Annotation of OpenXM_contrib/pari/src/basemath/polarit1.c, Revision 1.1.1.1
1.1 maekawa 1: /***********************************************************************/
2: /** **/
3: /** ARITHMETIC OPERATIONS ON POLYNOMIALS **/
4: /** (first part) **/
5: /** **/
6: /***********************************************************************/
7: /* $Id: polarit1.c,v 1.2 1999/09/23 17:50:56 karim Exp $ */
8: #include "pari.h"
9: GEN get_mul_table(GEN x,GEN bas,GEN *T);
10: GEN pol_to_monic(GEN pol, GEN *lead);
11: GEN sort_factor(GEN y, int (*cmp)(GEN,GEN));
12: GEN bsrch(GEN p, GEN fa, long Ka, GEN eta, long Ma);
13: GEN eleval(GEN f,GEN h,GEN a);
14: GEN respm(GEN f1,GEN f2,GEN pm);
15: GEN setup(GEN p,GEN f,GEN theta,GEN nut, long *La, long *Ma);
16: GEN vstar(GEN p,GEN h);
17:
18: /* see splitgen() for how to use these two */
19: GEN
20: setloop(GEN a)
21: {
22: a=icopy(a); new_chunk(2); /* dummy to get two cells of extra space */
23: return a;
24: }
25:
26: /* assume a > 0 */
27: GEN
28: incpos(GEN a)
29: {
30: long i,l=lgefint(a);
31:
32: for (i=l-1; i>1; i--)
33: if (++a[i]) return a;
34: i=l+1; a--; /* use extra cell */
35: a[0]=evaltyp(1) | evallg(i);
36: a[1]=evalsigne(1) | evallgefint(i);
37: return a;
38: }
39:
40: GEN
41: incloop(GEN a)
42: {
43: long i,l;
44:
45: switch(signe(a))
46: {
47: case 0:
48: a--; /* use extra cell */
49: a[0]=evaltyp(t_INT) | evallg(3);
50: a[1]=evalsigne(1) | evallgefint(3);
51: a[2]=1; return a;
52:
53: case -1:
54: l=lgefint(a);
55: for (i=l-1; i>1; i--)
56: if (a[i]--) break;
57: if (a[2] == 0)
58: {
59: a++; /* save one cell */
60: a[0] = evaltyp(t_INT) | evallg(2);
61: a[1] = evalsigne(0) | evallgefint(2);
62: }
63: return a;
64:
65: default:
66: return incpos(a);
67: }
68: }
69:
70: /*******************************************************************/
71: /* */
72: /* DIVISIBILITE */
73: /* Renvoie 1 si y divise x, 0 sinon . */
74: /* */
75: /*******************************************************************/
76:
77: int
78: gdivise(GEN x, GEN y)
79: {
80: long av=avma;
81: x=gmod(x,y); avma=av; return gcmp0(x);
82: }
83:
84: int
85: poldivis(GEN x, GEN y, GEN *z)
86: {
87: long av = avma;
88: GEN p1 = poldivres(x,y,ONLY_DIVIDES);
89: if (p1) { *z = p1; return 1; }
90: avma=av; return 0;
91: }
92:
93: /*******************************************************************/
94: /* */
95: /* REDUCTION */
96: /* Do the transformation t_FRACN/t_RFRACN --> t_FRAC/t_RFRAC */
97: /* */
98: /*******************************************************************/
99:
100: /* x[1] is scalar, non-zero */
101: static GEN
102: gred_simple(GEN x)
103: {
104: GEN p1,p2,x2,x3;
105:
106: x2=content((GEN)x[2]); if (gcmp1(x2)) return gcopy(x);
107: x3=gdiv((GEN)x[1],x2); p2=denom(x3);
108: x2=gdiv((GEN)x[2],x2);
109:
110: p1=cgetg(3,t_RFRAC);
111: p1[1]=(long)numer(x3);
112: p1[2]=lmul(x2,p2); return p1;
113: }
114:
115: GEN
116: gred_rfrac(GEN x)
117: {
