Annotation of OpenXM_contrib2/asir2000/lib/bfct, Revision 1.17
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,
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8: * computer program, "Risa/Asir" ("SOFTWARE"), subject to the terms and
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12: * copyrights, in and to the SOFTWARE.
13: *
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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
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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.
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30: * THE SOFTWARE IS PROVIDED AS IS WITHOUT ANY WARRANTY OF ANY KIND. FLL
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45: * DEVELOPER SHALL HAVE NO LIABILITY IN CONNECTION WITH THE USE,
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47: *
1.17 ! noro 48: * $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.16 2001/01/18 00:52:32 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:
1.16 noro 215: /* If W is a list, convert it to a vector */
216: if ( type(W) == 4 )
217: W = newvect(length(W),W);
1.15 noro 218: dp_weyl_set_weight(W);
219:
1.14 noro 220: /* create a term order M in D<x,d> (DRL) */
1.13 noro 221: M = newmat(N2,N2);
222: for ( J = 0; J < N2; J++ )
223: M[0][J] = 1;
224: for ( I = 1; I < N2; I++ )
225: M[I][N2-I] = -1;
226:
227: VDV = append(V,DV);
228:
229: /* create a non-term order MW in D<x,d> */
230: MW = newmat(N2+1,N2);
231: for ( J = 0; J < N; J++ )
232: MW[0][J] = -W[J];
233: for ( ; J < N2; J++ )
234: MW[0][J] = W[J-N];
235: for ( I = 1; I <= N2; I++ )
236: for ( J = 0; J < N2; J++ )
237: MW[I][J] = M[I-1][J];
238:
239: /* create a homogenized term order MWH in D<x,d,h> */
240: MWH = newmat(N2+2,N2+1);
241: for ( J = 0; J <= N2; J++ )
242: MWH[0][J] = 1;
243: for ( I = 1; I <= N2+1; I++ )
244: for ( J = 0; J < N2; J++ )
245: MWH[I][J] = MW[I-1][J];
246:
247: /* homogenize F */
248: VDVH = append(VDV,[h]);
249: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
250:
251: /* compute a groebner basis of FH w.r.t. MWH */
1.15 noro 252: dp_gr_flags(["Top",1,"NoRA",1]);
253: GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
254: dp_gr_flags(["Top",0,"NoRA",0]);
1.13 noro 255:
256: /* dehomigenize GH */
257: G = map(subst,GH,h,1);
258:
259: /* G is a groebner basis w.r.t. a non term order MW */
260: /* take the initial part w.r.t. (-W,W) */
261: GIN = map(initial_part,G,VDV,MW,W);
262:
263: /* GIN is a groebner basis w.r.t. a term order M */
264: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
265:
266: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
267: for ( I = 0, T = 0; I < N; I++ )
268: T += W[I]*V[I]*DV[I];
1.14 noro 269: B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
1.13 noro 270: return B;
271: }
272:
273: def initial_part(F,V,MW,W)
274: {
275: N2 = length(V);
276: N = N2/2;
277: dp_ord(MW);
278: DF = dp_ptod(F,V);
