Annotation of OpenXM_contrib/pari-2.2/src/basemath/polarit1.c, Revision 1.1.1.1
1.1 noro 1: /* $Id: polarit1.c,v 1.66 2001/10/01 17:19:06 bill Exp $
2:
3: Copyright (C) 2000 The PARI group.
4:
5: This file is part of the PARI/GP package.
6:
7: PARI/GP is free software; you can redistribute it and/or modify it under the
8: terms of the GNU General Public License as published by the Free Software
9: Foundation. It is distributed in the hope that it will be useful, but WITHOUT
10: ANY WARRANTY WHATSOEVER.
11:
12: Check the License for details. You should have received a copy of it, along
13: with the package; see the file 'COPYING'. If not, write to the Free Software
14: Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */
15:
16: /***********************************************************************/
17: /** **/
18: /** ARITHMETIC OPERATIONS ON POLYNOMIALS **/
19: /** (first part) **/
20: /** **/
21: /***********************************************************************/
22: #include "pari.h"
23: extern GEN get_bas_den(GEN bas);
24: extern GEN get_mul_table(GEN x,GEN bas,GEN invbas,GEN *T);
25: extern GEN pol_to_monic(GEN pol, GEN *lead);
26:
27: /* see splitgen() for how to use these two */
28: GEN
29: setloop(GEN a)
30: {
31: a=icopy(a); new_chunk(2); /* dummy to get two cells of extra space */
32: return a;
33: }
34:
35: /* assume a > 0 */
36: GEN
37: incpos(GEN a)
38: {
39: long i,l=lgefint(a);
40:
41: for (i=l-1; i>1; i--)
42: if (++a[i]) return a;
43: i=l+1; a--; /* use extra cell */
44: a[0]=evaltyp(1) | evallg(i);
45: a[1]=evalsigne(1) | evallgefint(i);
46: return a;
47: }
48:
49: GEN
50: incloop(GEN a)
51: {
52: long i,l;
53:
54: switch(signe(a))
55: {
56: case 0:
57: a--; /* use extra cell */
58: a[0]=evaltyp(t_INT) | evallg(3);
59: a[1]=evalsigne(1) | evallgefint(3);
60: a[2]=1; return a;
61:
62: case -1:
63: l=lgefint(a);
64: for (i=l-1; i>1; i--)
65: if (a[i]--) break;
66: if (a[2] == 0)
67: {
68: a++; /* save one cell */
69: a[0] = evaltyp(t_INT) | evallg(2);
70: a[1] = evalsigne(0) | evallgefint(2);
71: }
72: return a;
73:
74: default:
75: return incpos(a);
76: }
77: }
78:
79: /*******************************************************************/
80: /* */
81: /* DIVISIBILITE */
82: /* Return 1 if y | x, 0 otherwise */
83: /* */
84: /*******************************************************************/
85:
86: int
87: gdivise(GEN x, GEN y)
88: {
89: long av=avma;
90: x=gmod(x,y); avma=av; return gcmp0(x);
91: }
92:
93: int
94: poldivis(GEN x, GEN y, GEN *z)
95: {
96: long av = avma;
97: GEN p1 = poldivres(x,y,ONLY_DIVIDES);
98: if (p1) { *z = p1; return 1; }
99: avma=av; return 0;
100: }
101:
102: /*******************************************************************/
103: /* */
104: /* POLYNOMIAL EUCLIDEAN DIVISION */
105: /* */
106: /*******************************************************************/
107: /* Polynomial division x / y:
108: * if z = ONLY_REM return remainder, otherwise return quotient
109: * if z != NULL set *z to remainder
110: * *z is the last object on stack (and thus can be disposed of with cgiv
111: * instead of gerepile)
112: */
113: GEN
114: poldivres(GEN x, GEN y, GEN *pr)
115: {
116: ulong avy,av,av1;
117: long ty=typ(y),tx,vx,vy,dx,dy,dz,i,j,sx,lrem;
118: int remainder;
119: GEN z,p1,rem,y_lead,mod;
120: GEN (*f)(GEN,GEN);
121:
122: if (pr == ONLY_DIVIDES_EXACT)
123: { f = gdivexact; pr = ONLY_DIVIDES; }
124: else
125: f = gdiv;
126: if (is_scalar_t(ty))
127: {
128: if (pr == ONLY_REM) return gzero;
129: if (pr && pr != ONLY_DIVIDES) *pr=gzero;
130: return f(x,y);
131: }
132: tx=typ(x); vy=gvar9(y);
133: if (is_scalar_t(tx) || gvar9(x)>vy)
134: {
135: if (pr == ONLY_REM) return gcopy(x);
136: if (pr == ONLY_DIVIDES) return gcmp0(x)? gzero: NULL;
137: if (pr) *pr=gcopy(x);
138: return gzero;
139: }
140: if (tx!=t_POL || ty!=t_POL) err(typeer,"euclidean division (poldivres)");
141:
142: vx=varn(x);
143: if (vx<vy)
144: {
145: if (pr && pr != ONLY_DIVIDES)
146: {
147: p1 = zeropol(vx); if (pr == ONLY_REM) return p1;
148: *pr = p1;
149: }
150: return f(x,y);
151: }
152: if (!signe(y)) err(talker,"euclidean division by zero (poldivres)");
153:
154: dy=degpol(y); y_lead = (GEN)y[dy+2];
155: if (gcmp0(y_lead)) /* normalize denominator if leading term is 0 */
156: {
157: err(warner,"normalizing a polynomial with 0 leading term");
158: for (dy--; dy>=0; dy--)
159: {
160: y_lead = (GEN)y[dy+2];
161: if (!gcmp0(y_lead)) break;
162: }
163: }
164: if (!dy) /* y is constant */
165: {
166: if (pr && pr != ONLY_DIVIDES)
167: {
168: if (pr == ONLY_REM) return zeropol(vx);
169: *pr = zeropol(vx);
170: }
171: return f(x, constant_term(y));
172: }
173: dx=degpol(x);
174: if (vx>vy || dx<dy)
175: {
176: if (pr)
177: {
178: if (pr == ONLY_DIVIDES) return gcmp0(x)? gzero: NULL;
179: if (pr == ONLY_REM) return gcopy(x);
180: *pr = gcopy(x);
181: }
182: return zeropol(vy);
183: }
184: dz=dx-dy; av=avma; /* to avoid gsub's later on */
185: p1 = new_chunk(dy+3);
186: for (i=2; i<dy+3; i++)
187: {
188: GEN p2 = (GEN)y[i];
189: p1[i] = isexactzero(p2)? 0: (long)gneg_i(p2);
190: }
191: y = p1;
192: switch(typ(y_lead))
193: {
194: case t_INTMOD:
195: case t_POLMOD: y_lead = ginv(y_lead);
196: f = gmul; mod = gmodulcp(gun, (GEN)y_lead[1]);
197: break;
198: default: if (gcmp1(y_lead)) y_lead = NULL;
199: mod = NULL;
200: }
201: avy=avma; z=cgetg(dz+3,t_POL);
202: z[1]=evalsigne(1) | evallgef(dz+3) | evalvarn(vx);
203: x += 2; y += 2; z += 2;
204:
205: p1 = (GEN)x[dx]; remainder = (pr == ONLY_REM);
206: z[dz]=y_lead? (long)f(p1,y_lead): lcopy(p1);
207: for (i=dx-1; i>=dy; i--)
208: {
209: av1=avma; p1=(GEN)x[i];
210: for (j=i-dy+1; j<=i && j<=dz; j++)
211: if (y[i-j]) p1 = gadd(p1, gmul((GEN)z[j],(GEN)y[i-j]));
212: if (y_lead) p1 = f(p1,y_lead);
213: if (!remainder) p1 = avma==av1? gcopy(p1): gerepileupto(av1,p1);
214: z[i-dy] = (long)p1;
215: }
216: if (!pr) return gerepileupto(av,z-2);
217:
218: rem = (GEN)avma; av1 = (long)new_chunk(dx+3);
219: for (sx=0; ; i--)
220: {
221: p1 = (GEN)x[i];
222: /* we always enter this loop at least once */
223: for (j=0; j<=i && j<=dz; j++)
224: if (y[i-j]) p1 = gadd(p1, gmul((GEN)z[j],(GEN)y[i-j]));
225: if (mod && avma==av1) p1 = gmul(p1,mod);
226: if (!gcmp0(p1)) { sx = 1; break; } /* remainder is non-zero */
227: if (!isinexactreal(p1) && !isexactzero(p1)) break;
228: if (!i) break;
229: avma=av1;
230: }
231: if (pr == ONLY_DIVIDES)
232: {
233: if (sx) { avma=av; return NULL; }
234: avma = (long)rem;
235: return gerepileupto(av,z-2);
236: }
237: lrem=i+3; rem -= lrem;
238: if (avma==av1) { avma = (long)rem; p1 = gcopy(p1); }
239: else p1 = gerepileupto((long)rem,p1);
240: rem[0]=evaltyp(t_POL) | evallg(lrem);
241: rem[1]=evalsigne(1) | evalvarn(vx) | evallgef(lrem);
242: rem += 2;
243: rem[i]=(long)p1;
244: for (i--; i>=0; i--)
245: {
246: av1=avma; p1 = (GEN)x[i];
247: for (j=0; j<=i && j<=dz; j++)
248: if (y[i-j]) p1 = gadd(p1, gmul((GEN)z[j],(GEN)y[i-j]));
249: if (mod && avma==av1) p1 = gmul(p1,mod);
250: rem[i]=avma==av1? lcopy(p1):lpileupto(av1,p1);
251: }
252: rem -= 2;
253: if (!sx) normalizepol_i(rem, lrem);
254: if (remainder) return gerepileupto(av,rem);
255: z -= 2;
256: {
257: GEN *gptr[2]; gptr[0]=&z; gptr[1]=&rem;
258: gerepilemanysp(av,avy,gptr,2); *pr = rem; return z;
259: }
260: }
261:
262: /*******************************************************************/
263: /* */
264: /* ROOTS MODULO a prime p (no multiplicities) */
265: /* */
266: /*******************************************************************/
267: static GEN
268: mod(GEN x, GEN y)
269: {
270: GEN z = cgetg(3,t_INTMOD);
271: z[1]=(long)y; z[2]=(long)x; return z;
272: }
273:
274: static long
275: factmod_init(GEN *F, GEN pp, long *p)
276: {
277: GEN f = *F;
278: long i,d;
279: if (typ(f)!=t_POL || typ(pp)!=t_INT) err(typeer,"factmod");
280: if (expi(pp) > BITS_IN_LONG - 3) *p = 0;
281: else
282: {
283: *p = itos(pp);
284: if (*p < 2) err(talker,"not a prime in factmod");
285: }
286: f = gmul(f, mod(gun,pp));
287: if (!signe(f)) err(zeropoler,"factmod");
288: f = lift_intern(f); d = lgef(f);
289: for (i=2; i <d; i++)
290: if (typ(f[i])!=t_INT) err(impl,"factormod for general polynomials");
291: *F = f; return d-3;
292: }
293:
294: #define mods(x,y) mod(stoi(x),y)
295: static GEN
296: root_mod_2(GEN f)
297: {
298: int z1, z0 = !signe(constant_term(f));
299: long i,n;
300: GEN y;
301:
302: for (i=2, n=1; i < lgef(f); i++)
303: if (signe(f[i])) n++;
304: z1 = n & 1;
305: y = cgetg(z0+z1+1, t_COL); i = 1;
306: if (z0) y[i++] = (long)mods(0,gdeux);
307: if (z1) y[i] = (long)mods(1,gdeux);
308: return y;
309: }
310:
311: #define i_mod4(x) (signe(x)? mod4((GEN)(x)): 0)
312: static GEN
313: root_mod_4(GEN f)
314: {
315: long no,ne;
316: int z0 = !signe(constant_term(f));
317: int z2 = ((i_mod4(constant_term(f)) + 2*i_mod4(f[3])) & 3) == 0;
318: int i,z1,z3;
319: GEN y,p;
320:
321: for (ne=0,i=2; i<lgef(f); i+=2)
322: if (signe(f[i])) ne += mael(f,i,2);
323: for (no=0,i=3; i<lgef(f); i+=2)
324: if (signe(f[i])) no += mael(f,i,2);
325: no &= 3; ne &= 3;
326: z3 = (no == ne);
327: z1 = (no == ((4-ne)&3));
328: y=cgetg(1+z0+z1+z2+z3,t_COL); i = 1; p = stoi(4);
329: if (z0) y[i++] = (long)mods(0,p);
330: if (z1) y[i++] = (long)mods(1,p);
331: if (z2) y[i++] = (long)mods(2,p);
332: if (z3) y[i] = (long)mods(3,p);
333: return y;
334: }
335: #undef i_mod4
336:
337: /* p even, accept p = 4 for p-adic stuff */
338: static GEN
339: root_mod_even(GEN f, long p)
340: {
341: switch(p)
342: {
343: case 2: return root_mod_2(f);
344: case 4: return root_mod_4(f);
345: }
346: err(talker,"not a prime in rootmod");
347: return NULL; /* not reached */
348: }
349:
350: /* by checking f(0..p-1) */
351: GEN
352: rootmod2(GEN f, GEN pp)
353: {
354: GEN g,y,ss,q,r, x_minus_s;
355: long p,av = avma,av1,d,i,nbrac;
356:
357: if (!(d = factmod_init(&f, pp, &p))) { avma=av; return cgetg(1,t_COL); }
358: if (!p) err(talker,"prime too big in rootmod2");
359: if ((p & 1) == 0) { avma = av; return root_mod_even(f,p); }
360: x_minus_s = gadd(polx[varn(f)], stoi(-1));
361:
362: nbrac=1;
363: y=(GEN)gpmalloc((d+1)*sizeof(long));
364: if (gcmp0(constant_term(f))) y[nbrac++] = 0;
365: ss = icopy(gun); av1 = avma;
366: do
367: {
368: mael(x_minus_s,2,2) = ss[2];
369: /* one might do a FFT-type evaluation */
370: q = FpX_divres(f, x_minus_s, pp, &r);
371: if (signe(r)) avma = av1;
372: else
373: {
374: y[nbrac++] = ss[2]; f = q; av1 = avma;
375: }
376: ss[2]++;
377: }
378: while (nbrac<d && p>ss[2]);
379: if (nbrac == 1) { avma=av; return cgetg(1,t_COL); }
380: if (nbrac == d && p != ss[2])
381: {
382: g = mpinvmod((GEN)f[3], pp); setsigne(g,-1);
383: ss = modis(mulii(g, (GEN)f[2]), p);
384: y[nbrac++]=ss[2];
385: }
386: avma=av; g=cgetg(nbrac,t_COL);
387: if (isonstack(pp)) pp = icopy(pp);
388: for (i=1; i<nbrac; i++) g[i]=(long)mods(y[i],pp);
389: free(y); return g;
390: }
391:
392: /* by splitting */
393: GEN
394: rootmod(GEN f, GEN p)
395: {
396: long av = avma,tetpil,n,i,j,la,lb;
397: GEN y,pol,a,b,q,pol0;
398:
399: if (!factmod_init(&f, p, &i)) { avma=av; return cgetg(1,t_COL); }
400: i = p[lgefint(p)-1];
401: if ((i & 1) == 0) { avma = av; return root_mod_even(f,i); }
402: i=2; while (!signe(f[i])) i++;
403: if (i == 2) j = 1;
404: else
405: {
