SUP ==> SparseUnivariatePolynomial F
QF ==> Fraction UP
GP ==> LaurentPolynomial(F, UP)
K ==> Kernel F
IR ==> IntegrationResult R
IRF ==> IntegrationResult F
UPR ==> SparseUnivariatePolynomial R
FD ==> FiniteDivisor(F, UP, UPUP, R)
LOG ==> Record(scalar : Q, coeff : UPR, logand : UPR)
DIV ==> Record(num : R, den : UP, derivden : UP, gd : UP)
U2 ==> Union(Record(ratpart:R, coeff:F),"failed")
FAIL0 ==> error "integrate: implementation incomplete (constant residues)"
FAIL1 ==> error "integrate: implementation incomplete (irrational residues)"
FAIL2 ==> error "integrate: implementation incomplete (residue poly has multiple non-linear factors)"
FAIL3 ==> error "integrate: implementation incomplete (has polynomial part)"
FAIL4 ==> error "integrate: implementation incomplete (trace 0)"
NOTI ==> error "Not integrable (provided residues have no relations)"
Exports ==> with
algintegrate : (R, UP -> UP, F -> IRF) -> IR
++ algintegrate(f, d, rec) integrates f with respect to the derivation d.
palgintegrate : (R, F, UP -> UP) -> Record(result1 : IR, result2 : F)
++ palgintegrate(f, x, d) integrates f with respect to the derivation d.
++ Argument f must be a pure algebraic function.
Implementation ==> add
import from FD
import from DoubleResultantPackage(F, UP, UPUP, R)
import from PointsOfFiniteOrder(R0, F, UP, UPUP, R)
import from AlgebraicHermiteIntegration(F, UP, UPUP, R)
import from InnerCommonDenominator(Z, Q, List Z, List Q)
import from PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R0, SparseMultivariatePolynomial(R0, K), F)
F2R : F -> R
F2UPR : F -> UPR
UP2SUP : UP -> SUP
SUP2UP : SUP -> UP
univ : (F, K) -> QF
pLogDeriv : (LOG, R -> R) -> R
mkLog : (UP, Q, R, F) -> List LOG
R2UP : (R, K) -> UPR
alglogint : (R, UP -> UP) -> Union(List LOG, "failed")
palglogint : (R, UP -> UP) -> Union(List LOG, "failed")
do_root : (UP, DIV) -> Union(List LOG, "failed")
trace1 : (UP, List F, List FD, FD, F, F) -> Union(List LOG, "failed")
nonQ : (DIV, UP) -> Union(List LOG, "failed")
rlift : (F, K, K) -> R
varRoot? : (UP, F -> F) -> Boolean
algintexp : (R, UP -> UP, F -> IRF) -> IR
algintprim : (R, UP -> UP, F -> IRF) -> IR
dummy : R := 0
dumx := kernel(new()$SY)$K
dumy := kernel(new()$SY)$K
F2UPR f == F2R(f)::UPR
F2R f == f::UP::QF::R
IRF_to_IR(irf : IRF) : IR ==
map(F2R, irf)$IntegrationResultFunctions2(F, R)
RF ==> Fraction UP
R_to_F(f : R) : Union(F, "failed") ==
(u1 := retractIfCan(f)@Union(RF, "failed")) case "failed" => "failed"
f1 := u1@RF
(u2 := retractIfCan(f1)@Union(UP, "failed")) case "failed" => "failed"
f2 := u2@UP
retractIfCan(f2)@Union(F, "failed")
algintexp(f, derivation, rec_int) ==
d := (c := integralCoordinates f).den
v := c.num
vp : Vector(GP) := new(n := #v, 0)
vf : Vector(QF) := new(n, 0)
for i in minIndex v .. maxIndex v repeat
r := separate(qelt(v, i) / d)$GP
qsetelt!(vf, i, r.fracPart)