118: GEN y,p1,xx1,xx2,x3, x1 = (GEN)x[1], x2 = (GEN)x[2];
119: long tx,ty;
120:
121: if (gcmp0(x1)) return gcopy(x1);
122:
123: tx=typ(x1); ty=typ(x2);
124: if (ty!=t_POL)
125: {
126: if (tx!=t_POL) return gcopy(x);
127: if (gvar2(x2) > varn(x1)) return gdiv(x1,x2);
128: err(talker,"incompatible variables in gred");
129: }
130: if (tx!=t_POL)
131: {
132: if (varn(x2) < gvar2(x1)) return gred_simple(x);
133: err(talker,"incompatible variables in gred");
134: }
135: if (varn(x2) < varn(x1)) return gred_simple(x);
136: if (varn(x2) > varn(x1)) return gdiv(x1,x2);
137:
138: /* now x1 and x2 are polynomials with the same variable */
139: xx1=content(x1); if (!gcmp1(xx1)) x1=gdiv(x1,xx1);
140: xx2=content(x2); if (!gcmp1(xx2)) x2=gdiv(x2,xx2);
141: x3=gdiv(xx1,xx2);
142: y = poldivres(x1,x2,&p1);
143: if (!signe(p1)) { cgiv(p1); return gmul(x3,y); }
144:
145: p1 = ggcd(x2,p1);
146: if (!isscalar(p1)) { x1=gdeuc(x1,p1); x2=gdeuc(x2,p1); }
147: xx1=numer(x3); xx2=denom(x3);
148: p1=cgetg(3,t_RFRAC);
149: p1[1]=lmul(x1,xx1);
150: p1[2]=lmul(x2,xx2); return p1;
151: }
152:
153: GEN
154: gred(GEN x)
155: {
156: long tx=typ(x),av=avma;
157: GEN y,p1,x1,x2;
158:
159: if (is_frac_t(tx))
160: {
161: x1=(GEN)x[1]; x2=(GEN)x[2];
162: y = dvmdii(x1,x2,&p1);
163: if (p1 == gzero) return y; /* gzero volontaire */
164: (void)new_chunk((lgefint(x1)+lgefint(x2))<<1);
165: p1=mppgcd(x2,p1);
166: if (is_pm1(p1)) { avma=av; return gcopy(x); }
167: avma=av; y=cgetg(3,t_FRAC);
168: y[1]=ldivii(x1,p1);
169: y[2]=ldivii(x2,p1); return y;
170: }
171: if (is_rfrac_t(tx))
172: return gerepileupto(av, gred_rfrac(x));
173: return gcopy(x);
174: }
175:
176: /*******************************************************************/
177: /* */
178: /* POLYNOMIAL EUCLIDEAN DIVISION */
179: /* */
180: /*******************************************************************/
181: GEN gdivexact(GEN x, GEN y);
182:
183: /* Polynomial division x / y:
184: * if z = ONLY_REM return remainder, otherwise return quotient
185: * if z != NULL set *z to remainder
186: * *z is the last object on stack (and thus can be disposed of with cgiv
187: * instead of gerepile)
188: */
189: GEN
190: poldivres(GEN x, GEN y, GEN *pr)
191: {
192: long ty=typ(y),tx,vx,vy,dx,dy,dz,i,j,avy,av,av1,sx,lrem;
193: int remainder;
194: GEN z,p1,rem,y_lead,mod;
195: GEN (*f)(GEN,GEN);
196:
197: if (pr == ONLY_DIVIDES_EXACT)
198: { f = gdivexact; pr = ONLY_DIVIDES; }
199: else
200: f = gdiv;
201: if (is_scalar_t(ty))
202: {
203: if (pr == ONLY_REM) return gzero;
204: if (pr && pr != ONLY_DIVIDES) *pr=gzero;
205: return f(x,y);
206: }
207: tx=typ(x); vy=gvar9(y);
208: if (is_scalar_t(tx) || gvar9(x)>vy)
209: {
210: if (pr == ONLY_REM) return gcopy(x);
211: if (pr == ONLY_DIVIDES) return gcmp0(x)? gzero: NULL;
212: if (pr) *pr=gcopy(x);
213: return gzero;
214: }
215: if (tx!=t_POL || ty!=t_POL) err(typeer,"euclidean division (poldivres)");
216:
217: vx=varn(x);
218: if (vx<vy)
219: {
