Annotation of OpenXM_contrib2/asir2000/lib/bfct, Revision 1.13
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: *
1.13 ! noro 48: * $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.12 2000/12/15 07:15:18 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:
! 215: /* create a term order M in D<x,d> */
! 216: M = newmat(N2,N2);
! 217: for ( J = 0; J < N2; J++ )
! 218: M[0][J] = 1;
! 219: for ( I = 1; I < N2; I++ )
! 220: M[I][N2-I] = -1;
! 221:
! 222: VDV = append(V,DV);
! 223:
! 224: /* create a non-term order MW in D<x,d> */
! 225: MW = newmat(N2+1,N2);
! 226: for ( J = 0; J < N; J++ )
! 227: MW[0][J] = -W[J];
! 228: for ( ; J < N2; J++ )
! 229: MW[0][J] = W[J-N];
! 230: for ( I = 1; I <= N2; I++ )
! 231: for ( J = 0; J < N2; J++ )
! 232: MW[I][J] = M[I-1][J];
! 233:
! 234: /* create a homogenized term order MWH in D<x,d,h> */
! 235: MWH = newmat(N2+2,N2+1);
! 236: for ( J = 0; J <= N2; J++ )
! 237: MWH[0][J] = 1;
! 238: for ( I = 1; I <= N2+1; I++ )
! 239: for ( J = 0; J < N2; J++ )
! 240: MWH[I][J] = MW[I-1][J];
! 241:
! 242: /* homogenize F */
! 243: VDVH = append(VDV,[h]);
! 244: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
! 245:
! 246: /* compute a groebner basis of FH w.r.t. MWH */
! 247: GH = dp_weyl_gr_main(FH,VDVH,0,0,MWH);
! 248:
! 249: /* dehomigenize GH */
! 250: G = map(subst,GH,h,1);
! 251:
! 252: /* G is a groebner basis w.r.t. a non term order MW */
! 253: /* take the initial part w.r.t. (-W,W) */
! 254: GIN = map(initial_part,G,VDV,MW,W);
! 255:
! 256: /* GIN is a groebner basis w.r.t. a term order M */
! 257: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
! 258:
! 259: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
! 260: for ( I = 0, T = 0; I < N; I++ )
! 261: T += W[I]*V[I]*DV[I];
! 262: B = weyl_minipoly(GIN,VDV,M,T);
! 263: return B;
! 264: }
! 265:
! 266: def initial_part(F,V,MW,W)
! 267: {
! 268: N2 = length(V);
! 269: N = N2/2;
! 270: dp_ord(MW);
! 271: DF = dp_ptod(F,V);
! 272: R = dp_hm(DF);
! 273: DF = dp_rest(DF);
! 274:
! 275: E = dp_etov(R);
! 276: for ( I = 0, TW = 0; I < N; I++ )
! 277: TW += W[I]*(-E[I]+E[N+I]);
! 278: RW = TW;
! 279:
! 280: for ( ; DF; DF = dp_rest(DF) ) {
! 281: E = dp_etov(DF);
! 282: for ( I = 0, TW = 0; I < N; I++ )
! 283: TW += W[I]*(-E[I]+E[N+I]);
! 284: if ( TW == RW )
! 285: R += dp_hm(DF);
! 286: else if ( TW < RW )
! 287: break;
! 288: else
! 289: error("initial_part : cannot happen");
! 290: }
! 291: return dp_dtop(R,V);
! 292:
! 293: }
! 294:
1.1 noro 295: /* b-function of F ? */
296:
297: def bfct(F)
298: {
299: V = vars(F);
300: N = length(V);
1.6 noro 301: D = newvect(N);
1.7 noro 302:
1.6 noro 303: for ( I = 0; I < N; I++ )
304: D[I] = [deg(F,V[I]),V[I]];
305: qsort(D,compare_first);
306: for ( V = [], I = 0; I < N; I++ )
307: V = cons(D[I][1],V);
1.1 noro 308: for ( I = N-1, DV = []; I >= 0; I-- )
309: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.6 noro 310: V1 = cons(s,V); DV1 = cons(ds,DV);
1.7 noro 311:
312: G0 = indicial1(F,reverse(V));
313: G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
314: Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
1.6 noro 315: return Minipoly;
316: }
317:
318: def weyl_minipolym(G,V,O,M,V0)
319: {
320: N = length(V);
321: Len = length(G);
322: dp_ord(O);
323: setmod(M);
324: PS = newvect(Len);
325: PS0 = newvect(Len);
326:
327: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
328: PS0[I] = dp_ptod(car(T),V);
329: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
330: PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
331:
332: for ( I = Len - 1, GI = []; I >= 0; I-- )
333: GI = cons(I,GI);
334:
335: U = dp_mod(dp_ptod(V0,V),M,[]);
336:
337: T = dp_mod(<<0>>,M,[]);
338: TT = dp_mod(dp_ptod(1,V),M,[]);
339: G = H = [[TT,T]];
340:
341: for ( I = 1; ; I++ ) {
342: T = dp_mod(<<I>>,M,[]);
343:
344: TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