qsetelt!(vp, i, r.polyPart)
ff := represents(vf, w := integralBasis())
h := HermiteIntegrate(ff, derivation)
p := represents(map((x1 : GP) : QF+->convert(x1)@QF,
vp)$VectorFunctions2(GP, QF), w)
not(zero?(p)) => FAIL3
zero?(h.logpart) => h.answer::IR
(p1u := R_to_F(h.logpart)) case F =>
res1 := rec_int(p1u@F)
mkAnswer(h.answer, empty(), empty()) + IRF_to_IR(res1)
(u := alglogint(h.logpart, derivation)) case "failed" =>
mkAnswer(h.answer, empty(), [[p + h.logpart, dummy]])
mkAnswer(h.answer, u@List(LOG), empty())
algintprim(f, derivation, rec_int) ==
h := HermiteIntegrate(f, derivation)
zero?(h.logpart) => h.answer::IR
(p1u := R_to_F(h.logpart)) case F =>
res1 := rec_int(p1u@F)
mkAnswer(h.answer, empty(), empty()) + IRF_to_IR(res1)
(u := alglogint(h.logpart, derivation)) case "failed" =>
mkAnswer(h.answer, empty(), [[h.logpart, dummy]])
mkAnswer(h.answer, u@List(LOG), empty())
FAC2 ==> Record(factor : UP, exponent : NonNegativeInteger)
PC ==> Record(factor : UP, exponent : NonNegativeInteger, coeff : Q)
RR ==> Record(result1 : List(PC), result2 : List(FAC2))
find_multiples(f1 : FAC2, lp : List(FAC2)) : RR ==
p1 := f1.factor
res : List(PC) := [[p1, f1.exponent, 1]$PC]
rr : List(FAC2) := []
k := degree(p1)
cp1 : F
l : NonNegativeInteger
rp1 := reductum(p1)
while rp1 ~= 0 repeat
cp1 := leadingCoefficient(rp1)
l := qcoerce(degree(rp1))@NonNegativeInteger
if odd?(k - l) then break
rp1 := reductum(rp1)
n := qcoerce(k - l)@NonNegativeInteger
for f in lp repeat
rr := cons(f, rr)
p := f.factor
degree(p) ~= k => iterate
cp := coefficient(p, l)
cp = 0 => iterate
fac := cp/cp1
facu := retractIfCan(fac)@Union(Fraction(Integer), "failed")
facu case "failed" => iterate
nfac := facu@Fraction(Integer)
nfac_ok := true
if n > 1 then
nnu := perfectNthRoot(numer(nfac), n)$IntegerRoots(Z)
nnu case "failed" => nfac_ok := false
ndu := perfectNthRoot(denom(nfac), n)$IntegerRoots(Z)
ndu case "failed" => nfac_ok := false
nfac := (nnu@Z)/(ndu@Z)
not(nfac_ok) => iterate
p1 ~= (nfac::F)^(-k)*elt(p, monomial(nfac::F, 1)$UP) => iterate
res := cons([p, f.exponent, nfac]$PC, res)
rr := rest(rr)
res := reverse!(res)
[res, rr]$RR
PC2 ==> Record(factor : UP, exponent : NonNegativeInteger, coeff : Q,
scalar : F, divisor : FD)
RR2 ==> Record(result1 : List(LOG), result2 : List(PC2),
result3 : List(PC))
handle_multiples1(rec : DIV, lp : List(PC), cc : Z) : RR2 ==
nlp : List(PC2) := []
llg : List(LOG) := []
pc1 := first(lp)
p := pc1.factor
alpha := rootOf(UP2SUP p)
for peq in lp repeat
nfac := peq.coeff
na := nfac::F * alpha
di1 := divisor(rec.num, rec.den, rec.derivden, rec.gd, na)
di2 := divisor(rec.num, rec.den, rec.derivden, rec.gd, -na)
di := di1 - di2
(rc := torsionIfCan di) case "failed" =>
pc2 := [peq.factor, peq.exponent, peq.coeff,
na/(cc::F), di]$PC2
nlp := cons(pc2, nlp)
nlog := mkLog(p, nfac*inv(cc*(rc.order::Q)), rc.function, alpha)