406: j = lgef(f) - (i-2);
407: if (j==3) /* f = x^n */
408: {
409: avma = av; y = cgetg(2,t_COL);
410: y[1] = (long)gmodulsg(0,p);
411: return y;
412: }
413: a = cgetg(j, t_POL); /* a = f / x^{v_x(f)} */
414: a[1] = evalsigne(1) | evalvarn(varn(f)) | evallgef(j);
415: f += i-2; for (i=2; i<j; i++) a[i]=f[i];
416: j = 2; f = a;
417: }
418: q = shifti(p,-1);
419: /* take gcd(x^(p-1) - 1, f) by splitting (x^q-1) * (x^q+1) */
420: b = FpXQ_pow(polx[varn(f)],q, f,p);
421: if (lgef(b)<3) err(talker,"not a prime in rootmod");
422: b = ZX_s_add(b,-1); /* b = x^((p-1)/2) - 1 mod f */
423: a = FpX_gcd(f,b, p);
424: b = ZX_s_add(b, 2); /* b = x^((p-1)/2) + 1 mod f */
425: b = FpX_gcd(f,b, p);
426: la = degpol(a);
427: lb = degpol(b); n = la + lb;
428: if (!n)
429: {
430: avma = av; y = cgetg(n+j,t_COL);
431: if (j>1) y[1] = (long)gmodulsg(0,p);
432: return y;
433: }
434: y = cgetg(n+j,t_COL);
435: if (j>1) { y[1] = zero; n++; }
436: y[j] = (long)FpX_normalize(b,p);
437: if (la) y[j+lb] = (long)FpX_normalize(a,p);
438: pol = gadd(polx[varn(f)], gun); pol0 = constant_term(pol);
439: while (j<=n)
440: {
441: a=(GEN)y[j]; la=degpol(a);
442: if (la==1)
443: y[j++] = lsubii(p, constant_term(a));
444: else if (la==2)
445: {
446: GEN d = subii(sqri((GEN)a[3]), shifti((GEN)a[2],2));
447: GEN e = mpsqrtmod(d,p), u = addis(q, 1); /* u = 1/2 */
448: y[j++] = lmodii(mulii(u,subii(e,(GEN)a[3])), p);
449: y[j++] = lmodii(mulii(u,negi(addii(e,(GEN)a[3]))), p);
450: }
451: else for (pol0[2]=1; ; pol0[2]++)
452: {
453: b = ZX_s_add(FpXQ_pow(pol,q, a,p), -1); /* pol^(p-1)/2 - 1 */
454: b = FpX_gcd(a,b, p); lb = degpol(b);
455: if (lb && lb<la)
456: {
457: b = FpX_normalize(b, p);
458: y[j+lb] = (long)FpX_div(a,b, p);
459: y[j] = (long)b; break;
460: }
461: }
462: }
463: tetpil = avma; y = gerepile(av,tetpil,sort(y));
464: if (isonstack(p)) p = icopy(p);
465: for (i=1; i<=n; i++) y[i] = (long)mod((GEN)y[i], p);
466: return y;
467: }
468:
469: GEN
470: rootmod0(GEN f, GEN p, long flag)
471: {
472: switch(flag)
473: {
474: case 0: return rootmod(f,p);
475: case 1: return rootmod2(f,p);
476: default: err(flagerr,"polrootsmod");
477: }
478: return NULL; /* not reached */
479: }
480:
481: /*******************************************************************/
482: /* */
483: /* FACTORISATION MODULO p */
484: /* */
485: /*******************************************************************/
486: static GEN spec_FpXQ_pow(GEN x, GEN p, GEN S);
487: /* Functions giving information on the factorisation. */
488:
489: /* u in Z[X], return kernel of (Frob - Id) over Fp[X] / u */
490: static GEN
491: Berlekamp_ker(GEN u, GEN p)
492: {
493: long i,j,d,N = degpol(u);
494: GEN vker,v,w,Q,p1,p2;
495: if (DEBUGLEVEL > 7) timer2();
496: Q = cgetg(N+1,t_MAT); Q[1] = (long)zerocol(N);
497: w = v = FpXQ_pow(polx[varn(u)],p,u,p);
498: for (j=2; j<=N; j++)
499: {
500: Q[j] = lgetg(N+1,t_COL); p1 = (GEN)Q[j];
501: d = lgef(w)-1; p2 = w+1;
502: for (i=1; i<d ; i++) p1[i] = p2[i];
503: for ( ; i<=N; i++) p1[i] = zero;
504: p1[j] = laddis((GEN)p1[j], -1);
505: if (j < N)
506: {
507: ulong av = avma;
508: w = gerepileupto(av, FpX_res(gmul(w,v), u, p));
509: }
510: }
511: if (DEBUGLEVEL > 7) msgtimer("frobenius");
512: vker = FpM_ker(Q,p);
513: if (DEBUGLEVEL > 7) msgtimer("kernel");
514: return vker;
515: }
516:
517: /* f in ZZ[X] and p a prime number. */
518: long
519: FpX_is_squarefree(GEN f, GEN p)
520: {
521: long av = avma;
522: GEN z;
523: z = FpX_gcd(f,derivpol(f),p);
524: avma = av;
525: return lgef(z)==3;
526: }
527: /* idem
528: * leading term of f must be prime to p.
529: */
530: /* Compute the number of roots in Fp without counting multiplicity
531: * return -1 for 0 polynomial.
532: */
533: long
534: FpX_nbroots(GEN f, GEN p)
535: {
536: long av = avma, n=lgef(f);
537: GEN z;
538: if (n <= 4) return n-3;
539: f = FpX_red(f, p);
540: z = FpXQ_pow(polx[varn(f)], p, f, p);
541: z = FpX_sub(z,polx[varn(f)],NULL);
542: z = FpX_gcd(z,f,p),
543: avma = av; return degpol(z);
544: }
545: long
546: FpX_is_totally_split(GEN f, GEN p)
547: {
548: long av = avma, n=lgef(f);
549: GEN z;
550: if (n <= 4) return 1;
551: if (!is_bigint(p) && n-3 > p[2]) return 0;
552: f = FpX_red(f, p);
553: z = FpXQ_pow(polx[varn(f)], p, f, p);
554: avma = av; return lgef(z)==4 && gcmp1((GEN)z[3]) && !signe(z[2]);
555: }
556: /* u in ZZ[X] and p a prime number.
557: * u must be squarefree mod p.
558: * leading term of u must be prime to p. */
559: long
560: FpX_nbfact(GEN u, GEN p)
561: {
562: ulong av = avma;
563: GEN vker = Berlekamp_ker(u,p);
564: avma = av; return lg(vker)-1;
565: }
566:
567: /* Please use only use this function when you it is false, or that there is a
568: * lot of factors. If you believe f is irreducible or that it has few factors,
569: * then use `FpX_nbfact(f,p)==1' instead (faster).
570: */
571: static GEN factcantor0(GEN f, GEN pp, long flag);
572: long FpX_is_irred(GEN f, GEN p) { return !!factcantor0(f,p,2); }
573:
574: static GEN modulo;
575: static GEN gsmul(GEN a,GEN b){return FpX_mul(a,b,modulo);}
576: GEN
577: FpV_roots_to_pol(GEN V, GEN p, long v)
578: {
579: ulong ltop=avma;
580: long i;
581: GEN g=cgetg(lg(V),t_VEC);
582: for(i=1;i<lg(V);i++)
583: g[i]=(long)deg1pol(gun,negi((GEN)V[i]),v);
584: modulo=p;
585: g=divide_conquer_prod(g,&gsmul);
586: return gerepileupto(ltop,g);
587: }
588:
589: /************************************************************/
590: GEN
591: trivfact(void)
592: {
593: GEN y=cgetg(3,t_MAT);
594: y[1]=lgetg(1,t_COL);
595: y[2]=lgetg(1,t_COL); return y;
596: }
597:
598: static void
599: fqunclone(GEN x, GEN a, GEN p)
600: {
601: long i,j,lx = lgef(x);
602: for (i=2; i<lx; i++)
603: {
604: GEN p1 = (GEN)x[i];
605: if (typ(p1) == t_POLMOD) { p1[1] = (long)a; p1 = (GEN)p1[2]; }
606: if (typ(p1) == t_INTMOD) p1[1] = (long)p;
607: else /* t_POL */
608: for (j = lgef(p1)-1; j > 1; j--)
609: {
610: GEN p2 = (GEN)p1[j];
611: if (typ(p2) == t_INTMOD) p2[1] = (long)p;
612: }
613: }
614: }
615:
616: static GEN
617: try_pow(GEN w0, GEN pol, GEN p, GEN q, long r)
618: {
619: GEN w2, w = FpXQ_pow(w0,q, pol,p);
620: long s;
621: if (gcmp1(w)) return w0;
622: for (s=1; s<r; s++,w=w2)
623: {
624: w2 = gsqr(w);
625: w2 = FpX_res(w2, pol, p);
626: if (gcmp1(w2)) break;
627: }
628: return gcmp_1(w)? NULL: w;
629: }
630:
631: /* INPUT:
632: * m integer (converted to polynomial w in Z[X] by stopoly)
633: * p prime; q = (p^d-1) / 2^r
634: * t[0] polynomial of degree k*d product of k irreducible factors of degree d
635: * t[0] is expected to be normalized (leading coeff = 1)
636: * OUTPUT:
637: * t[0],t[1]...t[k-1] the k factors, normalized
638: */
639: static void
640: split(long m, GEN *t, long d, GEN p, GEN q, long r, GEN S)
641: {
642: long ps,l,v,dv,av0,av;
643: GEN w,w0;
644:
645: dv=degpol(*t); if (dv==d) return;
646: v=varn(*t); av0=avma; ps = p[2];
647: for(av=avma;;avma=av)
648: {
649: if (ps==2)
650: {
651: w0=w=gpuigs(polx[v],m-1); m+=2;
652: for (l=1; l<d; l++)
653: w = gadd(w0, spec_FpXQ_pow(w, p, S));
654: }
655: else
656: {
657: w = FpX_res(stopoly(m,ps,v),*t, p);
658: m++; w = try_pow(w,*t,p,q,r);
659: if (!w) continue;
660: w = ZX_s_add(w, -1);
661: }
662: w = FpX_gcd(*t,w, p);
663: l = degpol(w); if (l && l!=dv) break;
664: }
665: w = FpX_normalize(w, p);
666: w = gerepileupto(av0, w);
667: l /= d; t[l]=FpX_div(*t,w,p); *t=w;
668: split(m,t+l,d,p,q,r,S);
669: split(m,t, d,p,q,r,S);
670: }
671:
672: static void
673: splitgen(GEN m, GEN *t, long d, GEN p, GEN q, long r)
674: {
675: long l,v,dv,av;
676: GEN w;
677:
678: dv=degpol(*t); if (dv==d) return;
679: v=varn(*t); m=setloop(m); m=incpos(m);
680: av=avma;
681: for(;; avma=av, m=incpos(m))
682: {
683: w = FpX_res(stopoly_gen(m,p,v),*t, p);
684: w = try_pow(w,*t,p,q,r);
685: if (!w) continue;
686: w = ZX_s_add(w,-1);
687: w = FpX_gcd(*t,w, p); l=degpol(w);
688: if (l && l!=dv) break;
689:
690: }
691: w = FpX_normalize(w, p);
692: w = gerepileupto(av, w);
693: l /= d; t[l]=FpX_div(*t,w,p); *t=w;
694: splitgen(m,t+l,d,p,q,r);
695: splitgen(m,t, d,p,q,r);
696: }
697:
698: /* return S = [ x^p, x^2p, ... x^(n-1)p ] mod (p, T), n = degree(T) > 0 */
699: static GEN
700: init_pow_p_mod_pT(GEN p, GEN T)
701: {
702: long i, n = degpol(T), v = varn(T);
703: GEN p1, S = cgetg(n, t_VEC);
704: if (n == 1) return S;
705: S[1] = (long)FpXQ_pow(polx[v], p, T, p);
706: /* use as many squarings as possible */
707: for (i=2; i < n; i+=2)
708: {
709: p1 = gsqr((GEN)S[i>>1]);
710: S[i] = (long)FpX_res(p1, T, p);
711: if (i == n-1) break;
712: p1 = gmul((GEN)S[i], (GEN)S[1]);
713: S[i+1] = (long)FpX_res(p1, T, p);
714: }
715: return S;
716: }
717:
718: /* compute x^p, S is as above */
719: static GEN
720: spec_FpXQ_pow(GEN x, GEN p, GEN S)
721: {
722: long av = avma, lim = stack_lim(av,1), i,dx = degpol(x);
723: GEN x0 = x+2, z;
724: z = (GEN)x0[0];
725: if (dx < 0) err(talker, "zero polynomial in FpXQ_pow. %Z not prime", p);
726: for (i = 1; i <= dx; i++)
727: {
728: GEN d, c = (GEN)x0[i]; /* assume coeffs in [0, p-1] */
729: if (!signe(c)) continue;
730: d = (GEN)S[i]; if (!gcmp1(c)) d = gmul(c,d);
731: z = gadd(z, d);
732: if (low_stack(lim, stack_lim(av,1)))
733: {
734: if(DEBUGMEM>1) err(warnmem,"spec_FpXQ_pow");
735: z = gerepileupto(av, z);
736: }
737: }
738: z = FpX_red(z, p);
739: return gerepileupto(av, z);
740: }
741:
742: /* factor f mod pp.
743: * If (flag = 1) return the degrees, not the factors
744: * If (flag = 2) return NULL if f is not irreducible
745: */
746: static GEN
747: factcantor0(GEN f, GEN pp, long flag)
748: {
749: long i,j,k,d,e,vf,p,nbfact,tetpil,av = avma;
750: GEN ex,y,f2,g,g1,u,v,pd,q;
751: GEN *t;
752:
753: if (!(d = factmod_init(&f, pp, &p))) { avma=av; return trivfact(); }
754: /* to hold factors and exponents */
755: t = (GEN*)cgetg(d+1,t_VEC); ex = new_chunk(d+1);
756: vf=varn(f); e = nbfact = 1;
757: for(;;)
758: {
759: f2 = FpX_gcd(f,derivpol(f), pp);
760: if (flag > 1 && lgef(f2) > 3) return NULL;
761: g1 = FpX_div(f,f2,pp);
762: k = 0;
763: while (lgef(g1)>3)
764: {
765: long du,dg;
766: GEN S;
767: k++; if (p && !(k%p)) { k++; f2 = FpX_div(f2,g1,pp); }
768: u = g1; g1 = FpX_gcd(f2,g1, pp);
769: if (lgef(g1)>3)
770: {
771: u = FpX_div( u,g1,pp);
772: f2= FpX_div(f2,g1,pp);
773: }
774: du = degpol(u);
775: if (du <= 0) continue;
776:
777: /* here u is square-free (product of irred. of multiplicity e * k) */
778: S = init_pow_p_mod_pT(pp, u);
779: pd=gun; v=polx[vf];
780: for (d=1; d <= du>>1; d++)
781: {
782: if (!flag) pd = mulii(pd,pp);
783: v = spec_FpXQ_pow(v, pp, S);
784: g = FpX_gcd(gadd(v, gneg(polx[vf])), u, pp);
785: dg = degpol(g);
786: if (dg <= 0) continue;
787:
788: /* Ici g est produit de pol irred ayant tous le meme degre d; */
789: j=nbfact+dg/d;
790:
791: if (flag)
792: {
793: if (flag > 1) return NULL;
794: for ( ; nbfact<j; nbfact++) { t[nbfact]=(GEN)d; ex[nbfact]=e*k; }
795: }
796: else
797: {
798: long r;
799: g = FpX_normalize(g, pp);
800: t[nbfact]=g; q = subis(pd,1); /* also ok for p=2: unused */
801: r = vali(q); q = shifti(q,-r);
802: /* le premier parametre est un entier variable m qui sera
803: * converti en un polynome w dont les coeff sont ses digits en
804: * base p (initialement m = p --> X) pour faire pgcd de g avec
805: * w^(p^d-1)/2 jusqu'a casser. p = 2 is treated separately.
