Annotation of OpenXM_contrib2/asir2000/lib/bfct, Revision 1.15
1.2 noro 1: /*
2: * Copyright (c) 1994-2000 FUJITSU LABORATORIES LIMITED
3: * All rights reserved.
4: *
5: * FUJITSU LABORATORIES LIMITED ("FLL") hereby grants you a limited,
6: * non-exclusive and royalty-free license to use, copy, modify and
7: * redistribute, solely for non-commercial and non-profit purposes, the
8: * computer program, "Risa/Asir" ("SOFTWARE"), subject to the terms and
9: * conditions of this Agreement. For the avoidance of doubt, you acquire
10: * only a limited right to use the SOFTWARE hereunder, and FLL or any
11: * third party developer retains all rights, including but not limited to
12: * copyrights, in and to the SOFTWARE.
13: *
14: * (1) FLL does not grant you a license in any way for commercial
15: * purposes. You may use the SOFTWARE only for non-commercial and
16: * non-profit purposes only, such as academic, research and internal
17: * business use.
18: * (2) The SOFTWARE is protected by the Copyright Law of Japan and
19: * international copyright treaties. If you make copies of the SOFTWARE,
20: * with or without modification, as permitted hereunder, you shall affix
21: * to all such copies of the SOFTWARE the above copyright notice.
22: * (3) An explicit reference to this SOFTWARE and its copyright owner
23: * shall be made on your publication or presentation in any form of the
24: * results obtained by use of the SOFTWARE.
25: * (4) In the event that you modify the SOFTWARE, you shall notify FLL by
1.3 noro 26: * e-mail at risa-admin@sec.flab.fujitsu.co.jp of the detailed specification
1.2 noro 27: * for such modification or the source code of the modified part of the
28: * SOFTWARE.
29: *
30: * THE SOFTWARE IS PROVIDED AS IS WITHOUT ANY WARRANTY OF ANY KIND. FLL
31: * MAKES ABSOLUTELY NO WARRANTIES, EXPRESSED, IMPLIED OR STATUTORY, AND
32: * EXPRESSLY DISCLAIMS ANY IMPLIED WARRANTY OF MERCHANTABILITY, FITNESS
33: * FOR A PARTICULAR PURPOSE OR NONINFRINGEMENT OF THIRD PARTIES'
34: * RIGHTS. NO FLL DEALER, AGENT, EMPLOYEES IS AUTHORIZED TO MAKE ANY
35: * MODIFICATIONS, EXTENSIONS, OR ADDITIONS TO THIS WARRANTY.
36: * UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL THEORY, TORT, CONTRACT,
37: * OR OTHERWISE, SHALL FLL BE LIABLE TO YOU OR ANY OTHER PERSON FOR ANY
38: * DIRECT, INDIRECT, SPECIAL, INCIDENTAL, PUNITIVE OR CONSEQUENTIAL
39: * DAMAGES OF ANY CHARACTER, INCLUDING, WITHOUT LIMITATION, DAMAGES
40: * ARISING OUT OF OR RELATING TO THE SOFTWARE OR THIS AGREEMENT, DAMAGES
41: * FOR LOSS OF GOODWILL, WORK STOPPAGE, OR LOSS OF DATA, OR FOR ANY
42: * DAMAGES, EVEN IF FLL SHALL HAVE BEEN INFORMED OF THE POSSIBILITY OF
43: * SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY. EVEN IF A PART
44: * OF THE SOFTWARE HAS BEEN DEVELOPED BY A THIRD PARTY, THE THIRD PARTY
45: * DEVELOPER SHALL HAVE NO LIABILITY IN CONNECTION WITH THE USE,
46: * PERFORMANCE OR NON-PERFORMANCE OF THE SOFTWARE.