llg := concat(nlog, llg)
[llg, nlp, []]
handle_multiples(rec : DIV, lp : List(PC)) : RR2 ==
pc1 := first(lp)
p := pc1.factor
p = elt(p, monomial(-1::F, 1)$UP) =>
-- even case
handle_multiples1(rec, lp, 2)
lp1 := [peq for peq in lp | peq.coeff >$Q 0$Q ]
#lp ~= 2*#lp1 => [[], [], lp]
for peq in lp1 repeat
nfac := peq.coeff
fu := find((x : PC) : Boolean +-> x.coeff = -nfac, lp)
fu case "failed" => return [[], [], lp]
ff := fu@PC
if ff.exponent ~= peq.exponent then return [[], [], lp]
handle_multiples1(rec, lp1, 1)
RR3 ==> Record(result1 : List(LOG), result2 : List(List(PC2)),
result3 : List(List(PC)))
get_coeffs0(lp : List(UP)) : List(F) ==
res : List(F) := []
for p in lp repeat
res := concat(coefficients(p), res)
res
get_coeffs1(pp : UPUP) : List(UP) ==
lp := coefficients(pp)
nlp := map(numer, lp)$ListFunctions2(Fraction UP, UP)
dlp := map(denom, lp)$ListFunctions2(Fraction UP, UP)
concat(nlp, dlp)
get_coeffs(rec : DIV) : List(F) ==
clp : List(UP) := [rec.den, rec.derivden, rec.gd]$List(UP)
nlp := get_coeffs1(lift(rec.num))
dlp := get_coeffs1(definingPolynomial()$R)
get_coeffs0(concat(clp, concat(nlp, dlp)))
classify_divisors(rec : DIV, r : UP) : RR3 ==
cl := get_coeffs(rec)
u0 := factor(makeSUP(primitivePart(r)), cl
)$ExpressionFactorPolynomial(R0, F)
u := map(SUP2UP, u0)$FactoredFunctions2(SUP, UP)
lf := factors(u)
lm : List(List(PC)) := []
while not(empty?(lf)) repeat
rp := find_multiples(first(lf), rest(lf))
lf := rp.result2
lm := cons(rp.result1, lm)
llg : List(LOG) := []
nlm1 : List(List(PC2)) := []
nlm2 : List(List(PC)) := []
for lp in lm repeat
rr2 := handle_multiples(rec, lp)
llg := concat(rr2.result1, llg)
if not(empty?(nlp1 := rr2.result2)) then
nlm1 := cons(nlp1, nlm1)
if not(empty?(nlp2 := rr2.result3)) then
nlm2 := cons(nlp2, nlm2)
[llg, nlm1, nlm2]
get_lf(ll1 : List(List(PC2)), ll2 : List(List(PC))) : List(UP) ==
empty?(ll1) and empty?(ll2) => []
#ll1 +$Z #ll2 >$Z 1$Z => FAIL1
empty?(ll1) => [peq.factor for peq in first(ll2)]
[peq.factor for peq in first(ll1)]
get_la(ll1 : List(List(PC2)), ll2 : List(List(PC))) : List(F) ==
[-coefficient(lfac, 0) for lfac in get_lf(ll1, ll2)]
get_ld(rec : DIV, ll1 : List(List(PC2)), ll2 : List(List(PC))
) : List(FD) ==
empty?(ll1) and empty?(ll2) => []
#ll1 +$Z #ll2 >$Z 1$Z => FAIL1
empty?(ll1) => [divisor(rec.num, rec.den, rec.derivden, rec.gd, a) for
a in get_la(ll1, ll2)]
[peq.divisor for peq in first(ll1)]
-- checks whether f = +/[ci (ui)'/(ui)]
-- f dx must have no pole at infinity
palglogint(f, derivation) ==
rec := algSplitSimple(f, derivation)
-- r(z) has roots which are the residues of f at all its poles
ground?(r := doubleResultant(f, derivation)) => []
ppr := classify_divisors(rec, r)
nfacs1 := ppr.result2
nfacs2 := ppr.result3
empty?(nfacs1) and empty?(nfacs2) => ppr.result1
-- We have to look at Q-linear relations between residues.