220: if (pr && pr != ONLY_DIVIDES)
221: {
222: p1 = zeropol(vx); if (pr == ONLY_REM) return p1;
223: *pr = p1;
224: }
225: return f(x,y);
226: }
227: if (!signe(y)) err(talker,"euclidean division by zero (poldivres)");
228:
229: dy=lgef(y)-3; y_lead = (GEN)y[dy+2];
230: if (gcmp0(y_lead)) /* normalize denominator if leading term is 0 */
231: {
232: err(warner,"normalizing a polynomial with 0 leading term");
233: for (dy--; dy>=0; dy--)
234: {
235: y_lead = (GEN)y[dy+2];
236: if (!gcmp0(y_lead)) break;
237: }
238: }
239: if (!dy) /* y is constant */
240: {
241: if (pr && pr != ONLY_DIVIDES)
242: {
243: if (pr == ONLY_REM) return zeropol(vx);
244: *pr = zeropol(vx);
245: }
246: return f(x,(GEN)y[2]);
247: }
248: dx=lgef(x)-3;
249: if (vx>vy || dx<dy)
250: {
251: if (pr)
252: {
253: if (pr == ONLY_DIVIDES) return gcmp0(x)? gzero: NULL;
254: if (pr == ONLY_REM) return gcopy(x);
255: *pr = gcopy(x);
256: }
257: return zeropol(vy);
258: }
259: dz=dx-dy; av=avma; /* to avoid gsub's later on */
260: p1 = new_chunk(dy+3);
261: for (i=2; i<dy+3; i++)
262: {
263: GEN p2 = (GEN)y[i];
264: p1[i] = isexactzero(p2)? 0: (long)gneg_i(p2);
265: }
266: y = p1;
267: switch(typ(y_lead))
268: {
269: case t_INTMOD:
270: case t_POLMOD: y_lead = ginv(y_lead);
271: f = gmul; mod = (GEN)y_lead[1];
272: break;
273: default: if (gcmp1(y_lead)) y_lead = NULL;
274: mod = NULL;
275: }
276: avy=avma; z=cgetg(dz+3,t_POL);
277: z[1]=evalsigne(1) | evallgef(dz+3) | evalvarn(vx);
278: x += 2; y += 2; z += 2;
279:
280: p1 = (GEN)x[dx]; remainder = (pr == ONLY_REM);
281: z[dz]=y_lead? (long)f(p1,y_lead): lcopy(p1);
282: for (i=dx-1; i>=dy; i--)
283: {
284: av1=avma; p1=(GEN)x[i];
285: for (j=i-dy+1; j<=i && j<=dz; j++)
286: if (y[i-j]) p1 = gadd(p1, gmul((GEN)z[j],(GEN)y[i-j]));
287: if (y_lead) p1 = f(p1,y_lead);
288: if (!remainder) p1 = avma==av1? gcopy(p1): gerepileupto(av1,p1);
289: z[i-dy] = (long)p1;
290: }
291: if (!pr) return gerepileupto(av,z-2);
292:
293: rem = (GEN)avma; av1 = (long)new_chunk(dx+3);
294: for (sx=0; ; i--)
295: {
296: p1 = (GEN)x[i];
297: /* we always enter this loop at least once */
298: for (j=0; j<=i && j<=dz; j++)
299: if (y[i-j]) p1 = gadd(p1, gmul((GEN)z[j],(GEN)y[i-j]));
300: if (mod && avma==av1) p1 = gmod(p1,mod);
301: if (!gcmp0(p1)) { sx = 1; break; } /* remainder is non-zero */
302: if (!isinexactreal(p1) && !isexactzero(p1)) break;
303: if (!i) break;
304: avma=av1;
305: }
306: if (pr == ONLY_DIVIDES)
307: {
308: if (sx) { avma=av; return NULL; }
309: avma = (long)rem;
310: return gerepileupto(av,z-2);
311: }
312: lrem=i+3; rem -= lrem;
313: if (avma==av1) { avma = (long)rem; p1 = gcopy(p1); }
314: else p1 = gerepileupto((long)rem,p1);
315: rem[0]=evaltyp(t_POL) | evallg(lrem);
316: rem[1]=evalsigne(1) | evalvarn(vx) | evallgef(lrem);
317: rem += 2;
318: rem[i]=(long)p1;
319: for (i--; i>=0; i--)
320: {
321: av1=avma; p1 = (GEN)x[i];
322: for (j=0; j<=i && j<=dz; j++)
323: if (y[i-j]) p1 = gadd(p1, gmul((GEN)z[j],(GEN)y[i-j]));
