Annotation of OpenXM_contrib2/asir2000/lib/bfct, Revision 1.23
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'
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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
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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: *
1.23 ! noro 48: * $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.22 2003/04/20 08:54:28 noro Exp $
1.10 noro 49: */
1.1 noro 50: /* requires 'primdec' */
1.22 noro 51:
1.23 ! noro 52: #define TMP_S ssssssss
! 53: #define TMP_T tttttttt
! 54: #define TMP_Y1 yyyyyyyy1
! 55: #define TMP_Y2 yyyyyyyy2
! 56:
1.22 noro 57: extern LIBRARY_GR_LOADED$
58: extern LIBRARY_PRIMDEC_LOADED$
59:
60: if(!LIBRARY_GR_LOADED) load("gr"); else ; LIBRARY_GR_LOADED = 1$
61: if(!LIBRARY_PRIMDEC_LOADED) load("primdec"); else ; LIBRARY_PRIMDEC_LOADED = 1$
62:
63: /* toplevel */
64:
65: def bfunction(F)
66: {
1.23 ! noro 67: /* XXX */
! 68: F = replace_vars_f(F);
! 69:
1.22 noro 70: V = vars(F);
71: N = length(V);
72: D = newvect(N);
73:
74: for ( I = 0; I < N; I++ )
75: D[I] = [deg(F,V[I]),V[I]];
76: qsort(D,compare_first);
77: for ( V = [], I = 0; I < N; I++ )
78: V = cons(D[I][1],V);
79: return bfct_via_gbfct_weight(F,V);
80: }
1.1 noro 81:
1.6 noro 82: /* annihilating ideal of F^s */
1.1 noro 83:
84: def ann(F)
85: {
1.23 ! noro 86: /* XXX */
! 87: F = replace_vars_f(F);
! 88:
1.1 noro 89: V = vars(F);
90: N = length(V);
1.8 noro 91: D = newvect(N);
92:
93: for ( I = 0; I < N; I++ )
94: D[I] = [deg(F,V[I]),V[I]];
95: qsort(D,compare_first);
96: for ( V = [], I = N-1; I >= 0; I-- )
97: V = cons(D[I][1],V);
98:
1.1 noro 99: for ( I = N-1, DV = []; I >= 0; I-- )
100: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.8 noro 101:
102: W = append([y1,y2,t],V);
1.1 noro 103: DW = append([dy1,dy2,dt],DV);
1.8 noro 104:
105: B = [1-y1*y2,t-y1*F];
1.1 noro 106: for ( I = 0; I < N; I++ ) {
107: B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
108: }
1.10 noro 109:
110: /* homogenized (heuristics) */
1.1 noro 111: dp_nelim(2);
1.10 noro 112: G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
1.1 noro 113: G1 = [];
114: for ( T = G0; T != []; T = cdr(T) ) {
115: E = car(T); VL = vars(E);
116: if ( !member(y1,VL) && !member(y2,VL) )
117: G1 = cons(E,G1);
118: }
1.12 noro 119: G2 = map(psi,G1,t,dt);
120: G3 = map(subst,G2,t,-1-s);
121: return G3;
1.1 noro 122: }
123:
1.10 noro 124: /*
125: * compute J_f|s=r, where r = the minimal integral root of global b_f(s)
126: * ann0(F) returns [MinRoot,Ideal]
127: */
128:
129: def ann0(F)
130: {
1.23 ! noro 131: /* XXX */
! 132: F = replace_vars_f(F);
! 133:
1.10 noro 134: V = vars(F);
135: N = length(V);
136: D = newvect(N);
137:
138: for ( I = 0; I < N; I++ )
139: D[I] = [deg(F,V[I]),V[I]];
140: qsort(D,compare_first);
141: for ( V = [], I = 0; I < N; I++ )