-- Below we handle a few easy cases.
nlins1 := [nfl for nfl in nfacs1 | degree(first(nfl).factor) >$Z 1]
nlins2 : List(List(PC)) := []
fcf : UP
root_fails : Integer := 0
for nfl in nfacs2 | degree(first(nfl).factor) >$Z 1 repeat
fcf := first(nfl).factor
degree(fcf) = 3 and degree(reductum(fcf)) = 0 =>
pp := do_root(fcf, rec)
pp case "failed" =>
nlins2 := cons(nfl, nlins2)
root_fails := root_fails + #nfl
ppr.result1 := concat(pp@List(LOG), ppr.result1)
nlins2 := cons(nfl, nlins2)
nlins2 := reverse!(nlins2)
root_fails = 1 and #nlins2 = 1 and #nlins1 = 0 => "failed"
#nlins1 + #nlins2 > 1 => FAIL2
lins1 := [nfl for nfl in nfacs1 | degree(first(nfl).factor) = 1]
lins2 := [nfl for nfl in nfacs2 | degree(first(nfl).factor) = 1]
empty?(nlins2) and empty?(lins2) and empty?(nfacs1) => ppr.result1
empty?(nfacs2) and empty?(nlins1) and #lins1 = 1
and #first(lins1) = 1 => "failed"
#nfacs1 > 0 and #nlins2 = 0 =>
fc1 : PC2
if #nlins1 = 1 then
nfl1 := first(nlins1)
not(empty?(rest(nfl1))) => FAIL2
fc1 := first(nfl1)
fcf := fc1.factor
else
#lins1 > 1 => FAIL1
nfl1 := first(lins1)
empty?(rest(nfl1)) => FAIL1
fc1 := first(nfl1)
fcf := fc1.factor
lins1 := [rest(nfl1)]
zero?(bb := coefficient(fcf,
(degree(fcf) - 1)::NonNegativeInteger)) =>
fcf = elt(fcf, monomial(-1::F, 1)$UP) =>
degree(fcf) = 2 => "failed"
empty?(nfacs2) and #nfacs1 = 1 and degree(fcf) = 4 =>
"failed"
NOTI
NOTI
la := get_la(lins1, lins2)
ld := get_ld(rec, lins1, lins2)
tr1u := trace1(fcf, la, ld, fc1.divisor, fc1.scalar, bb)
tr1u case "failed" => "failed"
concat(ppr.result1, tr1u@List(LOG))
#lins1 + #lins2 > 1 => FAIL1
fc2 : PC
if #nlins2 = 1 then
nfl2 := first(nlins2)
not(empty?(rest(nfl2))) => FAIL2
fc2 := first(nfl2)
else
#lins2 > 1 => FAIL1
nfl2 := first(lins2)
fc2 := first(nfl2)
lins2 := [rest(nfl2)]
fcf := fc2.factor
zero?(bb := coefficient(fcf,
(degree(fcf) - 1)::NonNegativeInteger)) => FAIL4
la := get_la(lins1, lins2)
ld := get_ld(rec, lins1, lins2)
alpha := rootOf UP2SUP fcf
-- v is the divisor corresponding to all the residues
v1 := divisor(rec.num, rec.den, rec.derivden, rec.gd, alpha)
tr1u := trace1(fcf, la, ld, v1, alpha, bb)
tr1u case "failed" => "failed"
concat(ppr.result1, tr1u@List(LOG))
do_root(q : UP, rec : DIV) : Union(List LOG, "failed") ==
alpha := rootOf UP2SUP q
beta := sqrt(-(3::F))
w1 := (beta - 1)/(2::F)
v1 := divisor(rec.num, rec.den, rec.derivden, rec.gd, alpha)
v2 := divisor(rec.num, rec.den, rec.derivden, rec.gd, w1*alpha)
v3 := divisor(rec.num, rec.den, rec.derivden, rec.gd, w1^2*alpha)
vp1 := 2*v1
vp2 := v2 + v3
vp := vp1 - vp2
(rc := torsionIfCan vp) case "failed" => "failed"
mkLog(q, inv((rc.order * 3)::Q), rc.function, alpha)
UP2SUP p ==
map((x : F) : F+->x, p)$UnivariatePolynomialCategoryFunctions2(F, UP, F, SUP)
SUP2UP p ==
map((x : F) : F+->x, p)$UnivariatePolynomialCategoryFunctions2(F, SUP, F, UP)
varRoot?(p, derivation) ==
for c in coefficients p repeat