279: R = dp_hm(DF);
280: DF = dp_rest(DF);
281:
282: E = dp_etov(R);
283: for ( I = 0, TW = 0; I < N; I++ )
284: TW += W[I]*(-E[I]+E[N+I]);
285: RW = TW;
286:
287: for ( ; DF; DF = dp_rest(DF) ) {
288: E = dp_etov(DF);
289: for ( I = 0, TW = 0; I < N; I++ )
290: TW += W[I]*(-E[I]+E[N+I]);
291: if ( TW == RW )
292: R += dp_hm(DF);
293: else if ( TW < RW )
294: break;
295: else
296: error("initial_part : cannot happen");
297: }
298: return dp_dtop(R,V);
299:
300: }
301:
1.1 noro 302: /* b-function of F ? */
303:
304: def bfct(F)
305: {
306: V = vars(F);
307: N = length(V);
1.6 noro 308: D = newvect(N);
1.7 noro 309:
1.6 noro 310: for ( I = 0; I < N; I++ )
311: D[I] = [deg(F,V[I]),V[I]];
312: qsort(D,compare_first);
313: for ( V = [], I = 0; I < N; I++ )
314: V = cons(D[I][1],V);
1.1 noro 315: for ( I = N-1, DV = []; I >= 0; I-- )
316: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.6 noro 317: V1 = cons(s,V); DV1 = cons(ds,DV);
1.7 noro 318:
319: G0 = indicial1(F,reverse(V));
320: G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
321: Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
1.6 noro 322: return Minipoly;
323: }
324:
1.14 noro 325: /* b-function computation via generic_bfct() (experimental) */
326:
327: def bfct_via_gbfct(F)
328: {
329: V = vars(F);
330: N = length(V);
331: D = newvect(N);
332:
333: for ( I = 0; I < N; I++ )
334: D[I] = [deg(F,V[I]),V[I]];
335: qsort(D,compare_first);
336: for ( V = [], I = 0; I < N; I++ )
337: V = cons(D[I][1],V);
338: V = reverse(V);
339: for ( I = N-1, DV = []; I >= 0; I-- )
340: DV = cons(strtov("d"+rtostr(V[I])),DV);
341:
342: B = [t-F];
343: for ( I = 0; I < N; I++ ) {
344: B = cons(DV[I]+diff(F,V[I])*dt,B);
345: }
346: V1 = cons(t,V); DV1 = cons(dt,DV);
347: W = newvect(N+1);
348: W[0] = 1;
349: R = generic_bfct(B,V1,DV1,W);
350:
351: return subst(R,s,-s-1);
352: }
353:
1.17 ! noro 354: /* use an order s.t. [t,x,y,z,...,dt,dx,dy,dz,...,h] */
! 355:
! 356: def bfct_via_gbfct_weight(F,V)
! 357: {
! 358: N = length(V);
! 359: D = newvect(N);
! 360: Wt = getopt(weight);
! 361: if ( type(Wt) == 4 ) {
! 362: Tdeg = w_tdeg(F,V,Wt);
! 363: WtV = newvect(2*(N+1)+1);
! 364: WtV[0] = Tdeg;
! 365: WtV[N+1] = 1;
! 366: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
! 367: for ( I = 1; I <= N; I++ ) {
! 368: WtV[I] = Wt[I-1];
! 369: WtV[N+1+I] = Tdeg-Wt[I-1]+1;
! 370: }
! 371: WtV[2*(N+1)] = 1;
! 372: dp_set_weight(WtV);
! 373: }
! 374: for ( I = N-1, DV = []; I >= 0; I-- )
! 375: DV = cons(strtov("d"+rtostr(V[I])),DV);
! 376:
! 377: B = [t-F];
! 378: for ( I = 0; I < N; I++ ) {
! 379: B = cons(DV[I]+diff(F,V[I])*dt,B);
! 380: }
! 381: V1 = cons(t,V); DV1 = cons(dt,DV);
! 382: W = newvect(N+1);
! 383: W[0] = 1;
! 384: R = generic_bfct(B,V1,DV1,W);
! 385:
! 386: return subst(R,s,-s-1);
! 387: }
! 388:
! 389: /* use an order s.t. [x,y,z,...,t,dx,dy,dz,...,dt,h] */
! 390:
! 391: def bfct_via_gbfct_weight_1(F,V)
! 392: {
! 393: N = length(V);