806: */
807: if (p)
808: split(p,t+nbfact,d,pp,q,r,S);
809: else
810: splitgen(pp,t+nbfact,d,pp,q,r);
811: for (; nbfact<j; nbfact++) ex[nbfact]=e*k;
812: }
813: du -= dg;
814: u = FpX_div(u,g,pp);
815: v = FpX_res(v,u,pp);
816: }
817: if (du)
818: {
819: t[nbfact] = flag? (GEN)du: FpX_normalize(u, pp);
820: ex[nbfact++]=e*k;
821: }
822: }
823: j = lgef(f2); if (j==3) break;
824:
825: e*=p; j=(j-3)/p+3; setlg(f,j); setlgef(f,j);
826: for (i=2; i<j; i++) f[i]=f2[p*(i-2)+2];
827: }
828: if (flag > 1) { avma = av; return gun; } /* irreducible */
829: tetpil=avma; y=cgetg(3,t_MAT);
830: if (!flag)
831: {
832: y[1]=(long)t; setlg(t, nbfact);
833: y[2]=(long)ex; (void)sort_factor(y,cmpii);
834: }
835: u=cgetg(nbfact,t_COL); y[1]=(long)u;
836: v=cgetg(nbfact,t_COL); y[2]=(long)v;
837: if (flag)
838: for (j=1; j<nbfact; j++)
839: {
840: u[j] = lstoi((long)t[j]);
841: v[j] = lstoi(ex[j]);
842: }
843: else
844: for (j=1; j<nbfact; j++)
845: {
846: u[j] = (long)FpX(t[j], pp);
847: v[j] = lstoi(ex[j]);
848: }
849: return gerepile(av,tetpil,y);
850: }
851:
852: GEN
853: factcantor(GEN f, GEN p)
854: {
855: return factcantor0(f,p,0);
856: }
857:
858: GEN
859: simplefactmod(GEN f, GEN p)
860: {
861: return factcantor0(f,p,1);
862: }
863:
864: /* vector of polynomials (in v) whose coeffs are given by the columns of x */
865: GEN
866: mat_to_vecpol(GEN x, long v)
867: {
868: long i,j, lx = lg(x), lcol = lg(x[1]);
869: GEN y = cgetg(lx, t_VEC);
870:
871: for (j=1; j<lx; j++)
872: {
873: GEN p1, col = (GEN)x[j];
874: long k = lcol;
875:
876: while (k-- && gcmp0((GEN)col[k]));
877: i=k+2; p1=cgetg(i,t_POL);
878: p1[1] = evalsigne(1) | evallgef(i) | evalvarn(v);
879: col--; for (k=2; k<i; k++) p1[k] = col[k];
880: y[j] = (long)p1;
881: }
882: return y;
883: }
884:
885: /* matrix whose entries are given by the coeffs of the polynomials in
886: * vector v (considered as degree n-1 polynomials) */
887: GEN
888: vecpol_to_mat(GEN v, long n)
889: {
890: long i,j,d,N = lg(v);
891: GEN p1,w, y = cgetg(N, t_MAT);
892: if (typ(v) != t_VEC) err(typeer,"vecpol_to_mat");
893: n++;
894: for (j=1; j<N; j++)
895: {
896: p1 = cgetg(n,t_COL); y[j] = (long)p1;
897: w = (GEN)v[j];
898: if (typ(w) != t_POL) { p1[1] = (long)w; i=2; }
899: else
900: {
901: d=lgef(w)-1; w++;
902: for (i=1; i<d; i++) p1[i] = w[i];
903: }
904: for ( ; i<n; i++) p1[i] = zero;
905: }
906: return y;
907: }
908: /* polynomial (in v) of polynomials (in w) whose coeffs are given by the columns of x */
909: GEN
910: mat_to_polpol(GEN x, long v,long w)
911: {
912: long i,j, lx = lg(x), lcol = lg(x[1]);
913: GEN y = cgetg(lx+1, t_POL);
914: y[1]=evalsigne(1) | evallgef(lx+1) | evalvarn(v);
915: y++;
916: for (j=1; j<lx; j++)
917: {
918: GEN p1, col = (GEN)x[j];
919: long k;
920: i=lcol+1; p1=cgetg(i,t_POL);
921: p1[1] = evalsigne(1) | evallgef(i) | evalvarn(w);
922: col--; for (k=2; k<i; k++) p1[k] = col[k];
923: y[j] = (long)normalizepol_i(p1,i);
924: }
925: return normalizepol_i(--y,lx+1);
926: }
927:
928: /* matrix whose entries are given by the coeffs of the polynomial v in
929: * two variables (considered as degree n polynomials) */
930: GEN
931: polpol_to_mat(GEN v, long n)
932: {
933: long i,j,d,N = lgef(v)-1;
934: GEN p1,w, y = cgetg(N, t_MAT);
935: if (typ(v) != t_POL) err(typeer,"polpol_to_mat");
936: n++;v++;
937: for (j=1; j<N; j++)
938: {
939: p1 = cgetg(n,t_COL); y[j] = (long)p1;
940: w = (GEN)v[j];
941: if (typ(w) != t_POL) { p1[1] = (long)w; i=2; }
942: else
943: {
944: d=lgef(w)-1; w++;
945: for (i=1; i<d; i++) p1[i] = w[i];
946: }
947: for ( ; i<n; i++) p1[i] = zero;
948: }
949: return y;
950: }
951:
952:
953: /* set x <-- x + c*y mod p */
954: static void
955: split_berlekamp_addmul(GEN x, GEN y, long c, long p)
956: {
957: long i,lx,ly,l;
958: if (!c) return;
959: lx = lgef(x); ly = lgef(y); l = min(lx,ly);
960: if (p & ~MAXHALFULONG)
961: {
962: for (i=2; i<l; i++) x[i] = ((ulong)x[i]+ (ulong)mulssmod(c,y[i],p)) % p;
963: for ( ; i<ly; i++) x[i] = mulssmod(c,y[i],p);
964: }
965: else
966: {
967: for (i=2; i<l; i++) x[i] = ((ulong)x[i] + (ulong)(c*y[i])) % p;
968: for ( ; i<ly; i++) x[i] = (c*y[i]) % p;
969: }
970: do i--; while (i>1 && !x[i]);
971: if (i==1) setsigne(x,0); else { setsigne(x,1); setlgef(x,i+1); }
972: }
973:
974: long
975: split_berlekamp(GEN *t, GEN pp, GEN pps2)
976: {
977: GEN u = *t, p1, p2, vker,pol;
978: long av,N = degpol(u), d,i,kk,l1,l2,p, vu = varn(u);
979: ulong av0 = avma;
980:
981: vker = Berlekamp_ker(u,pp);
982: vker = mat_to_vecpol(vker,vu);
983: d = lg(vker)-1;
984: p = is_bigint(pp)? 0: pp[2];
985: if (p)
986: {
987: avma = av0; p1 = cgetg(d+1, t_VEC); /* hack: hidden gerepile */
988: for (i=1; i<=d; i++) p1[i] = (long)pol_to_small((GEN)vker[i]);
989: vker = p1;
990: }
991: pol = cgetg(N+3,t_POL);
992:
993: for (kk=1; kk<d; )
994: {
995: GEN polt;
996: if (p)
997: {
998: if (p==2)
999: {
1000: pol[2] = ((mymyrand() & 0x1000) == 0);
1001: pol[1] = evallgef(pol[2]? 3: 2);
1002: for (i=2; i<=d; i++)
1003: split_berlekamp_addmul(pol,(GEN)vker[i],(mymyrand()&0x1000)?0:1, p);
1004: }
1005: else
1006: {
1007: pol[2] = mymyrand()%p; /* vker[1] = 1 */
1008: pol[1] = evallgef(pol[2]? 3: 2);
1009: for (i=2; i<=d; i++)
1010: split_berlekamp_addmul(pol,(GEN)vker[i],mymyrand()%p, p);
1011: }
1012: polt = small_to_pol(pol,vu);
1013: }
1014: else
1015: {
1016: pol[2] = (long)genrand(pp);
1017: pol[1] = evallgef(signe(pol[2])? 3: 2) | evalvarn(vu);
1018: for (i=2; i<=d; i++)
1019: pol = gadd(pol, gmul((GEN)vker[i], genrand(pp)));
1020: polt = FpX_red(pol,pp);
1021: }
1022: for (i=1; i<=kk && kk<d; i++)
1023: {
1024: p1=t[i-1]; l1=degpol(p1);
1025: if (l1>1)
1026: {
1027: av = avma; p2 = FpX_res(polt, p1, pp);
1028: if (lgef(p2) <= 3) { avma=av; continue; }
1029: p2 = FpXQ_pow(p2,pps2, p1,pp);
1030: if (!signe(p2)) err(talker,"%Z not a prime in split_berlekamp",pp);
1031: p2 = ZX_s_add(p2, -1);
1032: p2 = FpX_gcd(p1,p2, pp); l2=degpol(p2);
1033: if (l2>0 && l2<l1)
1034: {
1035: p2 = FpX_normalize(p2, pp);
1036: t[i-1] = p2; kk++;
1037: t[kk-1] = FpX_div(p1,p2,pp);
1038: if (DEBUGLEVEL > 7) msgtimer("new factor");
1039: }
1040: else avma = av;
1041: }
1042: }
1043: }
1044: return d;
1045: }
1046:
1047: GEN
1048: factmod0(GEN f, GEN pp)
1049: {
1050: long i,j,k,e,p,N,nbfact,av = avma,tetpil,d;
1051: GEN pps2,ex,y,f2,p1,g1,u, *t;
1052:
1053: if (!(d = factmod_init(&f, pp, &p))) { avma=av; return trivfact(); }
1054: /* to hold factors and exponents */
1055: t = (GEN*)cgetg(d+1,t_VEC); ex = cgetg(d+1,t_VECSMALL);
1056: e = nbfact = 1;
1057: pps2 = shifti(pp,-1);
1058:
1059: for(;;)
1060: {
1061: f2 = FpX_gcd(f,derivpol(f), pp);
1062: g1 = lgef(f2)==3? f: FpX_div(f,f2,pp);
1063: k = 0;
1064: while (lgef(g1)>3)
1065: {
1066: k++; if (p && !(k%p)) { k++; f2 = FpX_div(f2,g1,pp); }
1067: p1 = FpX_gcd(f2,g1, pp); u = g1; g1 = p1;
1068: if (lgef(p1)!=3)
1069: {
1070: u = FpX_div( u,p1,pp);
1071: f2= FpX_div(f2,p1,pp);
1072: }
1073: N = degpol(u);
1074: if (N)
1075: {
1076: /* here u is square-free (product of irred. of multiplicity e * k) */
1077: t[nbfact] = FpX_normalize(u,pp);
1078: d = (N==1)? 1: split_berlekamp(t+nbfact, pp, pps2);
1079: for (j=0; j<d; j++) ex[nbfact+j] = e*k;
1080: nbfact += d;
1081: }
1082: }
1083: if (!p) break;
1084: j=(degpol(f2))/p+3; if (j==3) break;
1085:
1086: e*=p; setlg(f,j); setlgef(f,j);
1087: for (i=2; i<j; i++) f[i] = f2[p*(i-2)+2];
1088: }
1089: tetpil=avma; y=cgetg(3,t_VEC);
1090: setlg((GEN)t, nbfact);
1091: setlg(ex, nbfact);
1092: y[1]=lcopy((GEN)t);
1093: y[2]=lcopy(ex);
1094: (void)sort_factor(y,cmpii);
1095: return gerepile(av,tetpil,y);
1096: }
1097: GEN
1098: factmod(GEN f, GEN pp)
1099: {
1100: long tetpil,av=avma;
1101: long nbfact;
1102: long j;
1103: GEN y,u,v;
1104: GEN z=factmod0(f,pp),t=(GEN)z[1],ex=(GEN)z[2];
1105: nbfact=lg(t);
1106: tetpil=avma; y=cgetg(3,t_MAT);
1107: u=cgetg(nbfact,t_COL); y[1]=(long)u;
1108: v=cgetg(nbfact,t_COL); y[2]=(long)v;
1109: for (j=1; j<nbfact; j++)
1110: {
1111: u[j] = (long)FpX((GEN)t[j], pp);
1112: v[j] = lstoi(ex[j]);
1113: }
1114: return gerepile(av,tetpil,y);
1115: }
1116: GEN
1117: factormod0(GEN f, GEN p, long flag)
1118: {
1119: switch(flag)
1120: {
1121: case 0: return factmod(f,p);
1122: case 1: return simplefactmod(f,p);
1123: default: err(flagerr,"factormod");
1124: }
1125: return NULL; /* not reached */
1126: }
1127:
1128: /*******************************************************************/
1129: /* */
1130: /* Recherche de racines p-adiques */
1131: /* */
1132: /*******************************************************************/
1133: /* make f suitable for [root|factor]padic */
1134: static GEN
1135: padic_pol_to_int(GEN f)
1136: {
1137: long i, l = lgef(f);
1138: f = gdiv(f,content(f));
1139: for (i=2; i<l; i++)
1140: switch(typ(f[i]))
1141: {
1142: case t_INT: break;
1143: case t_PADIC: f[i] = ltrunc((GEN)f[i]); break;
1144: default: err(talker,"incorrect coeffs in padic_pol_to_int");
1145: }
1146: return f;
1147: }
1148:
1149: /* return invlead * (x + O(pr)), x in Z or Z_p, pr = p^r */
1150: static GEN
1151: int_to_padic(GEN x, GEN p, GEN pr, long r, GEN invlead)
1152: {
1153: GEN p1,y;
1154: long v,sx, av = avma;
1155:
1156: if (typ(x) == t_PADIC)
1157: {
1158: v = valp(x);
1159: if (r >= precp(x) + v) return invlead? gmul(x, invlead): gcopy(x);
1160: sx = !gcmp0(x);
1161: p1 = (GEN)x[4];
1162: }
1163: else
1164: {
1165: sx = signe(x);
1166: if (!sx) return gzero;
1167: v = pvaluation(x,p,&p1);
1168: }
1169: y = cgetg(5,t_PADIC);
1170: if (sx && v < r)
1171: {
1172: y[4] = lmodii(p1,pr); r -= v;
1173: }
1174: else
1175: {
1176: y[4] = zero; v = r; r = 0;
1177: }
1178: y[3] = (long)pr;
1179: y[2] = (long)p;
1180: y[1] = evalprecp(r)|evalvalp(v);
1181: return invlead? gerepileupto(av, gmul(invlead,y)): y;
1182: }
1183:
1184: /* return (x + O(p^r)) normalized (multiply by a unit such that leading coeff
1185: * is a power of p), x in Z[X] (or Z_p[X]) */
1186: static GEN
1187: pol_to_padic(GEN x, GEN pr, GEN p, long r)
1188: {
1189: long v = 0,i,lx = lgef(x);
1190: GEN z = cgetg(lx,t_POL), lead = leading_term(x);
1191:
1192: if (gcmp1(lead)) lead = NULL;
1193: else
1194: {
1195: long av = avma;
1196: v = ggval(lead,p);
1197: if (v) lead = gdiv(lead, gpowgs(p,v));
1198: lead = int_to_padic(lead,p,pr,r,NULL);
1199: lead = gerepileupto(av, ginv(lead));
1200: }
1201: for (i=lx-1; i>1; i--)
1202: z[i] = (long)int_to_padic((GEN)x[i],p,pr,r,lead);
1203: z[1] = x[1]; return z;
1204: }
1205:
1206: static GEN
1207: padic_trivfact(GEN x, GEN p, long r)
1208: {
1209: GEN p1, y = cgetg(3,t_MAT);
1210: p1=cgetg(2,t_COL); y[1]=(long)p1; p = icopy(p);
1211: p1[1]=(long)pol_to_padic(x,gpowgs(p,r),p,r);
1212: p1=cgetg(2,t_COL); y[2]=(long)p1;
1213: p1[1]=un; return y;
1214: }
1215:
1216: /* a etant un p-adique, retourne le vecteur des racines p-adiques de f
1217: * congrues a a modulo p dans le cas ou on suppose f(a) congru a 0 modulo p
1218: * (ou a 4 si p=2).