324: if (mod && avma==av1) p1 = gmod(p1,mod);
325: rem[i]=avma==av1? lcopy(p1):lpileupto(av1,p1);
326: }
327: rem -= 2;
328: if (!sx) normalizepol_i(rem, lrem);
329: if (remainder) return gerepileupto(av,rem);
330: z -= 2;
331: {
332: GEN *gptr[2]; gptr[0]=&z; gptr[1]=&rem;
333: gerepilemanysp(av,avy,gptr,2); *pr = rem; return z;
334: }
335: }
336:
337: /*******************************************************************/
338: /* */
339: /* ROOTS MODULO a prime p (no multiplicities) */
340: /* */
341: /*******************************************************************/
342: static GEN
343: mod(GEN x, GEN y)
344: {
345: GEN z = cgetg(3,t_INTMOD);
346: z[1]=(long)y; z[2]=(long)x; return z;
347: }
348:
349: static long
350: factmod_init(GEN *F, GEN pp, long *p)
351: {
352: GEN f = *F;
353: long i,d;
354: if (typ(f)!=t_POL || typ(pp)!=t_INT) err(typeer,"factmod");
355: if (expi(pp) > BITS_IN_LONG - 3) *p = 0;
356: else
357: {
358: *p = itos(pp);
359: if (*p < 2) err(talker,"not a prime in factmod");
360: }
361: f = gmul(f, mod(gun,pp));
362: if (!signe(f)) err(zeropoler,"factmod");
363: f = lift_intern(f); d = lgef(f);
364: for (i=2; i <d; i++)
365: if (typ(f[i])!=t_INT) err(impl,"factormod for general polynomials");
366: *F = f; return d-3;
367: }
368:
369: #define mods(x,y) mod(stoi(x),y)
370: static GEN
371: root_mod_2(GEN f)
372: {
373: int z1, z0 = !signe(f[2]);
374: long i,n;
375: GEN y;
376:
377: for (i=2, n=1; i < lgef(f); i++)
378: if (signe(f[i])) n++;
379: z1 = n & 1;
380: y = cgetg(z0+z1+1, t_COL); i = 1;
381: if (z0) y[i++] = (long)mods(0,gdeux);
382: if (z1) y[i] = (long)mods(1,gdeux);
383: return y;
384: }
385:
386: #define i_mod4(x) (signe(x)? mod4((GEN)(x)): 0)
387: static GEN
388: root_mod_4(GEN f)
389: {
390: long no,ne;
391: int z0 = !signe(f[2]);
392: int z2 = ((i_mod4(f[2]) + 2*i_mod4(f[3])) & 3) == 0;
393: int i,z1,z3;
394: GEN y,p;
395:
396: for (ne=0,i=2; i<lgef(f); i+=2)
397: if (signe(f[i])) ne += mael(f,i,2);
398: for (no=0,i=3; i<lgef(f); i+=2)
399: if (signe(f[i])) no += mael(f,i,2);
400: no &= 3; ne &= 3;
401: z3 = (no == ne);
402: z1 = (no == ((4-ne)&3));
403: y=cgetg(1+z0+z1+z2+z3,t_COL); i = 1; p = stoi(4);
404: if (z0) y[i++] = (long)mods(0,p);
405: if (z1) y[i++] = (long)mods(1,p);
406: if (z2) y[i++] = (long)mods(2,p);
407: if (z3) y[i] = (long)mods(3,p);
408: return y;
409: }
410: #undef i_mod4
411:
412: /* p even, accept p = 4 for p-adic stuff */
413: static GEN
414: root_mod_even(GEN f, long p)
415: {
416: switch(p)
417: {
418: case 2: return root_mod_2(f);
419: case 4: return root_mod_4(f);
420: }
421: err(talker,"not a prime in rootmod");
422: return NULL; /* not reached */
423: }
424:
425: /* by checking f(0..p-1) */
426: GEN
427: rootmod2(GEN f, GEN pp)
428: {
429: GEN g,y,ss,q,r, x_minus_s;
430: long p,av = avma,av1,d,i,nbrac;
431:
432: if (!(d = factmod_init(&f, pp, &p))) { avma=av; return cgetg(1,t_COL); }