142: V = cons(D[I][1],V);
143:
144: for ( I = N-1, DV = []; I >= 0; I-- )
145: DV = cons(strtov("d"+rtostr(V[I])),DV);
146:
147: /* XXX : heuristics */
148: W = append([y1,y2,t],reverse(V));
149: DW = append([dy1,dy2,dt],reverse(DV));
150: WDW = append(W,DW);
151:
152: B = [1-y1*y2,t-y1*F];
153: for ( I = 0; I < N; I++ ) {
154: B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
155: }
156:
157: /* homogenized (heuristics) */
158: dp_nelim(2);
159: G0 = dp_weyl_gr_main(B,WDW,1,0,6);
160: G1 = [];
161: for ( T = G0; T != []; T = cdr(T) ) {
162: E = car(T); VL = vars(E);
163: if ( !member(y1,VL) && !member(y2,VL) )
164: G1 = cons(E,G1);
165: }
1.12 noro 166: G2 = map(psi,G1,t,dt);
167: G3 = map(subst,G2,t,-1-s);
1.10 noro 168:
1.12 noro 169: /* G3 = J_f(s) */
1.10 noro 170:
171: V1 = cons(s,V); DV1 = cons(ds,DV); V1DV1 = append(V1,DV1);
1.12 noro 172: G4 = dp_weyl_gr_main(cons(F,G3),V1DV1,0,1,0);
173: Bf = weyl_minipoly(G4,V1DV1,0,s);
1.10 noro 174:
175: FList = cdr(fctr(Bf));
176: for ( T = FList, Min = 0; T != []; T = cdr(T) ) {
177: LF = car(car(T));
178: Root = -coef(LF,0)/coef(LF,1);
179: if ( dn(Root) == 1 && Root < Min )
180: Min = Root;
181: }
1.12 noro 182: return [Min,map(subst,G3,s,Min)];
1.10 noro 183: }
184:
1.7 noro 185: def indicial1(F,V)
1.6 noro 186: {
187: W = append([y1,t],V);
188: N = length(V);
189: B = [t-y1*F];
190: for ( I = N-1, DV = []; I >= 0; I-- )
191: DV = cons(strtov("d"+rtostr(V[I])),DV);
192: DW = append([dy1,dt],DV);
193: for ( I = 0; I < N; I++ ) {
194: B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
195: }
196: dp_nelim(1);
1.10 noro 197:
198: /* homogenized (heuristics) */
1.7 noro 199: G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
1.6 noro 200: G1 = map(subst,G0,y1,1);
201: G2 = map(psi,G1,t,dt);
202: G3 = map(subst,G2,t,-s-1);
203: return G3;
204: }
205:
206: def psi(F,T,DT)
207: {
208: D = dp_ptod(F,[T,DT]);
209: Wmax = weight(D);
210: D1 = dp_rest(D);
211: for ( ; D1; D1 = dp_rest(D1) )
212: if ( weight(D1) > Wmax )
213: Wmax = weight(D1);
214: for ( D1 = D, Dmax = 0; D1; D1 = dp_rest(D1) )
215: if ( weight(D1) == Wmax )
216: Dmax += dp_hm(D1);
217: if ( Wmax >= 0 )
218: Dmax = dp_weyl_mul(<<Wmax,0>>,Dmax);
219: else
220: Dmax = dp_weyl_mul(<<0,-Wmax>>,Dmax);
221: Rmax = dp_dtop(Dmax,[T,DT]);
222: R = b_subst(subst(Rmax,DT,1),T);
223: return R;
224: }
225:
226: def weight(D)
227: {
228: V = dp_etov(D);
229: return V[1]-V[0];
230: }
231:
232: def compare_first(A,B)
233: {
234: A0 = car(A);
235: B0 = car(B);
236: if ( A0 > B0 )
237: return 1;
238: else if ( A0 < B0 )
239: return -1;
240: else
241: return 0;
242: }
243:
1.13 noro 244: /* generic b-function w.r.t. weight vector W */
245:
246: def generic_bfct(F,V,DV,W)
247: {
248: N = length(V);
249: N2 = N*2;
250:
1.16 noro 251: /* If W is a list, convert it to a vector */