345: H = cons([TT,T],H);
346: L = dp_lnf_mod([TT,T],G,M);
347: if ( !L[0] )
1.13 ! noro 348: return dp_dtop(L[1],[t]); /* XXX */
1.6 noro 349: else
350: G = insert(G,L);
351: }
352: }
353:
1.13 ! noro 354: def weyl_minipoly(G0,V0,O0,P)
1.6 noro 355: {
1.11 noro 356: HM = hmlist(G0,V0,O0);
1.13 ! noro 357:
! 358: N = length(V0);
! 359: Len = length(G0);
! 360: dp_ord(O0);
! 361: PS = newvect(Len);
! 362: for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
! 363: PS[I] = dp_ptod(car(T),V0);
! 364: for ( I = Len - 1, GI = []; I >= 0; I-- )
! 365: GI = cons(I,GI);
! 366: DP = dp_ptod(P,V0);
! 367:
1.6 noro 368: for ( I = 0; ; I++ ) {
369: Prime = lprime(I);
1.11 noro 370: if ( !valid_modulus(HM,Prime) )
371: continue;
1.13 ! noro 372: MP = weyl_minipolym(G0,V0,O0,Prime,P);
! 373: D = deg(MP,var(MP));
! 374:
! 375: NFP = weyl_nf(GI,DP,1,PS);
! 376: NF = [[dp_ptod(1,V0),1]];
! 377: LCM = 1;
! 378:
! 379: for ( J = 1; J <= D; J++ ) {
! 380: NFPrev = car(NF);
! 381: NFJ = weyl_nf(GI,
! 382: dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
! 383: NFJ = remove_cont(NFJ);
! 384: NF = cons(NFJ,NF);
! 385: LCM = ilcm(LCM,NFJ[1]);
! 386: }
! 387: U = NF[0][0]*idiv(LCM,NF[0][1]);
! 388: Coef = [];
! 389: for ( J = D-1; J >= 0; J-- ) {
! 390: Coef = cons(strtov("u"+rtostr(J)),Coef);
! 391: U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
! 392: }
1.6 noro 393:
1.13 ! noro 394: for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
! 395: Eq = cons(dp_hc(UU),Eq);
! 396: M = etom([Eq,Coef]);
! 397: B = henleq(M,Prime);
! 398: if ( dp_gr_print() )
! 399: print("");
1.6 noro 400: if ( B ) {
1.13 ! noro 401: R = 0;
! 402: for ( I = 0; I < D; I++ )
! 403: R += B[0][I]*s^I;
! 404: R += B[1]*s^D;
1.6 noro 405: return R;
406: }
407: }
408: }
409:
410: def weyl_nf(B,G,M,PS)
411: {
412: for ( D = 0; G; ) {
413: for ( U = 0, L = B; L != []; L = cdr(L) ) {
414: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
415: GCD = igcd(dp_hc(G),dp_hc(R));
416: CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
417: U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
418: if ( !U )
419: return [D,M];
420: D *= CG; M *= CG;
421: break;
422: }
423: }
424: if ( U )
425: G = U;
426: else {
427: D += dp_hm(G); G = dp_rest(G);
428: }
429: }
430: return [D,M];
431: }
432:
433: def weyl_nf_mod(B,G,PS,Mod)
434: {
435: for ( D = 0; G; ) {
436: for ( U = 0, L = B; L != []; L = cdr(L) ) {
437: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
438: CR = dp_hc(G)/dp_hc(R);
439: U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
440: if ( !U )
441: return D;
1.1 noro 442: break;
1.6 noro 443: }
444: }
445: if ( U )
446: G = U;
447: else {
448: D += dp_hm(G); G = dp_rest(G);
1.1 noro 449: }
450: }
1.6 noro 451: return D;
1.1 noro 452: }
453:
454: def remove_zero(L)
455: {
456: for ( R = []; L != []; L = cdr(L) )
457: if ( car(L) )
458: R = cons(car(L),R);
459: return R;
460: }
461:
462: def z_subst(F,V)
463: {
464: for ( ; V != []; V = cdr(V) )
465: F = subst(F,car(V),0);
466: return F;
467: }
468:
469: def flatmf(L) {
470: for ( S = []; L != []; L = cdr(L) )
471: if ( type(F=car(car(L))) != NUM )
472: S = append(S,[F]);
473: return S;
474: }
475:
476: def member(A,L) {
477: for ( ; L != []; L = cdr(L) )
478: if ( A == car(L) )
479: return 1;
480: return 0;
481: }
482:
483: def intersection(A,B)
484: {
485: for ( L = []; A != []; A = cdr(A) )
486: if ( member(car(A),B) )
487: L = cons(car(A),L);
488: return L;
489: }
490:
491: def b_subst(F,V)
492: {
493: D = deg(F,V);
494: C = newvect(D+1);
495: for ( I = D; I >= 0; I-- )
496: C[I] = coef(F,I,V);
497: for ( I = 0, R = 0; I <= D; I++ )
498: if ( C[I] )
499: R += C[I]*v_factorial(V,I);
500: return R;
501: }
502:
503: def v_factorial(V,N)
504: {
505: for ( J = N-1, R = 1; J >= 0; J-- )
506: R *= V-J;
507: return R;
508: }
509: end$
510:
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