433: if (!p) err(talker,"prime too big in rootmod2");
434: if ((p & 1) == 0) { avma = av; return root_mod_even(f,p); }
435: x_minus_s = gadd(polx[varn(f)], stoi(-1));
436:
437: nbrac=1;
438: y=(GEN)gpmalloc((d+1)*sizeof(long));
439: if (gcmp0((GEN)f[2])) y[nbrac++] = 0;
440: ss = icopy(gun); av1 = avma;
441: do
442: {
443: mael(x_minus_s,2,2) = ss[2];
444: /* one might do a FFT-type evaluation */
445: q = Fp_poldivres(f, x_minus_s, pp, &r);
446: if (signe(r)) avma = av1;
447: else
448: {
449: y[nbrac++] = ss[2]; f = q; av1 = avma;
450: }
451: ss[2]++;
452: }
453: while (nbrac<d && p>ss[2]);
454: if (nbrac == 1) { avma=av; return cgetg(1,t_COL); }
455: if (nbrac == d && p != ss[2])
456: {
457: g = mpinvmod((GEN)f[3], pp); setsigne(g,-1);
458: ss = modis(mulii(g, (GEN)f[2]), p);
459: y[nbrac++]=ss[2];
460: }
461: avma=av; g=cgetg(nbrac,t_COL);
462: if (isonstack(pp)) pp = icopy(pp);
463: for (i=1; i<nbrac; i++) g[i]=(long)mods(y[i],pp);
464: free(y); return g;
465: }
466:
467: /* by splitting */
468: GEN
469: rootmod(GEN f, GEN p)
470: {
471: long av = avma,tetpil,n,i,j,la,lb;
472: GEN y,pol,a,b,q,pol0;
473:
474: if (!factmod_init(&f, p, &i)) { avma=av; return cgetg(1,t_COL); }
475: i = p[lgefint(p)-1];
476: if ((i & 1) == 0) { avma = av; return root_mod_even(f,i); }
477: i=2; while (!signe(f[i])) i++;
478: if (i == 2) j = 1;
479: else
480: {
481: j = lgef(f) - (i-2);
482: if (j==3) /* f = x^n */
483: {
484: avma = av; y = cgetg(2,t_COL);
485: y[1] = (long)gmodulsg(0,p);
486: return y;
487: }
488: a = cgetg(j, t_POL); /* a = f / x^{v_x(f)} */
489: a[1] = evalsigne(1) | evalvarn(varn(f)) | evallgef(j);
490: f += i-2; for (i=2; i<j; i++) a[i]=f[i];
491: j = 2; f = a;
492: }
493: q = shifti(p,-1);
494: /* take gcd(x^(p-1) - 1, f) by splitting (x^q-1) * (x^q+1) */
495: b = Fp_pow_mod_pol(polx[varn(f)],q, f,p);
496: if (lgef(b)<3) err(talker,"not a prime in rootmod");
497: b[2] = laddis((GEN)b[2],-1); /* b = x^((p-1)/2) - 1 mod f */
498: a = Fp_pol_gcd(f,b, p);
499: b[2] = laddis((GEN)b[2], 2); /* b = x^((p-1)/2) + 1 mod f */
500: b = Fp_pol_gcd(f,b, p);
501: la = lgef(a)-3;
502: lb = lgef(b)-3; n = la + lb;
503: if (!n)
504: {
505: avma = av; y = cgetg(n+j,t_COL);
506: if (j>1) y[1] = (long)gmodulsg(0,p);
507: return y;
508: }
509: y = cgetg(n+j,t_COL);
510: if (j>1) { y[1] = zero; n++; }
511: y[j] = (long)normalize_mod_p(b,p);
512: if (la) y[j+lb] = (long)normalize_mod_p(a,p);
513: pol = gadd(polx[varn(f)], gun); pol0 = (GEN)pol[2];
514: while (j<=n)
515: {
516: a=(GEN)y[j]; la=lgef(a)-3;
517: if (la==1)
518: y[j++] = lsubii(p, (GEN)a[2]);
519: else if (la==2)
520: {
521: GEN d = subii(sqri((GEN)a[3]), shifti((GEN)a[2],2));
522: GEN e = mpsqrtmod(d,p), u = addis(q, 1); /* u = 1/2 */
523: y[j++] = lmodii(mulii(u,subii(e,(GEN)a[3])), p);
524: y[j++] = lmodii(mulii(u,negi(addii(e,(GEN)a[3]))), p);
525: }
526: else for (pol0[2]=1; ; pol0[2]++)
527: {
528: b = Fp_pow_mod_pol(pol,q, a,p);
529: b[2] = laddis((GEN)b[2], -1);