252: if ( type(W) == 4 )
253: W = newvect(length(W),W);
1.15 noro 254: dp_weyl_set_weight(W);
255:
1.14 noro 256: /* create a term order M in D<x,d> (DRL) */
1.13 noro 257: M = newmat(N2,N2);
258: for ( J = 0; J < N2; J++ )
259: M[0][J] = 1;
260: for ( I = 1; I < N2; I++ )
261: M[I][N2-I] = -1;
262:
263: VDV = append(V,DV);
264:
265: /* create a non-term order MW in D<x,d> */
266: MW = newmat(N2+1,N2);
267: for ( J = 0; J < N; J++ )
268: MW[0][J] = -W[J];
269: for ( ; J < N2; J++ )
270: MW[0][J] = W[J-N];
271: for ( I = 1; I <= N2; I++ )
272: for ( J = 0; J < N2; J++ )
273: MW[I][J] = M[I-1][J];
274:
275: /* create a homogenized term order MWH in D<x,d,h> */
276: MWH = newmat(N2+2,N2+1);
277: for ( J = 0; J <= N2; J++ )
278: MWH[0][J] = 1;
279: for ( I = 1; I <= N2+1; I++ )
280: for ( J = 0; J < N2; J++ )
281: MWH[I][J] = MW[I-1][J];
282:
283: /* homogenize F */
284: VDVH = append(VDV,[h]);
285: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
286:
287: /* compute a groebner basis of FH w.r.t. MWH */
1.21 noro 288: dp_gr_flags(["Top",1,"NoRA",1]);
1.15 noro 289: GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
1.21 noro 290: dp_gr_flags(["Top",0,"NoRA",0]);
1.13 noro 291:
292: /* dehomigenize GH */
293: G = map(subst,GH,h,1);
294:
295: /* G is a groebner basis w.r.t. a non term order MW */
296: /* take the initial part w.r.t. (-W,W) */
297: GIN = map(initial_part,G,VDV,MW,W);
298:
299: /* GIN is a groebner basis w.r.t. a term order M */
300: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
301:
302: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
303: for ( I = 0, T = 0; I < N; I++ )
304: T += W[I]*V[I]*DV[I];
1.14 noro 305: B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
1.13 noro 306: return B;
307: }
308:
1.18 noro 309: /* all term reduction + interreduce */
310: def generic_bfct_1(F,V,DV,W)
311: {
312: N = length(V);
313: N2 = N*2;
314:
315: /* If W is a list, convert it to a vector */
316: if ( type(W) == 4 )
317: W = newvect(length(W),W);
318: dp_weyl_set_weight(W);
319:
320: /* create a term order M in D<x,d> (DRL) */
321: M = newmat(N2,N2);
322: for ( J = 0; J < N2; J++ )
323: M[0][J] = 1;
324: for ( I = 1; I < N2; I++ )
325: M[I][N2-I] = -1;
326:
327: VDV = append(V,DV);
328:
329: /* create a non-term order MW in D<x,d> */
330: MW = newmat(N2+1,N2);
331: for ( J = 0; J < N; J++ )
332: MW[0][J] = -W[J];
333: for ( ; J < N2; J++ )
334: MW[0][J] = W[J-N];
335: for ( I = 1; I <= N2; I++ )
336: for ( J = 0; J < N2; J++ )
337: MW[I][J] = M[I-1][J];
338:
339: /* create a homogenized term order MWH in D<x,d,h> */
340: MWH = newmat(N2+2,N2+1);
341: for ( J = 0; J <= N2; J++ )
342: MWH[0][J] = 1;
343: for ( I = 1; I <= N2+1; I++ )
344: for ( J = 0; J < N2; J++ )
345: MWH[I][J] = MW[I-1][J];
346:
347: /* homogenize F */
348: VDVH = append(VDV,[h]);
349: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