1219: */
1220: GEN
1221: apprgen(GEN f, GEN a)
1222: {
1223: GEN fp,p1,p,P,pro,x,x2,u,ip;
1224: long av=avma,tetpil,v,Ps,i,j,k,lu,n,fl2;
1225:
1226: if (typ(f)!=t_POL) err(notpoler,"apprgen");
1227: if (gcmp0(f)) err(zeropoler,"apprgen");
1228: if (typ(a) != t_PADIC) err(rootper1);
1229: f = padic_pol_to_int(f);
1230: fp=derivpol(f); p1=ggcd(f,fp);
1231: if (lgef(p1)>3) { f=gdeuc(f,p1); fp=derivpol(f); }
1232: p=(GEN)a[2]; p1=poleval(f,a);
1233: v=ggval(p1,p); if (v <= 0) err(rootper2);
1234: fl2=egalii(p,gdeux);
1235: if (fl2 && v==1) err(rootper2);
1236: v=ggval(poleval(fp,a),p);
1237: if (!v) /* simple zero */
1238: {
1239: while (!gcmp0(p1))
1240: {
1241: a = gsub(a,gdiv(p1,poleval(fp,a)));
1242: p1 = poleval(f,a);
1243: }
1244: tetpil=avma; pro=cgetg(2,t_VEC); pro[1]=lcopy(a);
1245: return gerepile(av,tetpil,pro);
1246: }
1247: n=degpol(f); pro=cgetg(n+1,t_VEC);
1248:
1249: if (is_bigint(p)) err(impl,"apprgen for p>=2^31");
1250: x = ggrandocp(p, valp(a) | precp(a));
1251: if (fl2)
1252: {
1253: x2=ggrandocp(p,2); P = stoi(4);
1254: }
1255: else
1256: {
1257: x2=ggrandocp(p,1); P = p;
1258: }
1259: f = poleval(f, gadd(a,gmul(P,polx[varn(f)])));
1260: if (!gcmp0(f)) f = gdiv(f,gpuigs(p,ggval(f, p)));
1261: Ps = itos(P);
1262: for (j=0,i=0; i<Ps; i++)
1263: {
1264: ip=stoi(i);
1265: if (gcmp0(poleval(f,gadd(ip,x2))))
1266: {
1267: u=apprgen(f,gadd(x,ip)); lu=lg(u);
1268: for (k=1; k<lu; k++)
1269: {
1270: j++; pro[j]=ladd(a,gmul(P,(GEN)u[k]));
1271: }
1272: }
1273: }
1274: setlg(pro,j+1); return gerepilecopy(av,pro);
1275: }
1276:
1277: /* Retourne le vecteur des racines p-adiques de f en precision r */
1278: GEN
1279: rootpadic(GEN f, GEN p, long r)
1280: {
1281: GEN fp,y,z,p1,pr,rac;
1282: long lx,i,j,k,n,av=avma,tetpil,fl2;
1283:
1284: if (typ(f)!=t_POL) err(notpoler,"rootpadic");
1285: if (gcmp0(f)) err(zeropoler,"rootpadic");
1286: if (r<=0) err(rootper4);
1287: f = padic_pol_to_int(f);
1288: fp=derivpol(f); p1=ggcd(f,fp);
1289: if (lgef(p1)>3) { f=gdeuc(f,p1); fp=derivpol(f); }
1290: fl2=egalii(p,gdeux); rac=(fl2 && r>=2)? rootmod(f,stoi(4)): rootmod(f,p);
1291: lx=lg(rac); p=gclone(p);
1292: if (r==1)
1293: {
1294: tetpil=avma; y=cgetg(lx,t_COL);
1295: for (i=1; i<lx; i++)
1296: {
1297: z=cgetg(5,t_PADIC); y[i]=(long)z;
1298: z[1] = evalprecp(1)|evalvalp(0);
1299: z[2] = z[3] = (long)p;
1300: z[4] = lcopy(gmael(rac,i,2));
1301: }
1302: return gerepile(av,tetpil,y);
1303: }
1304: n=degpol(f); y=cgetg(n+1,t_COL);
1305: j=0; pr = NULL;
1306: z = cgetg(5,t_PADIC);
1307: z[2] = (long)p;
1308: for (i=1; i<lx; i++)
1309: {
1310: p1 = gmael(rac,i,2);
1311: if (signe(p1))
1312: {
1313: if (!fl2 || mod2(p1))
1314: {
1315: z[1] = evalvalp(0)|evalprecp(r);
1316: z[4] = (long)p1;
1317: }
1318: else
1319: {
1320: z[1] = evalvalp(1)|evalprecp(r);
1321: z[4] = un;
1322: }
1323: if (!pr) pr=gpuigs(p,r);
1324: z[3] = (long)pr;
1325: }
1326: else
1327: {
1328: z[1] = evalvalp(r);
1329: z[3] = un;
1330: z[4] = (long)p1;
1331: }
1332: p1 = apprgen(f,z);
1333: for (k=1; k<lg(p1); k++) y[++j]=p1[k];
1334: }
1335: setlg(y,j+1); return gerepilecopy(av,y);
1336: }
1337: /*************************************************************************/
1338: /* rootpadicfast */
1339: /*************************************************************************/
1340:
1341: /*lift accelerator. The author of the idea is unknown.*/
1342: long hensel_lift_accel(long n, long *pmask)
1343: {
1344: long a,j;
1345: long mask;
1346: mask=0;
1347: for(j=BITS_IN_LONG-1, a=n ;; j--)
1348: {
1349: mask|=(a&1)<<j;
1350: a=(a>>1)+(a&1);
1351: if (a==1) break;
1352: }
1353: *pmask=mask>>j;
1354: return BITS_IN_LONG-j;
1355: }
1356: /*
1357: SPEC:
1358: q is an integer > 1
1359: e>=0
1360: f in ZZ[X], with leading term prime to q.
1361: S must be a simple root mod p for all p|q.
1362:
1363: return roots of f mod q^e, as integers (implicitly mod Q)
1364: */
1365:
1366: /* STANDARD USE
1367: There exists p a prime number and a>0 such that
1368: q=p^a
1369: f in ZZ[X], with leading term prime to p.
1370: S must be a simple root mod p.
1371:
1372: return p-adics roots of f with prec b, as integers (implicitly mod q^e)
1373: */
1374:
1375: GEN
1376: rootpadiclift(GEN T, GEN S, GEN p, long e)
1377: {
1378: ulong ltop=avma;
1379: long x;
1380: GEN qold, q, qm1;
1381: GEN W, Tr, Sr, Wr = gzero;
1382: long i, nb, mask;
1383: x = varn(T);
1384: qold = p ; q = p; qm1 = gun;
1385: nb=hensel_lift_accel(e, &mask);
1386: Tr = FpX_red(T,q);
1387: W=FpX_eval(deriv(Tr, x),S,q);
1388: W=mpinvmod(W,q);
1389: for(i=0;i<nb;i++)
1390: {
1391: qm1 = (mask&(1<<i))?sqri(qm1):mulii(qm1, q);
1392: q = mulii(qm1, p);
1393: Tr = FpX_red(T,q);
1394: Sr = S;
1395: if (i)
1396: {
1397: W = modii(mulii(Wr,FpX_eval(deriv(Tr,x),Sr,qold)),qold);
1398: W = subii(gdeux,W);
1399: W = modii(mulii(Wr, W),qold);
1400: }
1401: Wr = W;
1402: S = subii(Sr, mulii(Wr, FpX_eval(Tr, Sr,q)));
1403: S = modii(S,q);
1404: qold = q;
1405: }
1406: return gerepileupto(ltop,S);
1407: }
1408: /*
1409: * Apply rootpadiclift to all roots in S and trace trick.
1410: * Elements of S must be distinct simple roots mod p for all p|q.
1411: */
1412:
1413: GEN
1414: rootpadicliftroots(GEN f, GEN S, GEN q, long e)
1415: {
1416: GEN y;
1417: long i,n=lg(S);
1418: if (n==1)
1419: return gcopy(S);
1420: y=cgetg(n,typ(S));
1421: for (i=1; i<n-1; i++)
1422: y[i]=(long) rootpadiclift(f, (GEN) S[i], q, e);
1423: if (n!=lgef(f)-2)/* non totally split*/
1424: y[n-1]=(long) rootpadiclift(f, (GEN) S[n-1], q, e);
1425: else/* distinct-->totally split-->use trace trick */
1426: {
1427: ulong av=avma;
1428: GEN z;
1429: z=(GEN)f[lgef(f)-2];/*-trace(roots)*/
1430: for(i=1; i<n-1;i++)
1431: z=addii(z,(GEN) y[i]);
1432: z=modii(negi(z),gpowgs(q,e));
1433: y[n-1]=lpileupto(av,z);
1434: }
1435: return y;
1436: }
1437: /*
1438: p is a prime number, pr a power of p,
1439:
1440: f in ZZ[X], with leading term prime to p.
1441: f must have no multiple roots mod p.
1442:
1443: return p-adics roots of f with prec pr, as integers (implicitly mod pr)
1444:
1445: */
1446: GEN
1447: rootpadicfast(GEN f, GEN p, long e)
1448: {
1449: ulong ltop=avma;
1450: GEN S,y;
1451: S=lift(rootmod(f,p));/*no multiplicity*/
1452: if (lg(S)==1)/*no roots*/
1453: {
1454: avma=ltop;
1455: return cgetg(1,t_COL);
1456: }
1457: S=gclone(S);
1458: avma=ltop;
1459: y=rootpadicliftroots(f,S,p,e);
1460: gunclone(S);
1461: return y;
1462: }
1463: /* Same as rootpadiclift for the polynomial X^n-a,
1464: * but here, n can be much larger.
1465: * TODO: generalize to sparse polynomials.
1466: */
1467: GEN
1468: padicsqrtnlift(GEN a, GEN n, GEN S, GEN p, long e)
1469: {
1470: ulong ltop=avma;
1471: GEN qold, q, qm1;
1472: GEN W, Sr, Wr = gzero;
1473: long i, nb, mask;
1474: qold = p ; q = p; qm1 = gun;
1475: nb = hensel_lift_accel(e, &mask);
1476: W = modii(mulii(n,powmodulo(S,subii(n,gun),q)),q);
1477: W = mpinvmod(W,q);
1478: for(i=0;i<nb;i++)
1479: {
1480: qm1 = (mask&(1<<i))?sqri(qm1):mulii(qm1, q);
1481: q = mulii(qm1, p);
1482: Sr = S;
1483: if (i)
1484: {
1485: W = modii(mulii(Wr,mulii(n,powmodulo(Sr,subii(n,gun),qold))),qold);
1486: W = subii(gdeux,W);
1487: W = modii(mulii(Wr, W),qold);
1488: }
1489: Wr = W;
1490: S = subii(Sr, mulii(Wr, subii(powmodulo(Sr,n,q),a)));
1491: S = modii(S,q);
1492: qold = q;
1493: }
1494: return gerepileupto(ltop,S);
1495: }
1496: /**************************************************************************/
1497: static long
1498: getprec(GEN x, long prec, GEN *p)
1499: {
1500: long i,e;
1501: GEN p1;
1502:
1503: for (i = lgef(x)-1; i>1; i--)
1504: {
1505: p1=(GEN)x[i];
1506: if (typ(p1)==t_PADIC)
1507: {
1508: e=valp(p1); if (signe(p1[4])) e += precp(p1);
1509: if (e<prec) prec = e; *p = (GEN)p1[2];
1510: }
1511: }
1512: return prec;
1513: }
1514:
1515: /* a appartenant a une extension finie de Q_p, retourne le vecteur des
1516: * racines de f congrues a a modulo p dans le cas ou on suppose f(a) congru a
1517: * 0 modulo p (ou a 4 si p=2).