! 394: D = newvect(N);
! 395: Wt = getopt(weight);
! 396: if ( type(Wt) == 4 ) {
! 397: Tdeg = w_tdeg(F,V,Wt);
! 398: WtV = newvect(2*(N+1));
! 399: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
! 400: for ( I = 0; I < N; I++ ) {
! 401: WtV[I] = Wt[I];
! 402: WtV[N+1+I] = Tdeg-Wt[I]+1;
! 403: }
! 404: WtV[N] = Tdeg;
! 405: WtV[2*N+1] = 1;
! 406: dp_set_weight(WtV);
! 407: }
! 408: for ( I = N-1, DV = []; I >= 0; I-- )
! 409: DV = cons(strtov("d"+rtostr(V[I])),DV);
! 410:
! 411: B = [t-F];
! 412: for ( I = 0; I < N; I++ ) {
! 413: B = cons(DV[I]+diff(F,V[I])*dt,B);
! 414: }
! 415: V1 = append(V,[t]); DV1 = append(DV,[dt]);
! 416: W = newvect(N+1);
! 417: W[N] = 1;
! 418: R = generic_bfct(B,V1,DV1,W);
! 419:
! 420: return subst(R,s,-s-1);
! 421: }
! 422:
1.6 noro 423: def weyl_minipolym(G,V,O,M,V0)
424: {
425: N = length(V);
426: Len = length(G);
427: dp_ord(O);
428: setmod(M);
429: PS = newvect(Len);
430: PS0 = newvect(Len);
431:
432: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
433: PS0[I] = dp_ptod(car(T),V);
434: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
435: PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
436:
437: for ( I = Len - 1, GI = []; I >= 0; I-- )
438: GI = cons(I,GI);
439:
440: U = dp_mod(dp_ptod(V0,V),M,[]);
1.17 ! noro 441: U = dp_weyl_nf_mod(GI,U,PS,1,M);
1.6 noro 442:
443: T = dp_mod(<<0>>,M,[]);
444: TT = dp_mod(dp_ptod(1,V),M,[]);
445: G = H = [[TT,T]];
446:
447: for ( I = 1; ; I++ ) {
1.14 noro 448: if ( dp_gr_print() )
449: print(".",2);
1.6 noro 450: T = dp_mod(<<I>>,M,[]);
451:
452: TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
453: H = cons([TT,T],H);
454: L = dp_lnf_mod([TT,T],G,M);
1.14 noro 455: if ( !L[0] ) {
456: if ( dp_gr_print() )
457: print("");
1.13 noro 458: return dp_dtop(L[1],[t]); /* XXX */
1.14 noro 459: } else
1.6 noro 460: G = insert(G,L);
461: }
462: }
463:
1.13 noro 464: def weyl_minipoly(G0,V0,O0,P)
1.6 noro 465: {
1.11 noro 466: HM = hmlist(G0,V0,O0);
1.13 noro 467:
468: N = length(V0);
469: Len = length(G0);
470: dp_ord(O0);
471: PS = newvect(Len);
472: for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
473: PS[I] = dp_ptod(car(T),V0);
474: for ( I = Len - 1, GI = []; I >= 0; I-- )
475: GI = cons(I,GI);
476: DP = dp_ptod(P,V0);
477:
1.6 noro 478: for ( I = 0; ; I++ ) {
479: Prime = lprime(I);
1.11 noro 480: if ( !valid_modulus(HM,Prime) )
481: continue;
1.13 noro 482: MP = weyl_minipolym(G0,V0,O0,Prime,P);
483: D = deg(MP,var(MP));
484:
485: NFP = weyl_nf(GI,DP,1,PS);
486: NF = [[dp_ptod(1,V0),1]];
487: LCM = 1;
488:
489: for ( J = 1; J <= D; J++ ) {
1.14 noro 490: if ( dp_gr_print() )
491: print(".",2);
1.13 noro 492: NFPrev = car(NF);
493: NFJ = weyl_nf(GI,
494: dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
495: NFJ = remove_cont(NFJ);
496: NF = cons(NFJ,NF);
497: LCM = ilcm(LCM,NFJ[1]);
498: }
1.14 noro 499: if ( dp_gr_print() )
500: print("");
1.13 noro 501: U = NF[0][0]*idiv(LCM,NF[0][1]);