350:
351: /* compute a groebner basis of FH w.r.t. MWH */
352: /* dp_gr_flags(["Top",1,"NoRA",1]); */
353: GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
354: /* dp_gr_flags(["Top",0,"NoRA",0]); */
355:
356: /* dehomigenize GH */
357: G = map(subst,GH,h,1);
358:
359: /* G is a groebner basis w.r.t. a non term order MW */
360: /* take the initial part w.r.t. (-W,W) */
361: GIN = map(initial_part,G,VDV,MW,W);
362:
363: /* GIN is a groebner basis w.r.t. a term order M */
364: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
365:
366: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
367: for ( I = 0, T = 0; I < N; I++ )
368: T += W[I]*V[I]*DV[I];
369: B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
370: return B;
371: }
372:
1.13 noro 373: def initial_part(F,V,MW,W)
374: {
375: N2 = length(V);
376: N = N2/2;
377: dp_ord(MW);
378: DF = dp_ptod(F,V);
379: R = dp_hm(DF);
380: DF = dp_rest(DF);
381:
382: E = dp_etov(R);
383: for ( I = 0, TW = 0; I < N; I++ )
384: TW += W[I]*(-E[I]+E[N+I]);
385: RW = TW;
386:
387: for ( ; DF; DF = dp_rest(DF) ) {
388: E = dp_etov(DF);
389: for ( I = 0, TW = 0; I < N; I++ )
390: TW += W[I]*(-E[I]+E[N+I]);
391: if ( TW == RW )
392: R += dp_hm(DF);
393: else if ( TW < RW )
394: break;
395: else
396: error("initial_part : cannot happen");
397: }
398: return dp_dtop(R,V);
399:
400: }
401:
1.1 noro 402: /* b-function of F ? */
403:
404: def bfct(F)
405: {
1.23 ! noro 406: /* XXX */
! 407: F = replace_vars_f(F);
! 408:
1.1 noro 409: V = vars(F);
410: N = length(V);
1.6 noro 411: D = newvect(N);
1.7 noro 412:
1.6 noro 413: for ( I = 0; I < N; I++ )
414: D[I] = [deg(F,V[I]),V[I]];
415: qsort(D,compare_first);
416: for ( V = [], I = 0; I < N; I++ )
417: V = cons(D[I][1],V);
1.1 noro 418: for ( I = N-1, DV = []; I >= 0; I-- )
419: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.6 noro 420: V1 = cons(s,V); DV1 = cons(ds,DV);
1.7 noro 421:
422: G0 = indicial1(F,reverse(V));
423: G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
424: Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
1.6 noro 425: return Minipoly;
426: }
427:
1.14 noro 428: /* b-function computation via generic_bfct() (experimental) */
429:
430: def bfct_via_gbfct(F)
431: {
1.23 ! noro 432: /* XXX */
! 433: F = replace_vars_f(F);
! 434:
1.14 noro 435: V = vars(F);
436: N = length(V);
437: D = newvect(N);
438:
439: for ( I = 0; I < N; I++ )
440: D[I] = [deg(F,V[I]),V[I]];
441: qsort(D,compare_first);
442: for ( V = [], I = 0; I < N; I++ )
443: V = cons(D[I][1],V);
444: V = reverse(V);
445: for ( I = N-1, DV = []; I >= 0; I-- )
446: DV = cons(strtov("d"+rtostr(V[I])),DV);
447:
448: B = [t-F];
449: for ( I = 0; I < N; I++ ) {
450: B = cons(DV[I]+diff(F,V[I])*dt,B);
451: }
452: V1 = cons(t,V); DV1 = cons(dt,DV);
453: W = newvect(N+1);
454: W[0] = 1;
1.21 noro 455: R = generic_bfct(B,V1,DV1,W);