1518: */
1519: GEN
1520: apprgen9(GEN f, GEN a)
1521: {
1522: GEN fp,p1,p,pro,x,x2,u,ip,t,vecg;
1523: long av=avma,tetpil,v,ps_1,i,j,k,lu,n,prec,d,va,fl2;
1524:
1525: if (typ(f)!=t_POL) err(notpoler,"apprgen9");
1526: if (gcmp0(f)) err(zeropoler,"apprgen9");
1527: if (typ(a)==t_PADIC) return apprgen(f,a);
1528: if (typ(a)!=t_POLMOD || typ(a[2])!=t_POL) err(rootper1);
1529: fp=derivpol(f); p1=ggcd(f,fp);
1530: if (lgef(p1)>3) { f=gdeuc(f,p1); fp=derivpol(f); }
1531: t=(GEN)a[1];
1532: prec = getprec((GEN)a[2], BIGINT, &p);
1533: prec = getprec(t, prec, &p);
1534: if (prec==BIGINT) err(rootper1);
1535:
1536: p1=poleval(f,a); v=ggval(lift_intern(p1),p); if (v<=0) err(rootper2);
1537: fl2=egalii(p,gdeux);
1538: if (fl2 && v==1) err(rootper2);
1539: v=ggval(lift_intern(poleval(fp,a)), p);
1540: if (!v)
1541: {
1542: while (!gcmp0(p1))
1543: {
1544: a = gsub(a, gdiv(p1,poleval(fp,a)));
1545: p1 = poleval(f,a);
1546: }
1547: tetpil=avma; pro=cgetg(2,t_COL); pro[1]=lcopy(a);
1548: return gerepile(av,tetpil,pro);
1549: }
1550: n=degpol(f); pro=cgetg(n+1,t_COL); j=0;
1551:
1552: if (is_bigint(p)) err(impl,"apprgen9 for p>=2^31");
1553: x=gmodulcp(ggrandocp(p,prec), t);
1554: if (fl2)
1555: {
1556: ps_1=3; x2=ggrandocp(p,2); p=stoi(4);
1557: }
1558: else
1559: {
1560: ps_1=itos(p)-1; x2=ggrandocp(p,1);
1561: }
1562: f = poleval(f,gadd(a,gmul(p,polx[varn(f)])));
1563: if (!gcmp0(f)) f=gdiv(f,gpuigs(p,ggval(f,p)));
1564: d=degpol(t); vecg=cgetg(d+1,t_COL);
1565: for (i=1; i<=d; i++)
1566: vecg[i] = (long)setloop(gzero);
1567: va=varn(t);
1568: for(;;) /* loop through F_q */
1569: {
1570: ip=gmodulcp(gtopoly(vecg,va),t);
1571: if (gcmp0(poleval(f,gadd(ip,x2))))
1572: {
1573: u=apprgen9(f,gadd(ip,x)); lu=lg(u);
1574: for (k=1; k<lu; k++)
1575: {
1576: j++; pro[j]=ladd(a,gmul(p,(GEN)u[k]));
1577: }
1578: }
1579: for (i=d; i; i--)
1580: {
1581: p1 = (GEN)vecg[i];
1582: if (p1[2] != ps_1) { (void)incloop(p1); break; }
1583: affsi(0, p1);
1584: }
1585: if (!i) break;
1586: }
1587: setlg(pro,j+1); return gerepilecopy(av,pro);
1588: }
1589:
1590: /*****************************************/
1591: /* Factorisation p-adique d'un polynome */
1592: /*****************************************/
1593: int
1594: cmp_padic(GEN x, GEN y)
1595: {
1596: long vx, vy;
1597: if (x == gzero) return -1;
1598: if (y == gzero) return 1;
1599: vx = valp(x);
1600: vy = valp(y);
1601: if (vx < vy) return 1;
1602: if (vx > vy) return -1;
1603: return cmpii((GEN)x[4], (GEN)y[4]);
1604: }
1605:
1606: /* factorise le polynome T=nf[1] dans Zp avec la precision pr */
1607: static GEN
1608: padicff2(GEN nf,GEN p,long pr)
1609: {
1610: long N=degpol(nf[1]),i,j,d,l;
1611: GEN mat,V,D,fa,p1,pk,dec_p,pke,a,theta;
1612:
1613: pk=gpuigs(p,pr); dec_p=primedec(nf,p);
1614: l=lg(dec_p); fa=cgetg(l,t_COL);
1615: for (i=1; i<l; i++)
1616: {
1617: p1 = (GEN)dec_p[i];
1618: pke = idealpows(nf,p1, pr * itos((GEN)p1[3]));
1619: p1=smith2(pke); V=(GEN)p1[3]; D=(GEN)p1[1];
1620: for (d=1; d<=N; d++)
1621: if (! egalii(gcoeff(V,d,d),pk)) break;
1622: a=ginv(D); theta=gmael(nf,8,2); mat=cgetg(d,t_MAT);
1623: for (j=1; j<d; j++)
1624: {
1625: p1 = gmul(D, element_mul(nf,theta,(GEN)a[j]));
1626: setlg(p1,d); mat[j]=(long)p1;
1627: }
1628: fa[i]=(long)caradj(mat,0,NULL);
1629: }
1630: a = cgetg(l,t_COL); pk = icopy(pk);
1631: for (i=1; i<l; i++)
1632: a[i] = (long)pol_to_padic((GEN)fa[i],pk,p,pr);
1633: return a;
1634: }
1635:
1636: static GEN
1637: padicff(GEN x,GEN p,long pr)
1638: {
1639: GEN q,basden,bas,invbas,mul,dx,nf,mat;
1640: long n=degpol(x),av=avma;
1641:
1642: nf=cgetg(10,t_VEC); nf[1]=(long)x; dx=discsr(x);
1643: mat=cgetg(3,t_MAT); mat[1]=lgetg(3,t_COL); mat[2]=lgetg(3,t_COL);
1644: coeff(mat,1,1)=(long)p; coeff(mat,1,2)=lstoi(pvaluation(dx,p,&q));
1645: coeff(mat,2,1)=(long)q; coeff(mat,2,2)=un;
1646: bas=allbase4(x,(long)mat,(GEN*)(nf+3),NULL);
1647: if (!carrecomplet(divii(dx,(GEN)nf[3]),(GEN*)(nf+4)))
1648: err(bugparier,"factorpadic2 (incorrect discriminant)");
1649: basden = get_bas_den(bas);
1650: invbas = QM_inv(vecpol_to_mat(bas,n), gun);
1651: mul = get_mul_table(x,basden,invbas,NULL);
1652: nf[7]=(long)bas;
1653: nf[8]=(long)invbas;
1654: nf[9]=(long)mul; nf[2]=nf[5]=nf[6]=zero;
1655: return gerepileupto(av,padicff2(nf,p,pr));
1656: }
1657:
1658: GEN
1659: factorpadic2(GEN x, GEN p, long r)
1660: {
1661: long av=avma,av2,k,i,j,i1,f,nbfac;
1662: GEN res,p1,p2,y,d,a,ap,t,v,w;
1663: GEN *fa;
1664:
1665: if (typ(x)!=t_POL) err(notpoler,"factorpadic");
1666: if (gcmp0(x)) err(zeropoler,"factorpadic");
1667: if (r<=0) err(rootper4);
1668:
1669: if (lgef(x)==3) return trivfact();
1670: if (!gcmp1(leading_term(x)))
1671: err(impl,"factorpadic2 for non-monic polynomial");
1672: if (lgef(x)==4) return padic_trivfact(x,p,r);
1673: y=cgetg(3,t_MAT);
1674: fa = (GEN*)new_chunk(lgef(x)-2);
1675: d=content(x); a=gdiv(x,d);
1676: ap=derivpol(a); t=ggcd(a,ap); v=gdeuc(a,t);
1677: w=gdeuc(ap,t); j=0; f=1; nbfac=0;
1678: while (f)
1679: {
1680: j++; w=gsub(w,derivpol(v)); f=signe(w);
1681: if (f) { res=ggcd(v,w); v=gdeuc(v,res); w=gdeuc(w,res); }
1682: else res=v;
1683: fa[j]=(lgef(res)>3) ? padicff(res,p,r) : cgetg(1,t_COL);
1684: nbfac += (lg(fa[j])-1);
1685: }
1686: av2=avma; y=cgetg(3,t_MAT);
1687: p1=cgetg(nbfac+1,t_COL); y[1]=(long)p1;
1688: p2=cgetg(nbfac+1,t_COL); y[2]=(long)p2;
1689: for (i=1,k=0; i<=j; i++)
1690: for (i1=1; i1<lg(fa[i]); i1++)
1691: {
1692: p1[++k]=lcopy((GEN)fa[i][i1]); p2[k]=lstoi(i);
1693: }
1694: y = gerepile(av,av2,y);
1695: sort_factor(y, cmp_padic); return y;
1696: }
1697:
1698: /*******************************************************************/
1699: /* */
1700: /* FACTORISATION P-adique avec ROUND 4 */
1701: /* */
1702: /*******************************************************************/
1703: extern GEN Decomp(GEN p,GEN f,long mf,GEN theta,GEN chi,GEN nu,long r);
1704: extern GEN nilord(GEN p, GEN fx, long mf, GEN gx, long flag);
1705: extern GEN hensel_lift_fact(GEN pol, GEN Q, GEN T, GEN p, GEN pev, long e);
1706:
1707: static GEN
1708: squarefree(GEN f, GEN *ex)
1709: {
1710: GEN T,V,W,A,B;
1711: long n,i,k;
1712:
1713: T=ggcd(derivpol(f),f); V=gdeuc(f,T);
1714: n=lgef(f)-2; A=cgetg(n,t_COL); B=cgetg(n,t_COL);
1715: k=1; i=1;
1716: do
1717: {
1718: W=ggcd(T,V); T=gdeuc(T,W);
1719: if (lgef(V) != lgef(W))
1720: {
1721: A[i]=ldeuc(V,W); B[i]=k; i++;
1722: }
1723: k++; V=W;
1724: }
1725: while (lgef(V)>3);
1726: setlg(A,i); *ex=B; return A;
1727: }
1728:
1729: #define swap(x,y) { long _t=x; x=y; y=_t; }
1730:
1731: /* reverse x in place */
1732: static void
1733: polreverse(GEN x)
1734: {
1735: long i, j;
1736: if (typ(x) != t_POL) err(typeer,"polreverse");
1737: for (i=2, j=lgef(x)-1; i<j; i++, j--) swap(x[i], x[j]);
1738: (void)normalizepol(x);
1739: }
1740:
1741: GEN
1742: factorpadic4(GEN f,GEN p,long prec)
1743: {
1744: GEN w,g,poly,fx,y,p1,p2,ex,pols,exps,ppow,lead;
1745: long v=varn(f),n=degpol(f),av,tetpil,mfx,i,k,j,r,pr;
1746: int reverse = 0;
1747:
1748: if (typ(f)!=t_POL) err(notpoler,"factorpadic");
1749: if (typ(p)!=t_INT) err(arither1);
1750: if (gcmp0(f)) err(zeropoler,"factorpadic");
1751: if (prec<=0) err(rootper4);
1752:
1753: if (n==0) return trivfact();
1754: av=avma; f = padic_pol_to_int(f);
1755: if (n==1) return gerepileupto(av, padic_trivfact(f,p,prec));
1756: lead = leading_term(f); pr = prec;
1757: if (!gcmp1(lead))
1758: {
1759: long val = ggval(lead,p), val1 = ggval(constant_term(f),p);
1760: if (val1 < val)
1761: {
1762: reverse = 1; polreverse(f);
1763: /* take care of loss of precision from leading coeff of factor
1764: * (whose valuation is <= val) */
1765: pr += val;
1766: val = val1;
1767: }
1768: pr += val * (n-1);
1769: }
1770: f = pol_to_monic(f, &lead);
1771:
1772: poly=squarefree(f,&ex);
1773: pols=cgetg(n+1,t_COL);
1774: exps=cgetg(n+1,t_COL); n = lg(poly);
1775: for (j=1,i=1; i<n; i++)
1776: {
1777: long av1 = avma;
1778: fx=(GEN)poly[i]; mfx=ggval(discsr(fx),p);
1779: w = (GEN)factmod(fx,p)[1];
1780: if (!mfx)
1781: { /* no repeated factors: Hensel lift */
1782: p1 = hensel_lift_fact(fx, lift_intern(w), NULL, p, gpowgs(p,pr), pr);
1783: p2 = stoi(ex[i]);
1784: for (k=1; k<lg(p1); k++,j++)
1785: {
1786: pols[j] = p1[k];
1787: exps[j] = (long)p2;
1788: }
1789: continue;
1790: }
1791: /* use Round 4 */
1792: r = lg(w)-1;
1793: g = lift_intern((GEN)w[r]);
1794: p2 = (r == 1)? nilord(p,fx,mfx,g,pr)
1795: : Decomp(p,fx,mfx,polx[v],fx,g, (pr<=mfx)? mfx+1: pr);
1796: if (p2)
1797: {
1798: p2 = gerepileupto(av1,p2);
1799: p1 = (GEN)p2[1];
1800: p2 = (GEN)p2[2];
1801: for (k=1; k<lg(p1); k++,j++)
1802: {
1803: pols[j]=p1[k];
1804: exps[j]=lmulis((GEN)p2[k],ex[i]);
1805: }
1806: }
1807: else
1808: {
1809: avma=av1;
1810: pols[j]=(long)fx;
1811: exps[j]=lstoi(ex[i]); j++;
1812: }
1813: }
1814: if (lead)
1815: {
1816: p1 = gmul(polx[v],lead);
1817: for (i=1; i<j; i++)
1818: {
1819: p2 = poleval((GEN)pols[i], p1);
1820: pols[i] = ldiv(p2, content(p2));
1821: }
1822: }
1823:
1824: tetpil=avma; y=cgetg(3,t_MAT);
1825: p1 = cgetg(j,t_COL); ppow = gpowgs(p,prec); p = icopy(p);
1826: for (i=1; i<j; i++)
1827: {
1828: if (reverse) polreverse((GEN)pols[i]);
1829: p1[i] = (long)pol_to_padic((GEN)pols[i],ppow,p,prec);
1830: }
1831: y[1]=(long)p1; setlg(exps,j);
1832: y[2]=lcopy(exps); y = gerepile(av,tetpil,y);
1833: sort_factor(y, cmp_padic); return y;
1834: }
1835:
1836: GEN
1837: factorpadic0(GEN f,GEN p,long r,long flag)
1838: {
1839: switch(flag)
1840: {
1841: case 0: return factorpadic4(f,p,r);
1842: case 1: return factorpadic2(f,p,r);
1843: default: err(flagerr,"factorpadic");
1844: }
1845: return NULL; /* not reached */
1846: }
1847:
1848: /*******************************************************************/
1849: /* */
1850: /* FACTORISATION DANS F_q */
1851: /* */
1852: /*******************************************************************/
1853: extern GEN to_Kronecker(GEN P, GEN Q);
1854: extern GEN from_Kronecker(GEN z, GEN pol);
1855: static GEN spec_Fq_pow_mod_pol(GEN x, GEN p, GEN a, GEN S);
1856:
1857: static GEN
1858: to_fq(GEN x, GEN a, GEN p)
1859: {
1860: long i,lx = lgef(x);
1861: GEN z = cgetg(3,t_POLMOD), pol = cgetg(lx,t_POL);