derivation(c) ~= 0 => return true
false
pLogDeriv(log, derivation) ==
import from UPR
map(derivation, log.coeff) ~= 0 =>
error "can only handle logs with constant coefficients"
((n := degree(log.coeff)) = 1) =>
c := - (leadingCoefficient reductum log.coeff)
/ (leadingCoefficient log.coeff)
ans := (log.logand) c
(log.scalar)::R * c * derivation(ans) / ans
numlog := map(derivation, log.logand)
(diflog := extendedEuclidean(log.logand, log.coeff, numlog)) case
"failed" => error "this shouldn't happen"
algans := diflog.coef1
ans : R := 0
for i in 0..n-1 repeat
algans := (algans * monomial(1, 1)) rem log.coeff
ans := ans + coefficient(algans, i)
(log.scalar)::R * ans
R2UP(f, k) ==
x := dumx :: F
g := (map((f1 : QF) : F+->f1(x),
lift f)$UnivariatePolynomialCategoryFunctions2(QF, UPUP, F, UP))
(y := dumy::F)
map((x1 : F) : R+->rlift(x1, dumx, dumy), univariate(g, k,
minPoly k))$UnivariatePolynomialCategoryFunctions2(F, SUP, R, UPR)
univ(f, k) ==
g := univariate(f, k)
(SUP2UP numer g) / (SUP2UP denom g)
rlift(f, kx, ky) ==
uf := univariate(f, ky)
reduce map(x1+->univ(x1, kx), retract(uf)@SUP
)$UnivariatePolynomialCategoryFunctions2(F, SUP, QF, UPUP)
-- case when the irreducible factor p has roots which sum ~= 0
-- the residues of f are of the form [a1, ..., ak]
-- plus all the roots of q(z), which is squarefree
-- la is the list of residues la := [a1, ..., ak]
-- ld is the list of divisors [D1, ...Dk] where Di is the sum of all the
-- places where f has residue ai
-- q(z) is assumed doubly transitive for now.
-- let [alpha_1, ..., alpha_m] be the roots of q(z)
-- in this function, b = - alpha_1 - ... - alpha_m is ~= 0
-- which implies only one generic log term
-- we assume that [a1, ..., ak] are b times rational numbers
trace1(q, la, ld, v1, alpha, b) ==
-- cd = [[b1, ..., bk], d] such that ai / b = bi / d
cd := splitDenominator [retract(a/b)@Q for a in la]
-- then, a basis for all the residues of f over the integers is
-- [beta_1 = - alpha_1 / d, ..., beta_m = - alpha_m / d], since:
-- alpha_i = - d beta_i
-- ai = (ai / b)*b = (bi / d)*b = b1*beta_1 + ... + bm*beta_m
-- linear independence is a consequence of the doubly transitive
-- assumption
-- v0 is the divisor +/[bi Di] corresponding to the residues
-- [a1, ..., ak]
v0 := +/[a * dv for a in cd.num for dv in ld]
v2 := v0 - cd.den * v1
(rc := torsionIfCan v2) case "failed" => -- non-torsion case
degree(q) <= 2 => "failed" -- guaranteed doubly-transitive
NOTI -- maybe doubly-transitive
mkLog(q, inv((- rc.order * cd.den)::Q), rc.function, alpha)
mkLog(q, scalr, lgd, alpha) ==
degree(q) <=$Integer 1 =>
[[scalr, monomial(1, 1)$UPR - F2UPR alpha, lgd::UPR]]
[[scalr,
map(F2R, q)$UnivariatePolynomialCategoryFunctions2(F, UP, R, UPR),
R2UP(lgd, retract(alpha)@K)]]
import from GenerateEllipticIntegrals(F, UP, UPUP, R)
-- Currently code for elliptic integrals can only handle
-- elliptic curves y^2 = P(x) where P is of degree 3 or 4.