1.14 noro 456:
457: return subst(R,s,-s-1);
458: }
459:
1.17 noro 460: /* use an order s.t. [t,x,y,z,...,dt,dx,dy,dz,...,h] */
461:
462: def bfct_via_gbfct_weight(F,V)
463: {
1.23 ! noro 464: /* XXX */
! 465: F = replace_vars_f(F);
! 466: V = replace_vars_v(V);
! 467:
1.17 noro 468: N = length(V);
469: D = newvect(N);
470: Wt = getopt(weight);
1.18 noro 471: if ( type(Wt) != 4 ) {
472: for ( I = 0, Wt = []; I < N; I++ )
473: Wt = cons(1,Wt);
474: }
475: Tdeg = w_tdeg(F,V,Wt);
476: WtV = newvect(2*(N+1)+1);
477: WtV[0] = Tdeg;
478: WtV[N+1] = 1;
479: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
480: for ( I = 1; I <= N; I++ ) {
481: WtV[I] = Wt[I-1];
482: WtV[N+1+I] = Tdeg-Wt[I-1]+1;
1.17 noro 483: }
1.18 noro 484: WtV[2*(N+1)] = 1;
485: dp_set_weight(WtV);
1.17 noro 486: for ( I = N-1, DV = []; I >= 0; I-- )
487: DV = cons(strtov("d"+rtostr(V[I])),DV);
488:
489: B = [t-F];
490: for ( I = 0; I < N; I++ ) {
491: B = cons(DV[I]+diff(F,V[I])*dt,B);
492: }
493: V1 = cons(t,V); DV1 = cons(dt,DV);
494: W = newvect(N+1);
495: W[0] = 1;
1.18 noro 496: R = generic_bfct_1(B,V1,DV1,W);
497: dp_set_weight(0);
1.17 noro 498: return subst(R,s,-s-1);
499: }
500:
501: /* use an order s.t. [x,y,z,...,t,dx,dy,dz,...,dt,h] */
502:
503: def bfct_via_gbfct_weight_1(F,V)
504: {
1.23 ! noro 505: /* XXX */
! 506: F = replace_vars_f(F);
! 507: V = replace_vars_v(V);
! 508:
1.17 noro 509: N = length(V);
510: D = newvect(N);
511: Wt = getopt(weight);
1.18 noro 512: if ( type(Wt) != 4 ) {
513: for ( I = 0, Wt = []; I < N; I++ )
514: Wt = cons(1,Wt);
515: }
516: Tdeg = w_tdeg(F,V,Wt);
517: WtV = newvect(2*(N+1)+1);
518: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
519: for ( I = 0; I < N; I++ ) {
520: WtV[I] = Wt[I];
521: WtV[N+1+I] = Tdeg-Wt[I]+1;
1.17 noro 522: }
1.18 noro 523: WtV[N] = Tdeg;
524: WtV[2*N+1] = 1;
525: WtV[2*(N+1)] = 1;
526: dp_set_weight(WtV);
1.17 noro 527: for ( I = N-1, DV = []; I >= 0; I-- )
528: DV = cons(strtov("d"+rtostr(V[I])),DV);
529:
530: B = [t-F];
531: for ( I = 0; I < N; I++ ) {
532: B = cons(DV[I]+diff(F,V[I])*dt,B);
533: }
534: V1 = append(V,[t]); DV1 = append(DV,[dt]);
535: W = newvect(N+1);
536: W[N] = 1;
1.21 noro 537: R = generic_bfct_1(B,V1,DV1,W);
1.19 noro 538: dp_set_weight(0);
539: return subst(R,s,-s-1);
540: }
541:
542: def bfct_via_gbfct_weight_2(F,V)
543: {
1.23 ! noro 544: /* XXX */
! 545: F = replace_vars_f(F);
! 546: V = replace_vars_v(V);
! 547:
1.19 noro 548: N = length(V);
549: D = newvect(N);
550: Wt = getopt(weight);
551: if ( type(Wt) != 4 ) {
552: for ( I = 0, Wt = []; I < N; I++ )
553: Wt = cons(1,Wt);
554: }
555: Tdeg = w_tdeg(F,V,Wt);
556:
557: /* a weight for the first GB computation */
558: /* [t,x1,...,xn,dt,dx1,...,dxn,h] */
559: WtV = newvect(2*(N+1)+1);
560: WtV[0] = Tdeg;
561: WtV[N+1] = 1;
562: WtV[2*(N+1)] = 1;