530: b = Fp_pol_gcd(a,b, p); lb = lgef(b)-3;
531: if (lb && lb<la)
532: {
533: b = normalize_mod_p(b, p);
534: y[j+lb] = (long)Fp_deuc(a,b, p);
535: y[j] = (long)b; break;
536: }
537: }
538: }
539: tetpil = avma; y = gerepile(av,tetpil,sort(y));
540: if (isonstack(p)) p = icopy(p);
541: for (i=1; i<=n; i++) y[i] = (long)mod((GEN)y[i], p);
542: return y;
543: }
544:
545: GEN
546: rootmod0(GEN f, GEN p, long flag)
547: {
548: switch(flag)
549: {
550: case 0: return rootmod(f,p);
551: case 1: return rootmod2(f,p);
552: default: err(flagerr,"polrootsmod");
553: }
554: return NULL; /* not reached */
555: }
556:
557: /*******************************************************************/
558: /* */
559: /* FACTORISATION MODULO p */
560: /* */
561: /*******************************************************************/
562: static GEN
563: trivfact(void)
564: {
565: GEN y=cgetg(3,t_MAT);
566: y[1]=lgetg(1,t_COL);
567: y[2]=lgetg(1,t_COL); return y;
568: }
569:
570: static void
571: fqunclone(GEN x, GEN p)
572: {
573: long i,lx = lgef(x);
574: for (i=2; i<lx; i++)
575: {
576: GEN p1 = (GEN)x[i];
577: if (typ(p1) == t_POLMOD) p1[1] = (long)p;
578: }
579: }
580:
581: static GEN
582: try_pow(GEN w0, GEN pol, GEN p, GEN q, long r)
583: {
584: GEN w2, w = Fp_pow_mod_pol(w0,q, pol,p);
585: long s;
586: if (gcmp1(w)) return w0;
587: for (s=1; s<r; s++,w=w2)
588: {
589: w2 = gsqr(w);
590: w2 = Fp_res(w2, pol, p);
591: if (gcmp1(w2)) break;
592: }
593: return gcmp_1(w)? NULL: w;
594: }
595:
596: /* INPUT:
597: * m integer (converted to polynomial w in Z[X] by stopoly)
598: * p prime; q = (p^d-1) / 2^r
599: * t[0] polynomial of degree k*d product of k irreducible factors of degree d
600: * t[0] is expected to be normalized (leading coeff = 1)
601: * OUTPUT:
602: * t[0],t[1]...t[k-1] the k factors, normalized
603: */
604: static void
605: split(long m, GEN *t, long d, GEN pg, GEN q, long r)
606: {
607: long p,l,v,dv,av0,av;
608: GEN w,w0;
609:
610: dv=lgef(*t)-3; if (dv==d) return;
611: v=varn(*t); av0=avma; p = pg[2];
612: for(av=avma;;avma=av)
613: {
614: if (p==2)
615: {
616: w0=w=gpuigs(polx[v],m-1); m+=2;
617: for (l=1; l<d; l++)
618: {
619: w = gadd(w0, gsqr(w));
620: w = Fp_pol_red(w, pg);
621: w = Fp_res(w, *t, pg);
622: }
623: }
624: else
625: {
626: w = Fp_res(stopoly(m,p,v),*t, pg);
627: m++; w = try_pow(w,*t,pg,q,r);
628: if (!w) continue;
629: /* set w = w - 1 */
630: w[2] = laddis((GEN)w[2], -1); /* w != 1 or -1 */
631: }
632: w = Fp_pol_gcd(*t,w, pg);
633: l = lgef(w)-3; if (l && l!=dv) break;
634: }
635: w = normalize_mod_p(w, pg);
636: w = gerepileupto(av0, w);
637: l /= d; t[l]=Fp_deuc(*t,w,pg); *t=w;
638: split(m,t+l,d,pg,q,r);
639: split(m,t, d,pg,q,r);
640: }
641:
642: static void
643: splitgen(GEN m, GEN *t, long d, GEN p, GEN q, long r)
644: {
645: long l,v,dv,av;
646: GEN w;
647:
648: dv=lgef(*t)-3; if (dv==d) return;
649: v=varn(*t); m=setloop(m); m=incpos(m);
650: av=avma;