47: *
1.15 ! noro 48: * $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.14 2001/01/10 04:30:35 noro Exp $
1.10 noro 49: */
1.1 noro 50: /* requires 'primdec' */
51:
1.6 noro 52: /* annihilating ideal of F^s */
1.1 noro 53:
54: def ann(F)
55: {
56: V = vars(F);
57: N = length(V);
1.8 noro 58: D = newvect(N);
59:
60: for ( I = 0; I < N; I++ )
61: D[I] = [deg(F,V[I]),V[I]];
62: qsort(D,compare_first);
63: for ( V = [], I = N-1; I >= 0; I-- )
64: V = cons(D[I][1],V);
65:
1.1 noro 66: for ( I = N-1, DV = []; I >= 0; I-- )
67: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.8 noro 68:
69: W = append([y1,y2,t],V);
1.1 noro 70: DW = append([dy1,dy2,dt],DV);
1.8 noro 71:
72: B = [1-y1*y2,t-y1*F];
1.1 noro 73: for ( I = 0; I < N; I++ ) {
74: B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
75: }
1.10 noro 76:
77: /* homogenized (heuristics) */
1.1 noro 78: dp_nelim(2);
1.10 noro 79: G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
1.1 noro 80: G1 = [];
81: for ( T = G0; T != []; T = cdr(T) ) {
82: E = car(T); VL = vars(E);
83: if ( !member(y1,VL) && !member(y2,VL) )
84: G1 = cons(E,G1);
85: }
1.12 noro 86: G2 = map(psi,G1,t,dt);
87: G3 = map(subst,G2,t,-1-s);
88: return G3;
1.1 noro 89: }
90:
1.10 noro 91: /*
92: * compute J_f|s=r, where r = the minimal integral root of global b_f(s)
93: * ann0(F) returns [MinRoot,Ideal]
94: */
95:
96: def ann0(F)
97: {
98: V = vars(F);
99: N = length(V);
100: D = newvect(N);
101:
102: for ( I = 0; I < N; I++ )
103: D[I] = [deg(F,V[I]),V[I]];
104: qsort(D,compare_first);
105: for ( V = [], I = 0; I < N; I++ )
106: V = cons(D[I][1],V);
107:
108: for ( I = N-1, DV = []; I >= 0; I-- )
109: DV = cons(strtov("d"+rtostr(V[I])),DV);
110:
111: /* XXX : heuristics */
112: W = append([y1,y2,t],reverse(V));
113: DW = append([dy1,dy2,dt],reverse(DV));
114: WDW = append(W,DW);
115:
116: B = [1-y1*y2,t-y1*F];
117: for ( I = 0; I < N; I++ ) {
118: B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
119: }
120:
121: /* homogenized (heuristics) */
122: dp_nelim(2);
123: G0 = dp_weyl_gr_main(B,WDW,1,0,6);
124: G1 = [];
125: for ( T = G0; T != []; T = cdr(T) ) {
126: E = car(T); VL = vars(E);
127: if ( !member(y1,VL) && !member(y2,VL) )
128: G1 = cons(E,G1);
129: }
1.12 noro 130: G2 = map(psi,G1,t,dt);
131: G3 = map(subst,G2,t,-1-s);
1.10 noro 132:
1.12 noro 133: /* G3 = J_f(s) */
1.10 noro 134:
135: V1 = cons(s,V); DV1 = cons(ds,DV); V1DV1 = append(V1,DV1);
1.12 noro 136: G4 = dp_weyl_gr_main(cons(F,G3),V1DV1,0,1,0);
137: Bf = weyl_minipoly(G4,V1DV1,0,s);
1.10 noro 138:
139: FList = cdr(fctr(Bf));
140: for ( T = FList, Min = 0; T != []; T = cdr(T) ) {
141: LF = car(car(T));
142: Root = -coef(LF,0)/coef(LF,1);
143: if ( dn(Root) == 1 && Root < Min )
144: Min = Root;
145: }
1.12 noro 146: return [Min,map(subst,G3,s,Min)];
1.10 noro 147: }
148:
1.7 noro 149: def indicial1(F,V)
1.6 noro 150: {
151: W = append([y1,t],V);
152: N = length(V);
153: B = [t-y1*F];
154: for ( I = N-1, DV = []; I >= 0; I-- )
155: DV = cons(strtov("d"+rtostr(V[I])),DV);
156: DW = append([dy1,dt],DV);
157: for ( I = 0; I < N; I++ ) {
158: B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
159: }
160: dp_nelim(1);
1.10 noro 161:
