Annotation of OpenXM_contrib2/asir2000/lib/bfct, Revision 1.18
1.2 noro 1: /*
2: * Copyright (c) 1994-2000 FUJITSU LABORATORIES LIMITED
3: * All rights reserved.
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5: * FUJITSU LABORATORIES LIMITED ("FLL") hereby grants you a limited,
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12: * copyrights, in and to the SOFTWARE.
13: *
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20: * with or without modification, as permitted hereunder, you shall affix
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1.2 noro 27: * for such modification or the source code of the modified part of the
28: * SOFTWARE.
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1.18 ! noro 48: * $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.17 2002/01/28 01:02:03 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:
1.18 ! noro 273: /* all term reduction + interreduce */
! 274: def generic_bfct_1(F,V,DV,W)
! 275: {
! 276: N = length(V);
! 277: N2 = N*2;
! 278:
! 279: /* If W is a list, convert it to a vector */
! 280: if ( type(W) == 4 )
! 281: W = newvect(length(W),W);
! 282: dp_weyl_set_weight(W);
! 283:
! 284: /* create a term order M in D<x,d> (DRL) */
! 285: M = newmat(N2,N2);
! 286: for ( J = 0; J < N2; J++ )
! 287: M[0][J] = 1;
! 288: for ( I = 1; I < N2; I++ )
! 289: M[I][N2-I] = -1;
! 290:
! 291: VDV = append(V,DV);
! 292:
! 293: /* create a non-term order MW in D<x,d> */
! 294: MW = newmat(N2+1,N2);
! 295: for ( J = 0; J < N; J++ )
! 296: MW[0][J] = -W[J];
! 297: for ( ; J < N2; J++ )
! 298: MW[0][J] = W[J-N];
! 299: for ( I = 1; I <= N2; I++ )
! 300: for ( J = 0; J < N2; J++ )
! 301: MW[I][J] = M[I-1][J];
! 302:
! 303: /* create a homogenized term order MWH in D<x,d,h> */
! 304: MWH = newmat(N2+2,N2+1);
! 305: for ( J = 0; J <= N2; J++ )
! 306: MWH[0][J] = 1;
! 307: for ( I = 1; I <= N2+1; I++ )
! 308: for ( J = 0; J < N2; J++ )
! 309: MWH[I][J] = MW[I-1][J];
! 310:
! 311: /* homogenize F */
! 312: VDVH = append(VDV,[h]);
! 313: FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
! 314:
! 315: /* compute a groebner basis of FH w.r.t. MWH */
! 316: /* dp_gr_flags(["Top",1,"NoRA",1]); */
! 317: GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
! 318: /* dp_gr_flags(["Top",0,"NoRA",0]); */
! 319:
! 320: /* dehomigenize GH */
! 321: G = map(subst,GH,h,1);
! 322:
! 323: /* G is a groebner basis w.r.t. a non term order MW */
! 324: /* take the initial part w.r.t. (-W,W) */
! 325: GIN = map(initial_part,G,VDV,MW,W);
! 326:
! 327: /* GIN is a groebner basis w.r.t. a term order M */
! 328: /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
! 329:
! 330: /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
! 331: for ( I = 0, T = 0; I < N; I++ )
! 332: T += W[I]*V[I]*DV[I];
! 333: B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
! 334: return B;
! 335: }
! 336:
1.13 noro 337: def initial_part(F,V,MW,W)
338: {
339: N2 = length(V);
340: N = N2/2;
341: dp_ord(MW);
342: DF = dp_ptod(F,V);