651: for(;; avma=av, m=incpos(m))
652: {
653: w = Fp_res(stopoly_gen(m,p,v),*t, p);
654: w = try_pow(w,*t,p,q,r);
655: if (!w) continue;
656: /* set w = w - 1 */
657: w[2] = laddis((GEN)w[2], -1); /* w != 1 or -1 */
658: w = Fp_pol_gcd(*t,w, p); l=lgef(w)-3;
659: if (l && l!=dv) break;
660:
661: }
662: w = normalize_mod_p(w, p);
663: w = gerepileupto(av, w);
664: l /= d; t[l]=Fp_deuc(*t,w,p); *t=w;
665: splitgen(m,t+l,d,p,q,r);
666: splitgen(m,t, d,p,q,r);
667: }
668:
669: /* factor f mod pp. If (simple) ouput only the degrees, not the factors */
670: static GEN
671: factcantor0(GEN f, GEN pp, long simple)
672: {
673: long i,j,k,d,e,vf,p,nbfact,tetpil,av = avma;
674: GEN ex,y,p1,f2,g,g1,u,v,pd,q;
675: GEN *t;
676:
677: if (!(d = factmod_init(&f, pp, &p))) { avma=av; return trivfact(); }
678: /* to hold factors and exponents */
679: t = (GEN*)cgetg(d+1,t_VEC); ex = new_chunk(d+1);
680: vf=varn(f); e = nbfact = 1;
681: f = lift_intern(f);
682: for(;;)
683: {
684: f2 = Fp_pol_gcd(f,derivpol(f), pp);
685: g1 = Fp_deuc(f,f2,pp);
686: k = 0;
687: while (lgef(g1)>3)
688: {
689: k++; if (p && !(k%p)) { k++; f2 = Fp_deuc(f2,g1,pp); }
690: p1 = Fp_pol_gcd(f2,g1, pp); u = g1; g1 = p1;
691: if (!gcmp1(p1))
692: {
693: u = Fp_deuc( u,p1,pp);
694: f2= Fp_deuc(f2,p1,pp);
695: }
696: if (lgef(u)>3)
697: {
698: /* here u is square-free (product of irred. of multiplicity e * k) */
699: pd=gun; v=polx[vf];
700: for (d=1; d <= (lgef(u)-3)>>1; d++)
701: {
702: pd=mulii(pd,pp); v=Fp_pow_mod_pol(v,pp, u,pp);
703: g = Fp_pol_gcd(gadd(v, gneg(polx[vf])), u, pp);
704:
705: if (lgef(g)>3)
706: {
707: /* Ici g est produit de pol irred ayant tous le meme degre d; */
708: j=nbfact+(lgef(g)-3)/d;
709:
710: if (simple)
711: for ( ; nbfact<j; nbfact++)
712: { t[nbfact]=(GEN)d; ex[nbfact]=e*k; }
713: else
714: {
715: long r;
716: g = normalize_mod_p(g, pp);
717: t[nbfact]=g; q = subis(pd,1); /* also ok for p=2: unused */
718: r = vali(q); q = shifti(q,-r);
719: /* le premier parametre est un entier variable m qui sera
720: * converti en un polynome w dont les coeff sont ses digits en
721: * base p (initialement m = p --> X) pour faire pgcd de g avec
722: * w^(p^d-1)/2 jusqu'a casser. p = 2 will be treated separately.
723: */
724: if (p)
725: split(p,t+nbfact,d,pp,q,r);
726: else
727: splitgen(pp,t+nbfact,d,pp,q,r);
728: for (; nbfact<j; nbfact++) ex[nbfact]=e*k;
729: }
730: u=Fp_deuc(u,g,pp);
731: v=Fp_res(v,u,pp);
732: }
733: }
734: if (lgef(u)>3)
735: {
736: t[nbfact] = simple? (GEN)(lgef(u)-3): u;
737: ex[nbfact++]=e*k;
738: }
739: }
740: }
741: j = lgef(f2); if (j==3) break;
742:
743: e*=p; j=(j-3)/p+3; setlg(f,j); setlgef(f,j);
744: for (i=2; i<j; i++) f[i]=f2[p*(i-2)+2];
745: }
746: tetpil=avma; y=cgetg(3,t_MAT);
747: if (!simple)
748: {
749: y[1]=(long)t; setlg(t, nbfact);
750: y[2]=(long)ex; (void)sort_factor(y,cmpii);
751: }
752: u=cgetg(nbfact,t_COL); y[1]=(long)u;
753: v=cgetg(nbfact,t_COL); y[2]=(long)v;
754: if (simple)