1862: pol[1] = x[1];
1863: if (lx == 2) setsigne(pol, 0);
1864: else
1865: for (i=2; i<lx; i++) pol[i] = (long)mod((GEN)x[i], p);
1866: /* assume deg(pol) < deg(a) */
1867: z[1] = (long)a;
1868: z[2] = (long)pol; return z;
1869: }
1870:
1871: /* x POLMOD over Fq, return lift(x^n) */
1872: static GEN
1873: Kronecker_powmod(GEN x, GEN mod, GEN n)
1874: {
1875: long lim,av,av0 = avma, i,j,m,v = varn(x);
1876: GEN y, p1, p = NULL, pol = NULL;
1877:
1878: for (i=lgef(mod)-1; i>1; i--)
1879: {
1880: p1 = (GEN)mod[i];
1881: if (typ(p1) == t_POLMOD) { pol = (GEN)p1[1] ; break; }
1882: }
1883: if (!pol) err(talker,"need POLMOD coeffs in Kronecker_powmod");
1884: for (i=lgef(pol)-1; i>1; i--)
1885: {
1886: p1 = (GEN)pol[i];
1887: if (typ(p1) == t_INTMOD) { p = (GEN)p1[1] ; break; }
1888: }
1889: if (!p) err(talker,"need Fq coeffs in Kronecker_powmod");
1890: x = lift_intern(to_Kronecker(x,pol));
1891:
1892: /* adapted from powgi */
1893: av=avma; lim=stack_lim(av,1);
1894: p1 = n+2; m = *p1;
1895:
1896: y=x; j=1+bfffo(m); m<<=j; j = BITS_IN_LONG-j;
1897: for (i=lgefint(n)-2;;)
1898: {
1899: for (; j; m<<=1,j--)
1900: {
1901: y = gsqr(y);
1902: y = from_Kronecker(FpX(y,p), pol);
1903: setvarn(y, v);
1904: y = gres(y, mod);
1905: y = lift_intern(to_Kronecker(y,pol));
1906:
1907: if (m<0)
1908: {
1909: y = gmul(y,x);
1910: y = from_Kronecker(FpX(y,p), pol);
1911: setvarn(y, v);
1912: y = gres(y, mod);
1913: y = lift_intern(to_Kronecker(y,pol));
1914: }
1915: if (low_stack(lim, stack_lim(av,1)))
1916: {
1917: if(DEBUGMEM>1) err(warnmem,"Kronecker_powmod");
1918: y = gerepilecopy(av, y);
1919: }
1920: }
1921: if (--i == 0) break;
1922: m = *++p1, j = BITS_IN_LONG;
1923: }
1924: y = from_Kronecker(FpX(y,p),pol);
1925: setvarn(y, v); return gerepileupto(av0, y);
1926: }
1927:
1928: GEN
1929: FpX_rand(long d1, long v, GEN p)
1930: {
1931: long i, d = d1+2;
1932: GEN y;
1933: y = cgetg(d,t_POL); y[1] = evalsigne(1) | evalvarn(v);
1934: for (i=2; i<d; i++) y[i] = (long)genrand(p);
1935: normalizepol_i(y,d); return y;
1936: }
1937:
1938: /* return a random polynomial in F_q[v], degree < d1 */
1939: static GEN
1940: FqX_rand(long d1, long v, GEN p, GEN a)
1941: {
1942: long i,j, d = d1+2, k = lgef(a)-1;
1943: GEN y,t;
1944:
1945: y = cgetg(d,t_POL); y[1] = evalsigne(1) | evalvarn(v);
1946: t = cgetg(k,t_POL); t[1] = a[1];
1947: for (i=2; i<d; i++)
1948: {
1949: for (j=2; j<k; j++) t[j] = (long)genrand(p);
1950: normalizepol_i(t,k); y[i]=(long)to_fq(t,a,p);
1951: }
1952: normalizepol_i(y,d); return y;
1953: }
1954:
1955: /* split into r factors of degree d */
1956: static void
1957: split9(GEN *t, long d, GEN p, GEN q, GEN a, GEN S)
1958: {
1959: long l,v,av,is2,cnt, dt = degpol(*t), da = degpol(a);
1960: GEN w,w0;
1961:
1962: if (dt == d) return;
1963: v = varn(*t);
1964: if (DEBUGLEVEL > 6) timer2();
1965: av = avma; is2 = egalii(p, gdeux);
1966: for(cnt = 1;;cnt++)
1967: { /* splits *t with probability ~ 1 - 2^(1-r) */
1968: w = w0 = FqX_rand(dt,v, p,a);
1969: for (l=1; l<d; l++) /* sum_{0<i<d} w^(q^i), result in (F_q)^r */
1970: w = gadd(w0, spec_Fq_pow_mod_pol(w, p, a, S));
1971: if (is2)
1972: {
1973: w0 = w;
1974: for (l=1; l<da; l++) /* sum_{0<i<k} w^(2^i), result in (F_2)^r */
1975: w = gadd(w0, gres(gsqr(w), *t));
1976: }
1977: else
1978: {
1979: w = Kronecker_powmod(w, *t, shifti(q,-1));
1980: /* w in {-1,0,1}^r */
1981: if (lgef(w) == 3) continue;
1982: w[2] = ladd((GEN)w[2], gun);
1983: }
1984: w = ggcd(*t,w); l = degpol(w);
1985: if (l && l != dt) break;
1986: avma = av;
1987: }
1988: w = gerepileupto(av,w);
1989: if (DEBUGLEVEL > 6)
1990: fprintferr("[split9] time for splitting: %ld (%ld trials)\n",timer2(),cnt);
1991: l /= d; t[l]=gdeuc(*t,w); *t=w;
1992: split9(t+l,d,p,q,a,S);
1993: split9(t ,d,p,q,a,S);
1994: }
1995:
1996: /* to "compare" (real) scalars and t_INTMODs */
1997: static int
1998: cmp_coeff(GEN x, GEN y)
1999: {
2000: if (typ(x) == t_INTMOD) x = (GEN)x[2];
2001: if (typ(y) == t_INTMOD) y = (GEN)y[2];
2002: return gcmp(x,y);
2003: }
2004:
2005: int
2006: cmp_pol(GEN x, GEN y)
2007: {
2008: long fx[3], fy[3];
2009: long i,lx,ly;
2010: int fl;
2011: if (typ(x) == t_POLMOD) x = (GEN)x[2];
2012: if (typ(y) == t_POLMOD) y = (GEN)y[2];
2013: if (typ(x) == t_POL) lx = lgef(x); else { lx = 3; fx[2] = (long)x; x = fx; }
2014: if (typ(y) == t_POL) ly = lgef(y); else { ly = 3; fy[2] = (long)y; y = fy; }
2015: if (lx > ly) return 1;
2016: if (lx < ly) return -1;
2017: for (i=lx-1; i>1; i--)
2018: if ((fl = cmp_coeff((GEN)x[i], (GEN)y[i]))) return fl;
2019: return 0;
2020: }
2021:
2022: /* assume n > 1, X a POLMOD over Fq */
2023: /* return S = [ X^q, X^2q, ... X^(n-1)q ] mod T (in Fq[X]) in Kronecker form */
2024: static GEN
2025: init_pow_q_mod_pT(GEN Xmod, GEN q, GEN a, GEN T)
2026: {
2027: long i, n = degpol(T);
2028: GEN p1, S = cgetg(n, t_VEC);
2029:
2030: S[1] = (long)Kronecker_powmod((GEN)Xmod[2], (GEN)Xmod[1], q);
2031: #if 1 /* use as many squarings as possible */
2032: for (i=2; i < n; i+=2)
2033: {
2034: p1 = gsqr((GEN)S[i>>1]);
2035: S[i] = lres(p1, T);
2036: if (i == n-1) break;
2037: p1 = gmul((GEN)S[i], (GEN)S[1]);
2038: S[i+1] = lres(p1, T);
2039: }
2040: #else
2041: for (i=2; i < n; i++)
2042: {
2043: p1 = gmul((GEN)S[i-1], (GEN)S[1]);
2044: S[i] = lres(p1, T);
2045: }
2046: #endif
2047: for (i=1; i < n; i++)
2048: S[i] = (long)lift_intern(to_Kronecker((GEN)S[i], a));
2049: return S;
2050: }
2051:
2052: /* compute x^q, S is as above */
2053: static GEN
2054: spec_Fq_pow_mod_pol(GEN x, GEN p, GEN a, GEN S)
2055: {
2056: long av = avma, lim = stack_lim(av,1), i,dx = degpol(x);
2057: GEN x0 = x+2, z,c;
2058:
2059: c = (GEN)x0[0];
2060: z = lift_intern(lift(c));
2061: for (i = 1; i <= dx; i++)
2062: {
2063: GEN d;
2064: c = (GEN)x0[i];
2065: if (gcmp0(c)) continue;
2066: d = (GEN)S[i];
2067: if (!gcmp1(c)) d = gmul(lift_intern(lift(c)),d);
2068: z = gadd(z, d);
2069: if (low_stack(lim, stack_lim(av,1)))
2070: {
2071: if(DEBUGMEM>1) err(warnmem,"spec_Fq_pow_mod_pol");
2072: z = gerepileupto(av, z);
2073: }
2074: }
2075: z = FpX(z, p);
2076: z = from_Kronecker(z, a);
2077: setvarn(z, varn(x)); return gerepileupto(av, z);
2078: }
2079:
2080: static long isabsolutepol(GEN f, GEN pp, GEN a)
2081: {
2082: int i,res=1;
2083: GEN c;
2084: for(i=2; i<lg(f); i++)
2085: {
2086: c = (GEN) f[i];
2087: switch(typ(c))
2088: {
2089: case t_INT: /* OK*/
2090: break;
2091: case t_INTMOD:
2092: if (gcmp((GEN)c[1],pp))
2093: err(typeer,"factmod9");
2094: break;
2095: case t_POLMOD:
2096: if (gcmp((GEN)c[1],a))
2097: err(typeer,"factmod9");
2098: isabsolutepol((GEN)c[1],pp,gzero);
2099: isabsolutepol((GEN)c[2],pp,gzero);
2100: if (degpol(c[1])>0)
2101: res = 0;
2102: break;
2103: case t_POL:
2104: isabsolutepol(c,pp,gzero);
2105: if (degpol(c)>0)
2106: res = 0;
2107: break;
2108: default:
2109: err(typeer,"factmod9");
2110: }
2111: }
2112: return res;
2113: }
2114:
2115: GEN
2116: factmod9(GEN f, GEN pp, GEN a)
2117: {
2118: long av = avma, tetpil,p,i,j,k,d,e,vf,va,nbfact,nbf,pk;
2119: GEN S,ex,y,f2,f3,df1,df2,g,g1,xmod,u,v,qqd,qq,unfp,unfq, *t;
2120: GEN frobinv,X;
2121:
2122: if (typ(a)!=t_POL || typ(f)!=t_POL || gcmp0(a)) err(typeer,"factmod9");
2123: vf=varn(f); va=varn(a);
2124: if (va<=vf) err(talker,"polynomial variable must be of higher priority than finite field\nvariable in factorff");
2125: if (isabsolutepol(f, pp, a))
2126: {
2127: GEN z= Fp_factor_rel0(simplify(lift(lift(f))), pp, lift(a));
2128: GEN t=(GEN)z[1],ex=(GEN)z[2];
2129: unfp = gmodulsg(1,pp);
2130: unfq = gmodulcp(gmul(unfp, polun[va]), gmul(unfp,a));
2131: nbfact=lg(t);
2132: tetpil=avma;
2133: y=cgetg(3,t_MAT);
2134: u=cgetg(nbfact,t_COL); y[1]=(long)u;
2135: v=cgetg(nbfact,t_COL); y[2]=(long)v;
2136: for (j=1; j<nbfact; j++)
2137: {
2138: u[j] = lmul((GEN)t[j],unfq);
2139: v[j] = lstoi(ex[j]);
2140: }
2141: return gerepile(av,tetpil,y);
2142: }
2143: p = is_bigint(pp)? 0: itos(pp);
2144: unfp=gmodulsg(1,pp); a=gmul(unfp,a);
2145: unfq=gmodulo(gmul(unfp,polun[va]), a); a = (GEN)unfq[1];
2146: f = gmul(unfq,f); if (!signe(f)) err(zeropoler,"factmod9");
2147: d = degpol(f); if (!d) { avma=av; gunclone(a); return trivfact(); }
2148:
2149: S = df2 = NULL; /* gcc -Wall */
2150: pp = gmael(a,2,1); /* out of the stack */
2151: t = (GEN*)cgetg(d+1,t_VEC); ex = new_chunk(d+1);
2152:
2153: frobinv = gpowgs(pp, lgef(a)-4);
2154: xmod = cgetg(3,t_POLMOD);
2155: X = gmul(polx[vf],unfq);
2156: xmod[2] = (long)X;
2157: qq=gpuigs(pp,degpol(a));
2158: e = nbfact = 1;
2159: pk=1; df1=derivpol(f); f3=NULL;
2160: for(;;)
2161: {
2162: long du,dg;
2163: while (gcmp0(df1))
2164: { /* needs d >= pp: p = 0 can't happen */
2165: pk *= p; e=pk;
2166: j=(degpol(f))/p+3; setlg(f,j); setlgef(f,j);
2167: for (i=2; i<j; i++) f[i] = (long)powgi((GEN)f[p*(i-2)+2], frobinv);
2168: df1=derivpol(f); f3=NULL;
2169: }
2170: f2 = f3? f3: ggcd(f,df1);
2171: if (lgef(f2)==3) u = f;
2172: else
2173: {
2174: g1=gdeuc(f,f2); df2=derivpol(f2);
2175: if (gcmp0(df2)) { u=g1; f3=f2; }
2176: else
2177: {
2178: f3=ggcd(f2,df2);
2179: if (lgef(f3)==3) u=gdeuc(g1,f2);
2180: else
2181: u=gdeuc(g1,gdeuc(f2,f3));
2182: }
2183: }
2184: /* u is square-free (product of irreducibles of multiplicity e) */
2185: qqd=gun; xmod[1]=(long)u;
2186:
2187: du = degpol(u); v = X;
2188: if (du > 1) S = init_pow_q_mod_pT(xmod, qq, a, u);
2189: for (d=1; d <= du>>1; d++)
2190: {
2191: qqd=mulii(qqd,qq);
2192: v = spec_Fq_pow_mod_pol(v, pp, a, S);
2193: g = ggcd(gsub(v,X),u);
2194: dg = degpol(g);
2195: if (dg <= 0) continue;
2196:
2197: /* all factors of g have degree d */
2198: j = nbfact+dg/d;
2199:
2200: t[nbfact] = g;
2201: split9(t+nbfact,d,pp,qq,a,S);
2202: for (; nbfact<j; nbfact++) ex[nbfact]=e;
2203: du -= dg;
2204: u = gdeuc(u,g);
2205: v = gres(v,u); xmod[1] = (long)u;
2206: }
2207: if (du) { t[nbfact]=u; ex[nbfact++]=e; }
2208: if (lgef(f2) == 3) break;
2209:
2210: f=f2; df1=df2; e += pk;
2211: }
2212:
2213: nbf=nbfact; tetpil=avma; y=cgetg(3,t_MAT);
2214: for (j=1; j<nbfact; j++)
2215: {
2216: t[j]=gdiv((GEN)t[j],leading_term(t[j]));
2217: for (k=1; k<j; k++)
2218: if (ex[k] && gegal(t[j],t[k]))
2219: {