343: R = dp_hm(DF);
344: DF = dp_rest(DF);
345:
346: E = dp_etov(R);
347: for ( I = 0, TW = 0; I < N; I++ )
348: TW += W[I]*(-E[I]+E[N+I]);
349: RW = TW;
350:
351: for ( ; DF; DF = dp_rest(DF) ) {
352: E = dp_etov(DF);
353: for ( I = 0, TW = 0; I < N; I++ )
354: TW += W[I]*(-E[I]+E[N+I]);
355: if ( TW == RW )
356: R += dp_hm(DF);
357: else if ( TW < RW )
358: break;
359: else
360: error("initial_part : cannot happen");
361: }
362: return dp_dtop(R,V);
363:
364: }
365:
1.1 noro 366: /* b-function of F ? */
367:
368: def bfct(F)
369: {
370: V = vars(F);
371: N = length(V);
1.6 noro 372: D = newvect(N);
1.7 noro 373:
1.6 noro 374: for ( I = 0; I < N; I++ )
375: D[I] = [deg(F,V[I]),V[I]];
376: qsort(D,compare_first);
377: for ( V = [], I = 0; I < N; I++ )
378: V = cons(D[I][1],V);
1.1 noro 379: for ( I = N-1, DV = []; I >= 0; I-- )
380: DV = cons(strtov("d"+rtostr(V[I])),DV);
1.6 noro 381: V1 = cons(s,V); DV1 = cons(ds,DV);
1.7 noro 382:
383: G0 = indicial1(F,reverse(V));
384: G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
385: Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
1.6 noro 386: return Minipoly;
387: }
388:
1.14 noro 389: /* b-function computation via generic_bfct() (experimental) */
390:
391: def bfct_via_gbfct(F)
392: {
393: V = vars(F);
394: N = length(V);
395: D = newvect(N);
396:
397: for ( I = 0; I < N; I++ )
398: D[I] = [deg(F,V[I]),V[I]];
399: qsort(D,compare_first);
400: for ( V = [], I = 0; I < N; I++ )
401: V = cons(D[I][1],V);
402: V = reverse(V);
403: for ( I = N-1, DV = []; I >= 0; I-- )
404: DV = cons(strtov("d"+rtostr(V[I])),DV);
405:
406: B = [t-F];
407: for ( I = 0; I < N; I++ ) {
408: B = cons(DV[I]+diff(F,V[I])*dt,B);
409: }
410: V1 = cons(t,V); DV1 = cons(dt,DV);
411: W = newvect(N+1);
412: W[0] = 1;
1.18 ! noro 413: R = generic_bfct_1(B,V1,DV1,W);
1.14 noro 414:
415: return subst(R,s,-s-1);
416: }
417:
1.17 noro 418: /* use an order s.t. [t,x,y,z,...,dt,dx,dy,dz,...,h] */
419:
420: def bfct_via_gbfct_weight(F,V)
421: {
422: N = length(V);
423: D = newvect(N);
424: Wt = getopt(weight);
1.18 ! noro 425: if ( type(Wt) != 4 ) {
! 426: for ( I = 0, Wt = []; I < N; I++ )
! 427: Wt = cons(1,Wt);
! 428: }
! 429: Tdeg = w_tdeg(F,V,Wt);
! 430: WtV = newvect(2*(N+1)+1);
! 431: WtV[0] = Tdeg;
! 432: WtV[N+1] = 1;
! 433: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
! 434: for ( I = 1; I <= N; I++ ) {
! 435: WtV[I] = Wt[I-1];
! 436: WtV[N+1+I] = Tdeg-Wt[I-1]+1;
1.17 noro 437: }
1.18 ! noro 438: WtV[2*(N+1)] = 1;
! 439: dp_set_weight(WtV);
1.17 noro 440: for ( I = N-1, DV = []; I >= 0; I-- )
441: DV = cons(strtov("d"+rtostr(V[I])),DV);
442:
443: B = [t-F];
444: for ( I = 0; I < N; I++ ) {
445: B = cons(DV[I]+diff(F,V[I])*dt,B);
446: }
447: V1 = cons(t,V); DV1 = cons(dt,DV);
448: W = newvect(N+1);
449: W[0] = 1;
1.18 ! noro 450: R = generic_bfct_1(B,V1,DV1,W);
! 451: dp_set_weight(0);
1.17 noro 452: return subst(R,s,-s-1);
453: }
454:
455: /* use an order s.t. [x,y,z,...,t,dx,dy,dz,...,dt,h] */
456:
457: def bfct_via_gbfct_weight_1(F,V)