502: Coef = [];
503: for ( J = D-1; J >= 0; J-- ) {
504: Coef = cons(strtov("u"+rtostr(J)),Coef);
505: U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
506: }
1.6 noro 507:
1.13 noro 508: for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
509: Eq = cons(dp_hc(UU),Eq);
510: M = etom([Eq,Coef]);
511: B = henleq(M,Prime);
512: if ( dp_gr_print() )
513: print("");
1.6 noro 514: if ( B ) {
1.13 noro 515: R = 0;
516: for ( I = 0; I < D; I++ )
517: R += B[0][I]*s^I;
518: R += B[1]*s^D;
1.6 noro 519: return R;
520: }
521: }
522: }
523:
524: def weyl_nf(B,G,M,PS)
525: {
526: for ( D = 0; G; ) {
527: for ( U = 0, L = B; L != []; L = cdr(L) ) {
528: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
529: GCD = igcd(dp_hc(G),dp_hc(R));
530: CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
531: U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
532: if ( !U )
533: return [D,M];
534: D *= CG; M *= CG;
535: break;
536: }
537: }
538: if ( U )
539: G = U;
540: else {
541: D += dp_hm(G); G = dp_rest(G);
542: }
543: }
544: return [D,M];
545: }
546:
547: def weyl_nf_mod(B,G,PS,Mod)
548: {
549: for ( D = 0; G; ) {
550: for ( U = 0, L = B; L != []; L = cdr(L) ) {
551: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
552: CR = dp_hc(G)/dp_hc(R);
553: U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
554: if ( !U )
555: return D;
1.1 noro 556: break;
1.6 noro 557: }
558: }
559: if ( U )
560: G = U;
561: else {
562: D += dp_hm(G); G = dp_rest(G);
1.1 noro 563: }
564: }
1.6 noro 565: return D;
1.1 noro 566: }
567:
568: def remove_zero(L)
569: {
570: for ( R = []; L != []; L = cdr(L) )
571: if ( car(L) )
572: R = cons(car(L),R);
573: return R;
574: }
575:
576: def z_subst(F,V)
577: {
578: for ( ; V != []; V = cdr(V) )
579: F = subst(F,car(V),0);
580: return F;
581: }
582:
583: def flatmf(L) {
584: for ( S = []; L != []; L = cdr(L) )
585: if ( type(F=car(car(L))) != NUM )
586: S = append(S,[F]);
587: return S;
588: }
589:
590: def member(A,L) {
591: for ( ; L != []; L = cdr(L) )
592: if ( A == car(L) )
593: return 1;
594: return 0;
595: }
596:
597: def intersection(A,B)
598: {
599: for ( L = []; A != []; A = cdr(A) )
600: if ( member(car(A),B) )
601: L = cons(car(A),L);
602: return L;
603: }
604:
605: def b_subst(F,V)
606: {
607: D = deg(F,V);
608: C = newvect(D+1);
609: for ( I = D; I >= 0; I-- )
610: C[I] = coef(F,I,V);
611: for ( I = 0, R = 0; I <= D; I++ )
612: if ( C[I] )
613: R += C[I]*v_factorial(V,I);
614: return R;
615: }
616:
617: def v_factorial(V,N)
618: {
619: for ( J = N-1, R = 1; J >= 0; J-- )
620: R *= V-J;
1.17 ! noro 621: return R;
! 622: }
! 623:
! 624: def w_tdeg(F,V,W)
! 625: {
! 626: dp_set_weight(newvect(length(W),W));
! 627: T = dp_ptod(F,V);
! 628: for ( R = 0; T; T = cdr(T) ) {
! 629: D = dp_td(T);
! 630: if ( D > R ) R = D;
! 631: }
1.1 noro 632: return R;
633: }
634: end$
635:
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