2220: ex[k] += ex[j]; ex[j]=0;
2221: nbf--; break;
2222: }
2223: }
2224: u=cgetg(nbf,t_COL); y[1]=(long)u;
2225: v=cgetg(nbf,t_COL); y[2]=(long)v;
2226: for (j=1,k=0; j<nbfact; j++)
2227: if (ex[j])
2228: {
2229: k++;
2230: u[k]=(long)t[j];
2231: v[k]=lstoi(ex[j]);
2232: }
2233: y = gerepile(av,tetpil,y);
2234: u=(GEN)y[1];
2235: { /* put a back on the stack */
2236: GEN tokill = a;
2237: a = forcecopy(a);
2238: gunclone(tokill);
2239: }
2240: pp = (GEN)leading_term(a)[1];
2241: for (j=1; j<nbf; j++) fqunclone((GEN)u[j], a, pp);
2242: (void)sort_factor(y, cmp_pol); return y;
2243: }
2244: /* See also: Isomorphisms between finite field and relative
2245: * factorization in polarit3.c */
2246:
2247: /*******************************************************************/
2248: /* */
2249: /* RACINES COMPLEXES */
2250: /* l represente la longueur voulue pour les parties */
2251: /* reelles et imaginaires des racines de x */
2252: /* */
2253: /*******************************************************************/
2254: GEN square_free_factorization(GEN pol);
2255: static GEN laguer(GEN pol,long N,GEN y0,GEN EPS,long PREC);
2256: GEN zrhqr(GEN a,long PREC);
2257:
2258: GEN
2259: rootsold(GEN x, long l)
2260: {
2261: long av1=avma,i,j,f,g,gg,fr,deg,l0,l1,l2,l3,l4,ln;
2262: long exc,expmin,m,deg0,k,ti,h,ii,e,e1,emax,v;
2263: GEN y,xc,xd0,xd,xdabs,p1,p2,p3,p4,p5,p6,p7,p8;
2264: GEN p9,p10,p11,p12,p14,p15,pa,pax,pb,pp,pq,ps;
2265:
2266: if (typ(x)!=t_POL) err(typeer,"rootsold");
2267: v=varn(x); deg0=degpol(x); expmin=12 - bit_accuracy(l);
2268: if (!signe(x)) err(zeropoler,"rootsold");
2269: y=cgetg(deg0+1,t_COL); if (!deg0) return y;
2270: for (i=1; i<=deg0; i++)
2271: {
2272: p1=cgetg(3,t_COMPLEX); p1[1]=lgetr(l); p1[2]=lgetr(l); y[i]=(long)p1;
2273: for (j=3; j<l; j++) ((GEN)p1[2])[j]=((GEN)p1[1])[j]=0;
2274: }
2275: g=1; gg=1;
2276: for (i=2; i<=deg0+2; i++)
2277: {
2278: ti=typ(x[i]);
2279: if (ti==t_REAL) gg=0;
2280: else if (ti==t_QUAD)
2281: {
2282: p2=gmael3(x,i,1,2);
2283: if (gsigne(p2)>0) g=0;
2284: } else if (ti != t_INT && ti != t_INTMOD && !is_frac_t(ti)) g=0;
2285: }
2286: l1=avma; p2=cgetg(3,t_COMPLEX);
2287: p2[1]=lmppi(DEFAULTPREC); p2[2]=ldivrs((GEN)p2[1],10);
2288: p11=cgetg(4,t_POL); p11[1]=evalsigne(1)+evallgef(4);
2289: setvarn(p11,v); p11[3]=un;
2290:
2291: p12=cgetg(5,t_POL); p12[1]=evalsigne(1)+evallgef(5);
2292: setvarn(p12,v); p12[4]=un;
2293: for (i=2; i<=deg0+2 && gcmp0((GEN)x[i]); i++) gaffsg(0,(GEN)y[i-1]);
2294: k=i-2;
2295: if (k!=deg0)
2296: {
2297: if (k)
2298: {
2299: j=deg0+3-k; pax=cgetg(j,t_POL);
2300: pax[1] = evalsigne(1) | evalvarn(v) | evallgef(j);
2301: for (i=2; i<j; i++) pax[i]=x[i+k];
2302: }
2303: else pax=x;
2304: xd0=deriv(pax,v); m=1; pa=pax;
2305: pq = NULL; /* for lint */
2306: if (gg) { pp=ggcd(pax,xd0); h=isnonscalar(pp); if (h) pq=gdeuc(pax,pp); }
2307: else{ pp=gun; h=0; }
2308: do
2309: {
2310: if (h)
2311: {
2312: pa=pp; pb=pq; pp=ggcd(pa,deriv(pa,v)); h=isnonscalar(pp);
2313: if (h) pq=gdeuc(pa,pp); else pq=pa; ps=gdeuc(pb,pq);
2314: }
2315: else ps=pa;
2316: /* calcul des racines d'ordre exactement m */
2317: deg=degpol(ps);
2318: if (deg)
2319: {
2320: l3=avma; e=gexpo((GEN)ps[deg+2]); emax=e;
2321: for (i=2; i<deg+2; i++)
2322: {
2323: p3=(GEN)(ps[i]);
2324: e1=gexpo(p3); if (e1>emax) emax=e1;
2325: }
2326: e=emax-e; if (e<0) e=0; avma=l3; if (ps!=pax) xd0=deriv(ps,v);
2327: xdabs=cgetg(deg+2,t_POL); xdabs[1]=xd0[1];
2328: for (i=2; i<deg+2; i++)
2329: {
2330: l3=avma; p3=(GEN)xd0[i];
2331: p4=gabs(greal(p3),l);
2332: p5=gabs(gimag(p3),l); l4=avma;
2333: xdabs[i]=lpile(l3,l4,gadd(p4,p5));
2334: }
2335: l0=avma; xc=gcopy(ps); xd=gcopy(xd0); l2=avma;
2336: for (i=1; i<=deg; i++)
2337: {
2338: if (i==deg)
2339: {
2340: p1=(GEN)y[k+m*i]; gdivz(gneg_i((GEN)xc[2]),(GEN)xc[3],p1);
2341: p14=(GEN)p1[1]; p15=(GEN)p1[2];
2342: }
2343: else
2344: {
2345: p3=gshift(p2,e); p4=poleval(xc,p3); p5=gnorm(p4); exc=0;
2346: while (exc >= -20)
2347: {
2348: p7 = gneg_i(gdiv(p4, poleval(xd,p3)));
2349: l3 = avma;
2350: if (gcmp0(p5)) exc = -32;
2351: else exc = expo(gnorm(p7))-expo(gnorm(p3));
2352: avma = l3;
2353: for (j=1; j<=10; j++)
2354: {
2355: p8=gadd(p3,p7); p9=poleval(xc,p8); p10=gnorm(p9);
2356: if (exc < -20 || cmprr(p10,p5) < 0)
2357: {
2358: GEN *gptr[3];
2359: p3=p8; p4=p9; p5=p10;
2360: gptr[0]=&p3; gptr[1]=&p4; gptr[2]=&p5;
2361: gerepilemanysp(l2,l3,gptr,3);
2362: break;
2363: }
2364: gshiftz(p7,-2,p7); avma=l3;
2365: }
2366: if (j > 10)
2367: {
2368: avma=av1;
2369: if (DEBUGLEVEL)
2370: {
2371: fprintferr("too many iterations in rootsold(): ");
2372: fprintferr("using roots2()\n"); flusherr();
2373: }
2374: return roots2(x,l);
2375: }
2376: }
2377: p1=(GEN)y[k+m*i]; setlg(p1[1],3); setlg(p1[2],3); gaffect(p3,p1);
2378: avma=l2; p14=(GEN)p1[1]; p15=(GEN)p1[2];
2379: for (ln=4; ln<=l; ln=(ln<<1)-2)
2380: {
2381: setlg(p14,ln); setlg(p15,ln);
2382: if (gcmp0(p14)) { settyp(p14,t_INT); p14[1]=2; }
2383: if (gcmp0(p15)) { settyp(p15,t_INT); p15[1]=2; }
2384: p4=poleval(xc,p1);
2385: p5=poleval(xd,p1); p6=gneg_i(gdiv(p4,p5));
2386: settyp(p14,t_REAL); settyp(p15,t_REAL);
2387: gaffect(gadd(p1,p6),p1); avma=l2;
2388: }
2389: }
2390: setlg(p14,l); setlg(p15,l);
2391: p7=gcopy(p1); p14=(GEN)(p7[1]); p15=(GEN)(p7[2]);
2392: setlg(p14,l+1); setlg(p15,l+1);
2393: if (gcmp0(p14)) { settyp(p14,t_INT); p14[1]=2; }
2394: if (gcmp0(p15)) { settyp(p15,t_INT); p15[1]=2; }
2395: for (ii=1; ii<=5; ii++)
2396: {
2397: p4=poleval(ps,p7); p5=poleval(xd0,p7);
2398: p6=gneg_i(gdiv(p4,p5)); p7=gadd(p7,p6);
2399: p14=(GEN)(p7[1]); p15=(GEN)(p7[2]);
2400: if (gcmp0(p14)) { settyp(p14,t_INT); p14[1]=2; }
2401: if (gcmp0(p15)) { settyp(p15,t_INT); p15[1]=2; }
2402: }
2403: gaffect(p7,p1); p4=poleval(ps,p7);
2404: p6=gdiv(p4,poleval(xdabs,gabs(p7,l)));
2405: if (gexpo(p6)>=expmin)
2406: {
2407: avma=av1;
2408: if (DEBUGLEVEL)
2409: {
2410: fprintferr("internal error in rootsold(): using roots2()\n");
2411: flusherr();
2412: }
2413: return roots2(x,l);
2414: }
2415: avma=l2;
2416: if (expo(p1[2])<expmin && g)
2417: {
2418: gaffect(gzero,(GEN)p1[2]);
2419: for (j=1; j<m; j++) gaffect(p1,(GEN)y[k+(i-1)*m+j]);
2420: p11[2]=lneg((GEN)p1[1]);
2421: l4=avma; xc=gerepile(l0,l4,gdeuc(xc,p11));
2422: }
2423: else
2424: {
2425: for (j=1; j<m; j++) gaffect(p1,(GEN)y[k+(i-1)*m+j]);
2426: if (g)
2427: {
2428: p1=gconj(p1);
2429: for (j=1; j<=m; j++) gaffect(p1,(GEN)y[k+i*m+j]);
2430: i++;
2431: p12[2]=lnorm(p1); p12[3]=lmulsg(-2,(GEN)p1[1]); l4=avma;
2432: xc=gerepile(l0,l4,gdeuc(xc,p12));
2433: }
2434: else
2435: {
2436: p11[2]=lneg(p1); l4=avma;
2437: xc=gerepile(l0,l4,gdeuc(xc,p11));
2438: }
2439: }
2440: xd=deriv(xc,v); l2=avma;
2441: }
2442: k += deg*m;
2443: }
2444: m++;
2445: }
2446: while (k!=deg0);
2447: }
2448: avma=l1;
2449: for (j=2; j<=deg0; j++)
2450: {
2451: p1 = (GEN)y[j];
2452: if (gcmp0((GEN)p1[2])) fr=0; else fr=1;
2453: for (k=j-1; k>=1; k--)
2454: {
2455: p2 = (GEN)y[k];
2456: if (gcmp0((GEN)p2[2])) f=0; else f=1;
2457: if (f<fr) break;
2458: if (f==fr && gcmp((GEN)p2[1],(GEN)p1[1]) <= 0) break;
2459: y[k+1]=y[k];
2460: }
2461: y[k+1]=(long)p1;
2462: }
2463: return y;
2464: }
2465:
2466: GEN
2467: roots2(GEN pol,long PREC)
2468: {
2469: ulong av = avma;
2470: long N,flagexactpol,flagrealpol,flagrealrac,ti,i,j;
2471: long nbpol,k,av1,multiqol,deg,nbroot,fr,f;
2472: GEN p1,p2,rr,EPS,qol,qolbis,x,b,c,*ad,v,tabqol;
2473:
2474: if (typ(pol)!=t_POL) err(typeer,"roots2");
2475: if (!signe(pol)) err(zeropoler,"roots2");
2476: N=degpol(pol);
2477: if (!N) return cgetg(1,t_COL);
2478: if (N==1)
2479: {
2480: p1=gmul(realun(PREC),(GEN)pol[3]);
2481: p2=gneg_i(gdiv((GEN)pol[2],p1));
2482: return gerepilecopy(av,p2);
2483: }
2484: EPS=realun(3); setexpo(EPS, 12 - bit_accuracy(PREC));
2485: flagrealpol=1; flagexactpol=1;
2486: for (i=2; i<=N+2; i++)
2487: {
2488: ti=typ(pol[i]);
2489: if (ti!=t_INT && ti!=t_INTMOD && !is_frac_t(ti))
2490: {
2491: flagexactpol=0;
2492: if (ti!=t_REAL) flagrealpol=0;
2493: }
2494: if (ti==t_QUAD)
2495: {
2496: p1=gmael3(pol,i,1,2);
2497: flagrealpol = (gsigne(p1)>0)? 0 : 1;
2498: }
2499: }
2500: rr=cgetg(N+1,t_COL);
2501: for (i=1; i<=N; i++)
2502: {
2503: p1 = cgetc(PREC); rr[i] = (long)p1;
2504: for (j=3; j<PREC; j++) mael(p1,2,j)=mael(p1,1,j)=0;
2505: }
2506: if (flagexactpol) tabqol=square_free_factorization(pol);
2507: else
2508: {
2509: tabqol=cgetg(3,t_MAT);
2510: tabqol[1]=lgetg(2,t_COL); mael(tabqol,1,1)=un;
2511: tabqol[2]=lgetg(2,t_COL); mael(tabqol,2,1)=lcopy(pol);
2512: }
2513: nbpol=lg(tabqol[1])-1; nbroot=0;
2514: for (k=1; k<=nbpol; k++)
2515: {
2516: av1=avma; qol=gmael(tabqol,2,k); qolbis=gcopy(qol);
2517: multiqol=itos(gmael(tabqol,1,k)); deg=degpol(qol);
2518: for (j=deg; j>=1; j--)
2519: {
2520: x=gzero; flagrealrac=0;
2521: if (j==1) x=gneg_i(gdiv((GEN)qolbis[2],(GEN)qolbis[3]));
2522: else
2523: {
2524: x=laguer(qolbis,j,x,EPS,PREC);
2525: if (x == NULL) goto RLAB;
2526: }
2527: if (flagexactpol)
2528: {
2529: x=gprec(x,(long)((PREC-1)*pariK));
2530: x=laguer(qol,deg,x,gmul2n(EPS,-32),PREC+1);
2531: }
2532: else x=laguer(qol,deg,x,EPS,PREC);
2533: if (x == NULL) goto RLAB;
2534:
2535: if (typ(x)==t_COMPLEX &&
2536: gcmp(gabs(gimag(x),PREC),gmul2n(gmul(EPS,gabs(greal(x),PREC)),1))<=0)
2537: { x[2]=zero; flagrealrac=1; }
2538: else if (j==1 && flagrealpol)
2539: { x[2]=zero; flagrealrac=1; }
2540: else if (typ(x)!=t_COMPLEX) flagrealrac=1;
2541:
2542: for (i=1; i<=multiqol; i++) gaffect(x,(GEN)rr[nbroot+i]);
2543: nbroot+=multiqol;
2544: if (!flagrealpol || flagrealrac)
2545: {
2546: ad = (GEN*) new_chunk(j+1);
2547: for (i=0; i<=j; i++) ad[i]=(GEN)qolbis[i+2];
2548: b=(GEN)ad[j];
2549: for (i=j-1; i>=0; i--)
2550: {
2551: c=(GEN)ad[i]; ad[i]=b;
2552: b=gadd(gmul((GEN)rr[nbroot],b),c);
2553: }
2554: v=cgetg(j+1,t_VEC); for (i=1; i<=j; i++) v[i]=(long)ad[j-i];
2555: qolbis=gtopoly(v,varn(qolbis));
2556: if (flagrealpol)
2557: for (i=2; i<=j+1; i++)
2558: if (typ(qolbis[i])==t_COMPLEX) mael(qolbis,i,2)=zero;
2559: }
2560: else
2561: {
2562: ad = (GEN*) new_chunk(j-1); ad[j-2]=(GEN)qolbis[j+2];
2563: p1=gmulsg(2,greal((GEN)rr[nbroot])); p2=gnorm((GEN)rr[nbroot]);