162: /* homogenized (heuristics) */
1.7 noro 163: G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
1.6 noro 164: G1 = map(subst,G0,y1,1);
165: G2 = map(psi,G1,t,dt);
166: G3 = map(subst,G2,t,-s-1);
167: return G3;
168: }
169:
170: def psi(F,T,DT)
171: {
172: D = dp_ptod(F,[T,DT]);
173: Wmax = weight(D);
174: D1 = dp_rest(D);
175: for ( ; D1; D1 = dp_rest(D1) )
176: if ( weight(D1) > Wmax )
177: Wmax = weight(D1);
178: for ( D1 = D, Dmax = 0; D1; D1 = dp_rest(D1) )
179: if ( weight(D1) == Wmax )
180: Dmax += dp_hm(D1);
181: if ( Wmax >= 0 )
182: Dmax = dp_weyl_mul(<<Wmax,0>>,Dmax);
183: else
184: Dmax = dp_weyl_mul(<<0,-Wmax>>,Dmax);
185: Rmax = dp_dtop(Dmax,[T,DT]);
186: R = b_subst(subst(Rmax,DT,1),T);
187: return R;
188: }
189:
190: def weight(D)
191: {
192: V = dp_etov(D);
193: return V[1]-V[0];
194: }
195:
196: def compare_first(A,B)
197: {
198: A0 = car(A);
199: B0 = car(B);
200: if ( A0 > B0 )
201: return 1;
202: else if ( A0 < B0 )
203: return -1;
204: else
205: return 0;
206: }
207:
1.13 noro 208: /* generic b-function w.r.t. weight vector W */
209:
210: def generic_bfct(F,V,DV,W)
211: {
212: N = length(V);
213: N2 = N*2;
214:
1.15 ! noro 215: dp_weyl_set_weight(W);
! 216:
1.14 noro 217: /* create a term order M in D<x,d> (DRL) */
1.13 noro 218: M = newmat(N2,N2);
219: for ( J = 0; J < N2; J++ )
220: M[0][J] = 1;
221: for ( I = 1; I < N2; I++ )
222: M[I][N2-I] = -1;
223:
224: VDV = append(V,DV);
225:
226: /* create a non-term order MW in D<x,d> */
227: MW = newmat(N2+1,N2);
228: for ( J = 0; J < N; J++ )
229: MW[0][J] = -W[J];
230: for ( ; J < N2; J++ )
231: MW[0][J] = W[J-N];
232: for ( I = 1; I <= N2; I++ )
233: for ( J = 0; J < N2; J++ )
234: MW[I][J] = M[I-1][J];
235:
236: /* create a homogenized term order MWH in D<x,d,h> */
237: MWH = newmat(N2+2,N2+1);
238: for ( J = 0; J <= N2; J++ )
239: MWH[0][J] = 1;
240: for ( I = 1; I <= N2+1; I++ )
241: for ( J = 0; J < N2; J++ )
242: MWH[I][J] = MW[I-1][J];
243:
244: /* homogenize F */
245: VDVH = append(VDV,[h]);
246: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
247:
248: /* compute a groebner basis of FH w.r.t. MWH */
1.15 ! noro 249: dp_gr_flags(["Top",1,"NoRA",1]);
! 250: GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
! 251: dp_gr_flags(["Top",0,"NoRA",0]);
1.13 noro 252:
253: /* dehomigenize GH */
254: G = map(subst,GH,h,1);
255:
256: /* G is a groebner basis w.r.t. a non term order MW */
257: /* take the initial part w.r.t. (-W,W) */
258: GIN = map(initial_part,G,VDV,MW,W);
259:
260: /* GIN is a groebner basis w.r.t. a term order M */
261: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
262:
263: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
264: for ( I = 0, T = 0; I < N; I++ )
265: T += W[I]*V[I]*DV[I];
1.14 noro 266: B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
1.13 noro 267: return B;
268: }
269:
270: def initial_part(F,V,MW,W)
271: {
272: N2 = length(V);
273: N = N2/2;
274: dp_ord(MW);
275: DF = dp_ptod(F,V);
276: R = dp_hm(DF);
277: DF = dp_rest(DF);
278:
279: E = dp_etov(R);
280: for ( I = 0, TW = 0; I < N; I++ )
281: TW += W[I]*(-E[I]+E[N+I]);
282: RW = TW;
283:
284: for ( ; DF; DF = dp_rest(DF) ) {
285: E = dp_etov(DF);
286: for ( I = 0, TW = 0; I < N; I++ )