563: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
564: for ( I = 1; I <= N; I++ ) {
565: WtV[I] = Wt[I-1];
566: WtV[N+1+I] = Tdeg-Wt[I-1]+1;
567: }
568: dp_set_weight(WtV);
569:
570: /* a weight for the second GB computation */
571: /* [x1,...,xn,t,dx1,...,dxn,dt,h] */
572: WtV2 = newvect(2*(N+1)+1);
573: WtV2[N] = Tdeg;
574: WtV2[2*N+1] = 1;
575: WtV2[2*(N+1)] = 1;
576: for ( I = 0; I < N; I++ ) {
577: WtV2[I] = Wt[I];
578: WtV2[N+1+I] = Tdeg-Wt[I]+1;
579: }
580:
581: for ( I = N-1, DV = []; I >= 0; I-- )
582: DV = cons(strtov("d"+rtostr(V[I])),DV);
583:
584: B = [t-F];
585: for ( I = 0; I < N; I++ ) {
586: B = cons(DV[I]+diff(F,V[I])*dt,B);
587: }
588: V1 = cons(t,V); DV1 = cons(dt,DV);
589: V2 = append(V,[t]); DV2 = append(DV,[dt]);
590: W = newvect(N+1,[1]);
591: dp_weyl_set_weight(W);
592:
593: VDV = append(V1,DV1);
594: N1 = length(V1);
595: N2 = N1*2;
596:
597: /* create a non-term order MW in D<x,d> */
598: MW = newmat(N2+1,N2);
599: for ( J = 0; J < N1; J++ ) {
600: MW[0][J] = -W[J]; MW[0][N1+J] = W[J];
601: }
602: for ( J = 0; J < N2; J++ ) MW[1][J] = 1;
603: for ( I = 2; I <= N2; I++ ) MW[I][N2-I+1] = -1;
604:
605: /* homogenize F */
606: VDVH = append(VDV,[h]);
607: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,B,VDV)),VDVH);
608:
609: /* compute a groebner basis of FH w.r.t. MWH */
610: GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
611:
612: /* dehomigenize GH */
613: G = map(subst,GH,h,1);
614:
615: /* G is a groebner basis w.r.t. a non term order MW */
616: /* take the initial part w.r.t. (-W,W) */
617: GIN = map(initial_part,G,VDV,MW,W);
618:
619: /* GIN is a groebner basis w.r.t. a term order M */
620: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
621:
622: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
623: for ( I = 0, T = 0; I < N1; I++ )
624: T += W[I]*V1[I]*DV1[I];
625:
626: /* change of ordering from VDV to VDV2 */
627: VDV2 = append(V2,DV2);
628: dp_set_weight(WtV2);
1.20 noro 629: for ( Pind = 0; ; Pind++ ) {
630: Prime = lprime(Pind);
631: GIN2 = dp_weyl_gr_main(GIN,VDV2,0,-Prime,0);
632: if ( GIN2 ) break;
633: }
1.19 noro 634:
635: R = weyl_minipoly(GIN2,VDV2,0,T); /* M represents DRL order */
1.18 noro 636: dp_set_weight(0);
1.17 noro 637: return subst(R,s,-s-1);
638: }
639:
1.6 noro 640: def weyl_minipolym(G,V,O,M,V0)
641: {
642: N = length(V);
643: Len = length(G);
644: dp_ord(O);
645: setmod(M);
646: PS = newvect(Len);
647: PS0 = newvect(Len);
648:
649: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
650: PS0[I] = dp_ptod(car(T),V);
651: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
652: PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
653:
654: for ( I = Len - 1, GI = []; I >= 0; I-- )
655: GI = cons(I,GI);
656:
657: U = dp_mod(dp_ptod(V0,V),M,[]);
1.17 noro 658: U = dp_weyl_nf_mod(GI,U,PS,1,M);
1.6 noro 659:
660: T = dp_mod(<<0>>,M,[]);