E_res ==> Union(List(F), "failed")
EF_REC ==> Record(result1 : RF, result2 : F)
pole_parts(lc : List(RF), ii : Integer) : List(UP) ==
nn := (ii - 1)::NonNegativeInteger
[(monomial(1, nn)$UP*numer(c)) quo denom(c) for c in lc]
get_e_coeffs(f : R, e_forms : List(R), der : UP -> UP) : E_res ==
#e_forms ~=$Integer 1 => []
i_mat := inverseIntegralMatrixAtInfinity()$R
f_c0 : Vector(RF) := coordinates(f)$R
f_c := parts(f_c0*i_mat)
inf_integral : Boolean := true
dx := der(monomial(1, 1)$UP)
dx ~= 1 => []
g := generator()$R
p := definingPolynomial()$R
ii : Integer := 2
if degree(p) = 2 and degree(r := reductum(p)) = 0 then
rf := retract(r)@RF
if degree(numer(rf)) = 3 then
ii := 1
for c in f_c repeat
if degree(denom(c))@Z - degree(numer(c))@Z < ii then
inf_integral := false
break
inf_integral => []
empty?(e_forms) => "failed"
fv := vector(pole_parts(f_c, ii))$Vector(UP)
zero?(fv) => []
ll := [pole_parts(parts(coordinates(e_f)*i_mat), ii) for e_f in e_forms]
e_mat := matrix(ll)$Matrix(UP)
r_sys := reducedSystem(transpose(e_mat), fv)
sol := solve(r_sys.mat, r_sys.vec)$LinearSystemMatrixPackage1(F)
pu := sol.particular
pu case "failed" => "failed"
cl := parts(pu@Vector(F))$Vector(F)
for c in cl repeat
if der(c::UP) ~= 0 then return "failed"
cl
gen_answer(h : Record(answer : R, logpart : R), u : List(LOG),
el_f : EF_REC) : Record(result1 : IR, result2 : F) ==
r1 := el_f.result1
ans1 : R :=
r1 = 0 => 0
nr1 := numer(r1)
dn1 := denom(r1)
-- FIXME: Hardcoded assumption that R is generated by square root
represents(vector([0, nr1])$Vector(UP), dn1)$R
[mkAnswer(h.answer + ans1, u, empty()), el_f.result2]
-- f dx must be locally integral at infinity
palgintegrate(f, x, derivation) ==
h := HermiteIntegrate(f, derivation)
zero?(h.logpart) => [h.answer::IR, 0]
ell_lst := get_elliptics(derivation, x)
e_forms := [el.e_form for el in ell_lst]
f_forms := [el.f_form for el in ell_lst]
e_cu := get_e_coeffs(h.logpart, e_forms, derivation)
e_cu case "failed" =>
[mkAnswer(h.answer, empty(), [[h.logpart, dummy]]), 0]
e_coeffs := e_cu@List(F)
1 <$Integer #e_coeffs =>
[mkAnswer(h.answer, empty(), [[h.logpart, dummy]]), 0]
h_lp :=
empty?(e_coeffs) => h.logpart
h.logpart - reduce(_+, [c_i::UP::RF*e_f for c_i in e_coeffs
for e_f in e_forms])
not(integralAtInfinity?(h_lp)) or
((u := palglogint(h_lp, derivation)) case "failed") =>
[mkAnswer(h.answer, empty(), [[h.logpart, dummy]]), 0]
difFirstKind := h_lp - +/[pLogDeriv(lg,
x1+->differentiate(x1, derivation)) for lg in u@List(LOG)]