458: {
459: N = length(V);
460: D = newvect(N);
461: Wt = getopt(weight);
1.18 ! noro 462: if ( type(Wt) != 4 ) {
! 463: for ( I = 0, Wt = []; I < N; I++ )
! 464: Wt = cons(1,Wt);
! 465: }
! 466: Tdeg = w_tdeg(F,V,Wt);
! 467: WtV = newvect(2*(N+1)+1);
! 468: /* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
! 469: for ( I = 0; I < N; I++ ) {
! 470: WtV[I] = Wt[I];
! 471: WtV[N+1+I] = Tdeg-Wt[I]+1;
1.17 noro 472: }
1.18 ! noro 473: WtV[N] = Tdeg;
! 474: WtV[2*N+1] = 1;
! 475: WtV[2*(N+1)] = 1;
! 476: dp_set_weight(WtV);
1.17 noro 477: for ( I = N-1, DV = []; I >= 0; I-- )
478: DV = cons(strtov("d"+rtostr(V[I])),DV);
479:
480: B = [t-F];
481: for ( I = 0; I < N; I++ ) {
482: B = cons(DV[I]+diff(F,V[I])*dt,B);
483: }
484: V1 = append(V,[t]); DV1 = append(DV,[dt]);
485: W = newvect(N+1);
486: W[N] = 1;
487: R = generic_bfct(B,V1,DV1,W);
1.18 ! noro 488: dp_set_weight(0);
1.17 noro 489: return subst(R,s,-s-1);
490: }
491:
1.6 noro 492: def weyl_minipolym(G,V,O,M,V0)
493: {
494: N = length(V);
495: Len = length(G);
496: dp_ord(O);
497: setmod(M);
498: PS = newvect(Len);
499: PS0 = newvect(Len);
500:
501: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
502: PS0[I] = dp_ptod(car(T),V);
503: for ( I = 0, T = G; T != []; T = cdr(T), I++ )
504: PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
505:
506: for ( I = Len - 1, GI = []; I >= 0; I-- )
507: GI = cons(I,GI);
508:
509: U = dp_mod(dp_ptod(V0,V),M,[]);
1.17 noro 510: U = dp_weyl_nf_mod(GI,U,PS,1,M);
1.6 noro 511:
512: T = dp_mod(<<0>>,M,[]);
513: TT = dp_mod(dp_ptod(1,V),M,[]);
514: G = H = [[TT,T]];
515:
516: for ( I = 1; ; I++ ) {
1.14 noro 517: if ( dp_gr_print() )
518: print(".",2);
1.6 noro 519: T = dp_mod(<<I>>,M,[]);
520:
521: TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
522: H = cons([TT,T],H);
523: L = dp_lnf_mod([TT,T],G,M);
1.14 noro 524: if ( !L[0] ) {
525: if ( dp_gr_print() )
526: print("");
1.13 noro 527: return dp_dtop(L[1],[t]); /* XXX */
1.14 noro 528: } else
1.6 noro 529: G = insert(G,L);
530: }
531: }
532:
1.13 noro 533: def weyl_minipoly(G0,V0,O0,P)
1.6 noro 534: {
1.11 noro 535: HM = hmlist(G0,V0,O0);
1.13 noro 536:
537: N = length(V0);
538: Len = length(G0);
539: dp_ord(O0);
540: PS = newvect(Len);
541: for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
542: PS[I] = dp_ptod(car(T),V0);
543: for ( I = Len - 1, GI = []; I >= 0; I-- )
544: GI = cons(I,GI);
545: DP = dp_ptod(P,V0);
546:
1.6 noro 547: for ( I = 0; ; I++ ) {
548: Prime = lprime(I);
1.11 noro 549: if ( !valid_modulus(HM,Prime) )
550: continue;
1.13 noro 551: MP = weyl_minipolym(G0,V0,O0,Prime,P);
552: D = deg(MP,var(MP));
553:
554: NFP = weyl_nf(GI,DP,1,PS);
555: NF = [[dp_ptod(1,V0),1]];
556: LCM = 1;
557:
558: for ( J = 1; J <= D; J++ ) {
1.14 noro 559: if ( dp_gr_print() )
560: print(".",2);
1.13 noro 561: NFPrev = car(NF);
562: NFJ = weyl_nf(GI,
563: dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
564: NFJ = remove_cont(NFJ);
565: NF = cons(NFJ,NF);
566: LCM = ilcm(LCM,NFJ[1]);