287: TW += W[I]*(-E[I]+E[N+I]);
288: if ( TW == RW )
289: R += dp_hm(DF);
290: else if ( TW < RW )
291: break;
292: else
293: error("initial_part : cannot happen");
294: }
295: return dp_dtop(R,V);
296:
297: }
298:
1.1 noro 299: /* b-function of F ? */
300:
301: def bfct(F)
302: {
303: V = vars(F);
304: N = length(V);
1.6 noro 305: D = newvect(N);
1.7 noro 306:
1.6 noro 307: for ( I = 0; I < N; I++ )
308: D[I] = [deg(F,V[I]),V[I]];
309: qsort(D,compare_first);
310: for ( V = [], I = 0; I < N; I++ )
311: V = cons(D[I][1],V);
1.1 noro 312: for ( I = N-1, DV = []; I >= 0; I-- )
313: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.6 noro 314: V1 = cons(s,V); DV1 = cons(ds,DV);
1.7 noro 315:
316: G0 = indicial1(F,reverse(V));
317: G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
318: Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
1.6 noro 319: return Minipoly;
320: }
321:
1.14 noro 322: /* b-function computation via generic_bfct() (experimental) */
323:
324: def bfct_via_gbfct(F)
325: {
326: V = vars(F);
327: N = length(V);
328: D = newvect(N);
329:
330: for ( I = 0; I < N; I++ )
331: D[I] = [deg(F,V[I]),V[I]];
332: qsort(D,compare_first);
333: for ( V = [], I = 0; I < N; I++ )
334: V = cons(D[I][1],V);
335: V = reverse(V);
336: for ( I = N-1, DV = []; I >= 0; I-- )
337: DV = cons(strtov("d"+rtostr(V[I])),DV);
338:
339: B = [t-F];
340: for ( I = 0; I < N; I++ ) {
341: B = cons(DV[I]+diff(F,V[I])*dt,B);
342: }
343: V1 = cons(t,V); DV1 = cons(dt,DV);
344: W = newvect(N+1);
345: W[0] = 1;
346: R = generic_bfct(B,V1,DV1,W);
347:
348: return subst(R,s,-s-1);
349: }
350:
1.6 noro 351: def weyl_minipolym(G,V,O,M,V0)
352: {
353: N = length(V);
354: Len = length(G);
355: dp_ord(O);
356: setmod(M);
357: PS = newvect(Len);
358: PS0 = newvect(Len);
359:
360: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
361: PS0[I] = dp_ptod(car(T),V);
362: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
363: PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
364:
365: for ( I = Len - 1, GI = []; I >= 0; I-- )
366: GI = cons(I,GI);
367:
368: U = dp_mod(dp_ptod(V0,V),M,[]);
369:
370: T = dp_mod(<<0>>,M,[]);
371: TT = dp_mod(dp_ptod(1,V),M,[]);
372: G = H = [[TT,T]];
373:
374: for ( I = 1; ; I++ ) {
1.14 noro 375: if ( dp_gr_print() )
376: print(".",2);
1.6 noro 377: T = dp_mod(<<I>>,M,[]);
378:
379: TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
380: H = cons([TT,T],H);
381: L = dp_lnf_mod([TT,T],G,M);
1.14 noro 382: if ( !L[0] ) {
383: if ( dp_gr_print() )
384: print("");
1.13 noro 385: return dp_dtop(L[1],[t]); /* XXX */
1.14 noro 386: } else
1.6 noro 387: G = insert(G,L);
388: }
389: }
390:
1.13 noro 391: def weyl_minipoly(G0,V0,O0,P)
1.6 noro 392: {
1.11 noro 393: HM = hmlist(G0,V0,O0);
1.13 noro 394:
395: N = length(V0);
396: Len = length(G0);
397: dp_ord(O0);
398: PS = newvect(Len);
399: for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
400: PS[I] = dp_ptod(car(T),V0);
401: for ( I = Len - 1, GI = []; I >= 0; I-- )
402: GI = cons(I,GI);
403: DP = dp_ptod(P,V0);
404:
1.6 noro 405: for ( I = 0; ; I++ ) {
406: Prime = lprime(I);
1.11 noro 407: if ( !valid_modulus(HM,Prime) )
408: continue;