zero?(difFirstKind) =>
empty?(e_coeffs) =>
[mkAnswer(h.answer, u@List(LOG), empty()), 0]
var := ell_lst(1).f_var
pol := ell_lst(1).f_pol
el_f := gen_ef(var, pol, 0, first(e_coeffs)::UP::RF::R)
el_f case "failed" =>
[mkAnswer(h.answer, empty(), [[h.logpart, dummy]]), 0]
gen_answer(h, u, el_f)
#f_forms ~=$Integer 1 =>
[mkAnswer(h.answer, u@List(LOG), [[difFirstKind, dummy]]), 0]
f_c := difFirstKind/f_forms(1)
differentiate(f_c) ~= 0 =>
empty?(e_coeffs) =>
[mkAnswer(h.answer, u@List(LOG), [[difFirstKind, dummy]]), 0]
[mkAnswer(h.answer, empty(), [[h.logpart, dummy]]), 0]
var := ell_lst(1).f_var
pol := ell_lst(1).f_pol
e_c : F :=
empty?(e_coeffs) => 0
first(e_coeffs)
el_f := gen_ef(var, pol, f_c, e_c::UP::RF::R)
el_f case "failed" =>
empty?(e_coeffs) =>
[mkAnswer(h.answer, u@List(LOG), [[difFirstKind, dummy]]), 0]
[mkAnswer(h.answer, empty(), [[h.logpart, dummy]]), 0]
gen_answer(h, u, el_f)
-- for mixed functions. f dx not assumed locally integral at infinity
algintegrate(f, derivation, rec_int) ==
x' := derivation(x := monomial(1, 1)$UP)
zero? degree(x') =>
algintprim(f, derivation, rec_int)
((xx := x' exquo x) case UP) and
(retractIfCan(xx::UP)@Union(F, "failed") case F) =>
algintexp(f, derivation, rec_int)
error "should not happen"
-- Just most naive heuristic, to check feasibility
try_logs(f : R, derivation : UP -> UP, r : UP, lp : List(UP)
) : Union(List(LOG), "failed") ==
rec := algSplitSimple(f, derivation)
-- FIXME: should get algebraic generators of constant
-- field
u0 := factor(makeSUP(r), []
)$ExpressionFactorPolynomial(R0, F)
u := map(SUP2UP, u0)$FactoredFunctions2(SUP, UP)
lf := factorList(u)
llg : List(LOG) := []
for fac in lf repeat
p := fac.factor
alpha := rootOf(UP2SUP p)
di := divisor(rec.num, rec.den, rec.derivden, rec.gd, alpha)
gu := generator(di, -fac.exponent, [])
gu case "failed" => return "failed"
nlog := mkLog(p, 1, gu, alpha)
llg := concat(nlog, llg)
llg
alglogint(f, derivation) ==
r := primitivePart(doubleResultant(f, derivation))
varRoot?(r, x1+->retract(derivation(x1::UP))@F) => "failed"
xx := monomial(1, 1)$UP
dxx := derivation(xx)
lp : List(UP) :=
degree(dxx) = 1 and reductum(dxx) = 0 => [xx]
[]
lu := try_logs(f, derivation, r, lp)
lu case "failed" => FAIL0
ll := lu@List(LOG)
diff1 := f - +/[pLogDeriv(lg,
x1+->differentiate(x1, derivation)) for lg in ll]
zero?(diff1) => ll
FAIL0
)boot $tryRecompileArguments := true
--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.