661: TT = dp_mod(dp_ptod(1,V),M,[]);
662: G = H = [[TT,T]];
663:
664: for ( I = 1; ; I++ ) {
1.14 noro 665: if ( dp_gr_print() )
666: print(".",2);
1.6 noro 667: T = dp_mod(<<I>>,M,[]);
668:
669: TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
670: H = cons([TT,T],H);
671: L = dp_lnf_mod([TT,T],G,M);
1.14 noro 672: if ( !L[0] ) {
673: if ( dp_gr_print() )
674: print("");
1.13 noro 675: return dp_dtop(L[1],[t]); /* XXX */
1.14 noro 676: } else
1.6 noro 677: G = insert(G,L);
678: }
679: }
680:
1.13 noro 681: def weyl_minipoly(G0,V0,O0,P)
1.6 noro 682: {
1.11 noro 683: HM = hmlist(G0,V0,O0);
1.13 noro 684:
685: N = length(V0);
686: Len = length(G0);
687: dp_ord(O0);
688: PS = newvect(Len);
689: for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
690: PS[I] = dp_ptod(car(T),V0);
691: for ( I = Len - 1, GI = []; I >= 0; I-- )
692: GI = cons(I,GI);
1.20 noro 693: PSM = newvect(Len);
1.13 noro 694: DP = dp_ptod(P,V0);
695:
1.20 noro 696: for ( Pind = 0; ; Pind++ ) {
697: Prime = lprime(Pind);
1.11 noro 698: if ( !valid_modulus(HM,Prime) )
699: continue;
1.20 noro 700: setmod(Prime);
701: for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
702: PSM[I] = dp_mod(dp_ptod(car(T),V0),Prime,[]);
1.13 noro 703:
704: NFP = weyl_nf(GI,DP,1,PS);
1.20 noro 705: NFPM = dp_mod(NFP[0],Prime,[])/ptomp(NFP[1],Prime);
706:
1.13 noro 707: NF = [[dp_ptod(1,V0),1]];
708: LCM = 1;
709:
1.20 noro 710: TM = dp_mod(<<0>>,Prime,[]);
711: TTM = dp_mod(dp_ptod(1,V0),Prime,[]);
712: GM = NFM = [[TTM,TM]];
713:
714: for ( D = 1; ; D++ ) {
1.14 noro 715: if ( dp_gr_print() )
716: print(".",2);
1.13 noro 717: NFPrev = car(NF);
718: NFJ = weyl_nf(GI,
719: dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
720: NFJ = remove_cont(NFJ);
721: NF = cons(NFJ,NF);
722: LCM = ilcm(LCM,NFJ[1]);
1.20 noro 723:
724: /* modular computation */
725: TM = dp_mod(<<D>>,Prime,[]);
726: TTM = dp_mod(NFJ[0],Prime,[])/ptomp(NFJ[1],Prime);
727: NFM = cons([TTM,TM],NFM);
728: LM = dp_lnf_mod([TTM,TM],GM,Prime);
729: if ( !LM[0] )
730: break;
731: else
732: GM = insert(GM,LM);
1.13 noro 733: }
1.20 noro 734:
1.14 noro 735: if ( dp_gr_print() )
736: print("");
1.13 noro 737: U = NF[0][0]*idiv(LCM,NF[0][1]);
738: Coef = [];
739: for ( J = D-1; J >= 0; J-- ) {
740: Coef = cons(strtov("u"+rtostr(J)),Coef);
741: U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
742: }
1.6 noro 743:
1.13 noro 744: for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
745: Eq = cons(dp_hc(UU),Eq);
746: M = etom([Eq,Coef]);
747: B = henleq(M,Prime);
748: if ( dp_gr_print() )
749: print("");
1.6 noro 750: if ( B ) {
1.13 noro 751: R = 0;
752: for ( I = 0; I < D; I++ )
753: R += B[0][I]*s^I;
754: R += B[1]*s^D;
1.6 noro 755: return R;
756: }
757: }
758: }
759:
760: def weyl_nf(B,G,M,PS)
761: {
762: for ( D = 0; G; ) {
763: for ( U = 0, L = B; L != []; L = cdr(L) ) {