2564: ad[j-3]=gadd((GEN)qolbis[j+1],gmul(p1,ad[j-2]));
2565: for (i=j-2; i>=2; i--)
2566: ad[i-2] = gadd((GEN)qolbis[i+2],gsub(gmul(p1,ad[i-1]),gmul(p2,ad[i])));
2567: v=cgetg(j,t_VEC); for (i=1; i<=j-1; i++) v[i]=(long)ad[j-1-i];
2568: qolbis=gtopoly(v,varn(qolbis));
2569: for (i=2; i<=j; i++)
2570: if (typ(qolbis[i])==t_COMPLEX) mael(qolbis,i,2)=zero;
2571: for (i=1; i<=multiqol; i++)
2572: gaffect(gconj((GEN)rr[nbroot]), (GEN)rr[nbroot+i]);
2573: nbroot+=multiqol; j--;
2574: }
2575: }
2576: avma=av1;
2577: }
2578: for (j=2; j<=N; j++)
2579: {
2580: x=(GEN)rr[j]; if (gcmp0((GEN)x[2])) fr=0; else fr=1;
2581: for (i=j-1; i>=1; i--)
2582: {
2583: if (gcmp0(gmael(rr,i,2))) f=0; else f=1;
2584: if (f<fr) break;
2585: if (f==fr && gcmp(greal((GEN)rr[i]),greal(x)) <= 0) break;
2586: rr[i+1]=rr[i];
2587: }
2588: rr[i+1]=(long)x;
2589: }
2590: return gerepilecopy(av,rr);
2591:
2592: RLAB:
2593: avma = av;
2594: for(i=2;i<=N+2;i++)
2595: {
2596: ti = typ(pol[i]);
2597: if (!is_intreal_t(ti)) err(talker,"too many iterations in roots");
2598: }
2599: if (DEBUGLEVEL)
2600: {
2601: fprintferr("too many iterations in roots2() ( laguer() ): \n");
2602: fprintferr(" real coefficients polynomial, using zrhqr()\n");
2603: flusherr();
2604: }
2605: return zrhqr(pol,PREC);
2606: }
2607:
2608: #define MR 8
2609: #define MT 10
2610:
2611: static GEN
2612: laguer(GEN pol,long N,GEN y0,GEN EPS,long PREC)
2613: {
2614: long av = avma, av1,MAXIT,iter,i,j;
2615: GEN rac,erre,I,x,abx,abp,abm,dx,x1,b,d,f,g,h,sq,gp,gm,g2,*ffrac;
2616:
2617: MAXIT=MR*MT; rac=cgetg(3,t_COMPLEX);
2618: rac[1]=lgetr(PREC); rac[2]=lgetr(PREC);
2619: av1 = avma;
2620: I=cgetg(3,t_COMPLEX); I[1]=un; I[2]=un;
2621: ffrac=(GEN*)new_chunk(MR+1); for (i=0; i<=MR; i++) ffrac[i]=cgetr(PREC);
2622: affrr(dbltor(0.0), ffrac[0]); affrr(dbltor(0.5), ffrac[1]);
2623: affrr(dbltor(0.25),ffrac[2]); affrr(dbltor(0.75),ffrac[3]);
2624: affrr(dbltor(0.13),ffrac[4]); affrr(dbltor(0.38),ffrac[5]);
2625: affrr(dbltor(0.62),ffrac[6]); affrr(dbltor(0.88),ffrac[7]);
2626: affrr(dbltor(1.0),ffrac[8]);
2627: x=y0;
2628: for (iter=1; iter<=MAXIT; iter++)
2629: {
2630: b=(GEN)pol[N+2]; erre=QuickNormL1(b,PREC);
2631: d=gzero; f=gzero; abx=QuickNormL1(x,PREC);
2632: for (j=N-1; j>=0; j--)
2633: {
2634: f=gadd(gmul(x,f),d); d=gadd(gmul(x,d),b);
2635: b=gadd(gmul(x,b),(GEN)pol[j+2]);
2636: erre=gadd(QuickNormL1(b,PREC),gmul(abx,erre));
2637: }
2638: erre=gmul(erre,EPS);
2639: if (gcmp(QuickNormL1(b,PREC),erre)<=0)
2640: {
2641: gaffect(x,rac); avma = av1; return rac;
2642: }
2643: g=gdiv(d,b); g2=gsqr(g); h=gsub(g2, gmul2n(gdiv(f,b),1));
2644: sq=gsqrt(gmulsg(N-1,gsub(gmulsg(N,h),g2)),PREC);
2645: gp=gadd(g,sq); gm=gsub(g,sq); abp=gnorm(gp); abm=gnorm(gm);
2646: if (gcmp(abp,abm)<0) gp=gcopy(gm);
2647: if (gsigne(gmax(abp,abm))==1)
2648: dx = gdivsg(N,gp);
2649: else
2650: dx = gmul(gadd(gun,abx),gexp(gmulgs(I,iter),PREC));
2651: x1=gsub(x,dx);
2652: if (gcmp(QuickNormL1(gsub(x,x1),PREC),EPS)<0)
2653: {
2654: gaffect(x,rac); avma = av1; return rac;
2655: }
2656: if (iter%MT) x=gcopy(x1); else x=gsub(x,gmul(ffrac[iter/MT],dx));
2657: }
2658: avma=av; return NULL;
2659: }
2660:
2661: #undef MR
2662: #undef MT
2663:
2664: /***********************************************************************/
2665: /** **/
2666: /** ROOTS of a polynomial with REAL coeffs **/
2667: /** **/
2668: /***********************************************************************/
2669: #define RADIX 1L
2670: #define COF 0.95
2671:
2672: /* ONLY FOR REAL COEFFICIENTS MATRIX : replace the matrix x with
2673: a symmetric matrix a with the same eigenvalues */
2674: static GEN
2675: balanc(GEN x)
2676: {
2677: ulong av = avma;
2678: long last,i,j, sqrdx = (RADIX<<1), n = lg(x);
2679: GEN r,c,cofgen,a;
2680:
2681: a = dummycopy(x);
2682: last = 0; cofgen = dbltor(COF);
2683: while (!last)
2684: {
2685: last = 1;
2686: for (i=1; i<n; i++)
2687: {
2688: r = c = gzero;
2689: for (j=1; j<n; j++)
2690: if (j!=i)
2691: {
2692: c = gadd(c, gabs(gcoeff(a,j,i),0));
2693: r = gadd(r, gabs(gcoeff(a,i,j),0));
2694: }
2695: if (!gcmp0(r) && !gcmp0(c))
2696: {
2697: GEN g, s = gmul(cofgen, gadd(c,r));
2698: long ex = 0;
2699: g = gmul2n(r,-RADIX); while (gcmp(c,g) < 0) {ex++; c=gmul2n(c, sqrdx);}
2700: g = gmul2n(r, RADIX); while (gcmp(c,g) > 0) {ex--; c=gmul2n(c,-sqrdx);}
2701: if (gcmp(gadd(c,r), gmul2n(s,ex)) < 0)
2702: {
2703: last = 0;
2704: for (j=1; j<n; j++) coeff(a,i,j)=lmul2n(gcoeff(a,i,j),-ex);
2705: for (j=1; j<n; j++) coeff(a,j,i)=lmul2n(gcoeff(a,j,i), ex);
2706: }
2707: }
2708: }
2709: }
2710: return gerepilecopy(av, a);
2711: }
2712:
2713: #define SIGN(a,b) ((b)>=0.0 ? fabs(a) : -fabs(a))
2714: static GEN
2715: hqr(GEN mat) /* find all the eigenvalues of the matrix mat */
2716: {
2717: long nn,n,m,l,k,j,its,i,mmin,flj,flk;
2718: double **a,p,q,r,s,t,u,v,w,x,y,z,anorm,*wr,*wi;
2719: const double eps = 0.000001;
2720: GEN eig;
2721:
2722: n=lg(mat)-1; a=(double**)gpmalloc(sizeof(double*)*(n+1));
2723: for (i=1; i<=n; i++) a[i]=(double*)gpmalloc(sizeof(double)*(n+1));
2724: for (j=1; j<=n; j++)
2725: for (i=1; i<=n; i++) a[i][j]=gtodouble((GEN)((GEN)mat[j])[i]);
2726: wr=(double*)gpmalloc(sizeof(double)*(n+1));
2727: wi=(double*)gpmalloc(sizeof(double)*(n+1));
2728:
2729: anorm=fabs(a[1][1]);
2730: for (i=2; i<=n; i++) for (j=(i-1); j<=n; j++) anorm+=fabs(a[i][j]);
2731: nn=n; t=0.0;
2732: if (DEBUGLEVEL>3) { fprintferr("* Finding eigenvalues\n"); flusherr(); }
2733: while (nn>=1)
2734: {
2735: its=0;
2736: do
2737: {
2738: for (l=nn; l>=2; l--)
2739: {
2740: s=fabs(a[l-1][l-1])+fabs(a[l][l]); if (s==0.0) s=anorm;
2741: if ((double)(fabs(a[l][l-1])+s)==s) break;
2742: }
2743: x=a[nn][nn];
2744: if (l==nn){ wr[nn]=x+t; wi[nn--]=0.0; }
2745: else
2746: {
2747: y=a[nn-1][nn-1];
2748: w=a[nn][nn-1]*a[nn-1][nn];
2749: if (l == nn-1)
2750: {
2751: p=0.5*(y-x); q=p*p+w; z=sqrt(fabs(q)); x+=t;
2752: if (q>=0.0 || fabs(q)<=eps)
2753: {
2754: z=p+SIGN(z,p); wr[nn-1]=wr[nn]=x+z;
2755: if (fabs(z)>eps) wr[nn]=x-w/z;
2756: wi[nn-1]=wi[nn]=0.0;
2757: }
2758: else{ wr[nn-1]=wr[nn]=x+p; wi[nn-1]=-(wi[nn]=z); }
2759: nn-=2;
2760: }
2761: else
2762: {
2763: p = q = r = 0.0; /* for lint */
2764: if (its==30) err(talker,"too many iterations in hqr");
2765: if (its==10 || its==20)
2766: {
2767: t+=x; for (i=1; i<=nn; i++) a[i][i]-=x;
2768: s = fabs(a[nn][nn-1]) + fabs(a[nn-1][nn-2]);
2769: y=x=0.75*s; w=-0.4375*s*s;
2770: }
2771: its++;
2772: for (m=nn-2; m>=l; m--)
2773: {
2774: z=a[m][m]; r=x-z; s=y-z;
2775: p=(r*s-w)/a[m+1][m]+a[m][m+1];
2776: q=a[m+1][m+1]-z-r-s;
2777: r=a[m+2][m+1]; s=fabs(p)+fabs(q)+fabs(r); p/=s; q/=s; r/=s;
2778: if (m==l) break;
2779: u=fabs(a[m][m-1])*(fabs(q)+fabs(r));
2780: v=fabs(p)*(fabs(a[m-1][m-1])+fabs(z)+fabs(a[m+1][m+1]));
2781: if ((double)(u+v)==v) break;
2782: }
2783: for (i=m+2; i<=nn; i++){ a[i][i-2]=0.0; if (i!=(m+2)) a[i][i-3]=0.0; }
2784: for (k=m; k<=nn-1; k++)
2785: {
2786: if (k!=m)
2787: {
2788: p=a[k][k-1]; q=a[k+1][k-1];
2789: r = (k != nn-1)? a[k+2][k-1]: 0.0;
2790: x = fabs(p)+fabs(q)+fabs(r);
2791: if (x != 0.0) { p/=x; q/=x; r/=x; }
2792: }
2793: s = SIGN(sqrt(p*p+q*q+r*r),p);
2794: if (s == 0.0) continue;
2795:
2796: if (k==m)
2797: { if (l!=m) a[k][k-1] = -a[k][k-1]; }
2798: else
2799: a[k][k-1] = -s*x;
2800: p+=s; x=p/s; y=q/s; z=r/s; q/=p; r/=p;
2801: for (j=k; j<=nn; j++)
2802: {
2803: p = a[k][j]+q*a[k+1][j];
2804: if (k != nn-1) { p+=r*a[k+2][j]; a[k+2][j]-=p*z; }
2805: a[k+1][j] -= p*y; a[k][j] -= p*x;
2806: }
2807: mmin = (nn < k+3)? nn: k+3;
2808: for (i=l; i<=mmin; i++)
2809: {
2810: p = x*a[i][k]+y*a[i][k+1];
2811: if (k != nn-1) { p+=z*a[i][k+2]; a[i][k+2]-=p*r; }
2812: a[i][k+1] -= p*q; a[i][k] -= p;
2813: }
2814: }
2815: }
2816: }
2817: }
2818: while (l<nn-1);
2819: }
2820: for (j=2; j<=n; j++) /* ordering the roots */
2821: {
2822: x=wr[j]; y=wi[j]; if (y) flj=1; else flj=0;
2823: for (k=j-1; k>=1; k--)
2824: {
2825: if (wi[k]) flk=1; else flk=0;
2826: if (flk<flj) break;
2827: if (!flk && !flj && wr[k]<=x) break;
2828: if (flk&&flj&& wr[k]<x) break;
2829: if (flk&&flj&& wr[k]==x && wi[k]>0) break;
2830: wr[k+1]=wr[k]; wi[k+1]=wi[k];
2831: }
2832: wr[k+1]=x; wi[k+1]=y;
2833: }
2834: if (DEBUGLEVEL>3) { fprintferr("* Eigenvalues computed\n"); flusherr(); }
2835: for (i=1; i<=n; i++) free(a[i]); free(a); eig=cgetg(n+1,t_COL);
2836: for (i=1; i<=n; i++)
2837: {
2838: if (wi[i])
2839: {
2840: GEN p1 = cgetg(3,t_COMPLEX);
2841: eig[i]=(long)p1;
2842: p1[1]=(long)dbltor(wr[i]);
2843: p1[2]=(long)dbltor(wi[i]);
2844: }
2845: else eig[i]=(long)dbltor(wr[i]);
2846: }
2847: free(wr); free(wi); return eig;
2848: }
2849:
2850: /* ONLY FOR POLYNOMIAL WITH REAL COEFFICIENTS : give the roots of the
2851: * polynomial a (first, the real roots, then the non real roots) in
2852: * increasing order of their real parts MULTIPLE ROOTS ARE FORBIDDEN.
2853: */
2854: GEN
2855: zrhqr(GEN a,long prec)
2856: {
2857: ulong av = avma;
2858: long i,j,prec2, n = degpol(a), ex = -bit_accuracy(prec);
2859: GEN aa,b,p1,rt,rr,hess,x,dx,y,newval,oldval;
2860:
2861: hess = cgetg(n+1,t_MAT);
2862: for (j=1; j<=n; j++)
2863: {
2864: p1 = cgetg(n+1,t_COL); hess[j] = (long)p1;
2865: p1[1] = lneg(gdiv((GEN)a[n-j+2],(GEN)a[n+2]));
2866: for (i=2; i<=n; i++) p1[i] = (i==(j+1))? un: zero;
2867: }
2868: rt = hqr(balanc(hess));
2869: prec2 = 2*prec; /* polishing the roots */
2870: aa = gprec_w(a, prec2);
2871: b = derivpol(aa); rr = cgetg(n+1,t_COL);
2872: for (i=1; i<=n; i++)
2873: {
2874: x = gprec_w((GEN)rt[i], prec2);
2875: for (oldval=NULL;; oldval=newval, x=y)
2876: { /* Newton iteration */
2877: dx = poleval(b,x);
2878: if (gexpo(dx) < ex)
2879: err(talker,"polynomial has probably multiple roots in zrhqr");
2880: y = gsub(x, gdiv(poleval(aa,x),dx));
2881: newval = gabs(poleval(aa,y),prec2);
2882: if (gexpo(newval) < ex || (oldval && gcmp(newval,oldval) > 0)) break;
2883: }
2884: if (DEBUGLEVEL>3) fprintferr("%ld ",i);
2885: rr[i] = (long)cgetc(prec); gaffect(y, (GEN)rr[i]);
2886: }
2887: if (DEBUGLEVEL>3) { fprintferr("\npolished roots = %Z",rr); flusherr(); }
2888: return gerepilecopy(av, rr);
2889: }
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