1.13 noro 409: MP = weyl_minipolym(G0,V0,O0,Prime,P);
410: D = deg(MP,var(MP));
411:
412: NFP = weyl_nf(GI,DP,1,PS);
413: NF = [[dp_ptod(1,V0),1]];
414: LCM = 1;
415:
416: for ( J = 1; J <= D; J++ ) {
1.14 noro 417: if ( dp_gr_print() )
418: print(".",2);
1.13 noro 419: NFPrev = car(NF);
420: NFJ = weyl_nf(GI,
421: dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
422: NFJ = remove_cont(NFJ);
423: NF = cons(NFJ,NF);
424: LCM = ilcm(LCM,NFJ[1]);
425: }
1.14 noro 426: if ( dp_gr_print() )
427: print("");
1.13 noro 428: U = NF[0][0]*idiv(LCM,NF[0][1]);
429: Coef = [];
430: for ( J = D-1; J >= 0; J-- ) {
431: Coef = cons(strtov("u"+rtostr(J)),Coef);
432: U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
433: }
1.6 noro 434:
1.13 noro 435: for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
436: Eq = cons(dp_hc(UU),Eq);
437: M = etom([Eq,Coef]);
438: B = henleq(M,Prime);
439: if ( dp_gr_print() )
440: print("");
1.6 noro 441: if ( B ) {
1.13 noro 442: R = 0;
443: for ( I = 0; I < D; I++ )
444: R += B[0][I]*s^I;
445: R += B[1]*s^D;
1.6 noro 446: return R;
447: }
448: }
449: }
450:
451: def weyl_nf(B,G,M,PS)
452: {
453: for ( D = 0; G; ) {
454: for ( U = 0, L = B; L != []; L = cdr(L) ) {
455: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
456: GCD = igcd(dp_hc(G),dp_hc(R));
457: CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
458: U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
459: if ( !U )
460: return [D,M];
461: D *= CG; M *= CG;
462: break;
463: }
464: }
465: if ( U )
466: G = U;
467: else {
468: D += dp_hm(G); G = dp_rest(G);
469: }
470: }
471: return [D,M];
472: }
473:
474: def weyl_nf_mod(B,G,PS,Mod)
475: {
476: for ( D = 0; G; ) {
477: for ( U = 0, L = B; L != []; L = cdr(L) ) {
478: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
479: CR = dp_hc(G)/dp_hc(R);
480: U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
481: if ( !U )
482: return D;
1.1 noro 483: break;
1.6 noro 484: }
485: }
486: if ( U )
487: G = U;
488: else {
489: D += dp_hm(G); G = dp_rest(G);
1.1 noro 490: }
491: }
1.6 noro 492: return D;
1.1 noro 493: }
494:
495: def remove_zero(L)
496: {
497: for ( R = []; L != []; L = cdr(L) )
498: if ( car(L) )
499: R = cons(car(L),R);
500: return R;
501: }
502:
503: def z_subst(F,V)
504: {
505: for ( ; V != []; V = cdr(V) )
506: F = subst(F,car(V),0);
507: return F;
508: }
509:
510: def flatmf(L) {
511: for ( S = []; L != []; L = cdr(L) )
512: if ( type(F=car(car(L))) != NUM )
513: S = append(S,[F]);
514: return S;
515: }
516:
517: def member(A,L) {
518: for ( ; L != []; L = cdr(L) )
519: if ( A == car(L) )
520: return 1;
521: return 0;
522: }
523:
524: def intersection(A,B)
525: {
526: for ( L = []; A != []; A = cdr(A) )
527: if ( member(car(A),B) )
528: L = cons(car(A),L);
529: return L;
530: }
531:
532: def b_subst(F,V)
533: {
534: D = deg(F,V);
535: C = newvect(D+1);
536: for ( I = D; I >= 0; I-- )
537: C[I] = coef(F,I,V);
538: for ( I = 0, R = 0; I <= D; I++ )
539: if ( C[I] )
540: R += C[I]*v_factorial(V,I);
541: return R;
542: }
543:
544: def v_factorial(V,N)
545: {
546: for ( J = N-1, R = 1; J >= 0; J-- )
547: R *= V-J;
548: return R;
549: }
550: end$
551:
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