764: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
765: GCD = igcd(dp_hc(G),dp_hc(R));
766: CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
767: U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
768: if ( !U )
769: return [D,M];
770: D *= CG; M *= CG;
771: break;
772: }
773: }
774: if ( U )
775: G = U;
776: else {
777: D += dp_hm(G); G = dp_rest(G);
778: }
779: }
780: return [D,M];
781: }
782:
783: def weyl_nf_mod(B,G,PS,Mod)
784: {
785: for ( D = 0; G; ) {
786: for ( U = 0, L = B; L != []; L = cdr(L) ) {
787: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
788: CR = dp_hc(G)/dp_hc(R);
789: U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
790: if ( !U )
791: return D;
1.1 noro 792: break;
1.6 noro 793: }
794: }
795: if ( U )
796: G = U;
797: else {
798: D += dp_hm(G); G = dp_rest(G);
1.1 noro 799: }
800: }
1.6 noro 801: return D;
1.1 noro 802: }
803:
804: def remove_zero(L)
805: {
806: for ( R = []; L != []; L = cdr(L) )
807: if ( car(L) )
808: R = cons(car(L),R);
809: return R;
810: }
811:
812: def z_subst(F,V)
813: {
814: for ( ; V != []; V = cdr(V) )
815: F = subst(F,car(V),0);
816: return F;
817: }
818:
819: def flatmf(L) {
820: for ( S = []; L != []; L = cdr(L) )
821: if ( type(F=car(car(L))) != NUM )
822: S = append(S,[F]);
823: return S;
824: }
825:
826: def member(A,L) {
827: for ( ; L != []; L = cdr(L) )
828: if ( A == car(L) )
829: return 1;
830: return 0;
831: }
832:
833: def intersection(A,B)
834: {
835: for ( L = []; A != []; A = cdr(A) )
836: if ( member(car(A),B) )
837: L = cons(car(A),L);
838: return L;
839: }
840:
841: def b_subst(F,V)
842: {
843: D = deg(F,V);
844: C = newvect(D+1);
845: for ( I = D; I >= 0; I-- )
846: C[I] = coef(F,I,V);
847: for ( I = 0, R = 0; I <= D; I++ )
848: if ( C[I] )
849: R += C[I]*v_factorial(V,I);
850: return R;
851: }
852:
853: def v_factorial(V,N)
854: {
855: for ( J = N-1, R = 1; J >= 0; J-- )
856: R *= V-J;
1.17 noro 857: return R;
858: }
859:
860: def w_tdeg(F,V,W)
861: {
862: dp_set_weight(newvect(length(W),W));
863: T = dp_ptod(F,V);
864: for ( R = 0; T; T = cdr(T) ) {
865: D = dp_td(T);
866: if ( D > R ) R = D;
1.23 ! noro 867: }
! 868: return R;
! 869: }
! 870:
! 871: def replace_vars_f(F)
! 872: {
! 873: return subst(F,s,TMP_S,t,TMP_T,y1,TMP_Y1,y2,TMP_Y2);
! 874: }
! 875:
! 876: def replace_vars_v(V)
! 877: {
! 878: V = replace_var(V,s,TMP_S);
! 879: V = replace_var(V,t,TMP_T);
! 880: V = replace_var(V,y1,TMP_Y1);
! 881: V = replace_var(V,y2,TMP_Y2);
! 882: return V;
! 883: }
! 884:
! 885: def replace_var(V,X,Y)
! 886: {
! 887: V = reverse(V);
! 888: for ( R = []; V != []; V = cdr(V) ) {
! 889: Z = car(V);
! 890: if ( Z == X )
! 891: R = cons(Y,R);
! 892: else
! 893: R = cons(Z,R);
1.17 noro 894: }
1.1 noro 895: return R;
896: }
897: end$
898:
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