567: }
1.14 noro 568: if ( dp_gr_print() )
569: print("");
1.13 noro 570: U = NF[0][0]*idiv(LCM,NF[0][1]);
571: Coef = [];
572: for ( J = D-1; J >= 0; J-- ) {
573: Coef = cons(strtov("u"+rtostr(J)),Coef);
574: U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
575: }
1.6 noro 576:
1.13 noro 577: for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
578: Eq = cons(dp_hc(UU),Eq);
579: M = etom([Eq,Coef]);
580: B = henleq(M,Prime);
581: if ( dp_gr_print() )
582: print("");
1.6 noro 583: if ( B ) {
1.13 noro 584: R = 0;
585: for ( I = 0; I < D; I++ )
586: R += B[0][I]*s^I;
587: R += B[1]*s^D;
1.6 noro 588: return R;
589: }
590: }
591: }
592:
593: def weyl_nf(B,G,M,PS)
594: {
595: for ( D = 0; G; ) {
596: for ( U = 0, L = B; L != []; L = cdr(L) ) {
597: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
598: GCD = igcd(dp_hc(G),dp_hc(R));
599: CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
600: U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
601: if ( !U )
602: return [D,M];
603: D *= CG; M *= CG;
604: break;
605: }
606: }
607: if ( U )
608: G = U;
609: else {
610: D += dp_hm(G); G = dp_rest(G);
611: }
612: }
613: return [D,M];
614: }
615:
616: def weyl_nf_mod(B,G,PS,Mod)
617: {
618: for ( D = 0; G; ) {
619: for ( U = 0, L = B; L != []; L = cdr(L) ) {
620: if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
621: CR = dp_hc(G)/dp_hc(R);
622: U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
623: if ( !U )
624: return D;
1.1 noro 625: break;
1.6 noro 626: }
627: }
628: if ( U )
629: G = U;
630: else {
631: D += dp_hm(G); G = dp_rest(G);
1.1 noro 632: }
633: }
1.6 noro 634: return D;
1.1 noro 635: }
636:
637: def remove_zero(L)
638: {
639: for ( R = []; L != []; L = cdr(L) )
640: if ( car(L) )
641: R = cons(car(L),R);
642: return R;
643: }
644:
645: def z_subst(F,V)
646: {
647: for ( ; V != []; V = cdr(V) )
648: F = subst(F,car(V),0);
649: return F;
650: }
651:
652: def flatmf(L) {
653: for ( S = []; L != []; L = cdr(L) )
654: if ( type(F=car(car(L))) != NUM )
655: S = append(S,[F]);
656: return S;
657: }
658:
659: def member(A,L) {
660: for ( ; L != []; L = cdr(L) )
661: if ( A == car(L) )
662: return 1;
663: return 0;
664: }
665:
666: def intersection(A,B)
667: {
668: for ( L = []; A != []; A = cdr(A) )
669: if ( member(car(A),B) )
670: L = cons(car(A),L);
671: return L;
672: }
673:
674: def b_subst(F,V)
675: {
676: D = deg(F,V);
677: C = newvect(D+1);
678: for ( I = D; I >= 0; I-- )
679: C[I] = coef(F,I,V);
680: for ( I = 0, R = 0; I <= D; I++ )
681: if ( C[I] )
682: R += C[I]*v_factorial(V,I);
683: return R;
684: }
685:
686: def v_factorial(V,N)
687: {
688: for ( J = N-1, R = 1; J >= 0; J-- )
689: R *= V-J;
1.17 noro 690: return R;
691: }
692:
693: def w_tdeg(F,V,W)
694: {
695: dp_set_weight(newvect(length(W),W));
696: T = dp_ptod(F,V);
697: for ( R = 0; T; T = cdr(T) ) {
698: D = dp_td(T);
699: if ( D > R ) R = D;
700: }
1.1 noro 701: return R;
702: }
703: end$
704:
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