755: for (j=1; j<nbfact; j++)
756: {
757: u[j] = lstoi((long)t[j]);
758: v[j] = lstoi(ex[j]);
759: }
760: else
761: for (j=1; j<nbfact; j++)
762: {
763: u[j] = (long)Fp_pol(t[j], pp);
764: v[j] = lstoi(ex[j]);
765: }
766: return gerepile(av,tetpil,y);
767: }
768:
769: GEN
770: factcantor(GEN f, GEN p)
771: {
772: return factcantor0(f,p,0);
773: }
774:
775: GEN
776: simplefactmod(GEN f, GEN p)
777: {
778: return factcantor0(f,p,1);
779: }
780:
781: /* vector of polynomials (in v) whose coeffs are given by the columns of x */
782: GEN
783: mat_to_vecpol(GEN x, long v)
784: {
785: long i,j, lx = lg(x), lcol = lg(x[1]);
786: GEN y = cgetg(lx, t_VEC);
787:
788: for (j=1; j<lx; j++)
789: {
790: GEN p1, col = (GEN)x[j];
791: long k = lcol;
792:
793: while (k-- && gcmp0((GEN)col[k]));
794: i=k+2; p1=cgetg(i,t_POL);
795: p1[1] = evalsigne(1) | evallgef(i) | evalvarn(v);
796: col--; for (k=2; k<i; k++) p1[k] = col[k];
797: y[j] = (long)p1;
798: }
799: return y;
800: }
801:
802: /* matrix whose entries are given by the coeffs of the polynomials in
803: * vector v (considered as degree n polynomials) */
804: GEN
805: vecpol_to_mat(GEN v, long n)
806: {
807: long i,j,d,N = lg(v);
808: GEN p1,w, y = cgetg(N, t_MAT);
809: n++;
810: for (j=1; j<N; j++)
811: {
812: p1 = cgetg(n,t_COL); y[j] = (long)p1;
813: w = (GEN)v[j];
814: if (typ(w) != t_POL) { p1[1] = (long)w; i=2; }
815: else
816: {
817: d=lgef(w)-1; w++;
818: for (i=1; i<d; i++) p1[i] = w[i];
819: }
820: for ( ; i<n; i++) p1[i] = zero;
821: }
822: return y;
823: }
824:
825: /* set x <-- x + c*y mod p */
826: static void
827: Fp_pol_addmul(GEN x, GEN y, long c, long p)
828: {
829: long i,lx = lgef(x), ly = lgef(y), l = min(lx,ly);
830: if (p & ~MAXHALFULONG)
831: {
832: for (i=2; i<l; i++) x[i] = ((ulong)x[i]+ (ulong)mulssmod(c,y[i],p)) % p;
833: for ( ; i<ly; i++) x[i] = mulssmod(c,y[i],p);
834: }
835: else
836: {
837: for (i=2; i<l; i++) x[i] = ((ulong)x[i] + (ulong)(c*y[i])) % p;
838: for ( ; i<ly; i++) x[i] = (c*y[i]) % p;
839: }
840: do i--; while (i>1 && !x[i]);
841: if (i==1) setsigne(x,0); else { setsigne(x,1); setlgef(x,i+1); }
842: }
843:
844: long
845: split_berlekamp(GEN Q, GEN *t, GEN pp, GEN pps2)
846: {
847: GEN u = *t, p1, p2, vker,v,w,pol;
848: long av,N = lgef(u)-3, d,i,j,kk,l1,l2,p, vu = varn(u);
849:
850: if (DEBUGLEVEL > 7) timer2();
851: p = is_bigint(pp)? 0: pp[2];
852: setlg(Q, N+1); setlg(Q[1], N+1);
853: w = v = Fp_pow_mod_pol(polx[vu],pp,u,pp);
854: for (j=2; j<=N; j++)
855: {
856: p1 = (GEN)Q[j]; setlg(p1, N+1);
857: d=lgef(w)-1; p2 = w+1;
858: for (i=1; i<d ; i++) p1[i] = p2[i];
859: for ( ; i<=N; i++) p1[i] = zero;
860: p1[j] = laddis((GEN)p1[j], -1);
861: if (j < N)
862: {
863: av = avma;
864: w = gerepileupto(av, Fp_res(gmul(w,v), u, pp));
865: }
866: }
867: if (DEBUGLEVEL > 7) msgtimer("frobenius");
868: vker = mat_to_vecpol(ker_mod_p(Q,pp), vu);
869: if (DEBUGLEVEL > 7) msgtimer("kernel");
870: d = lg(vker)-1;
871: if (p)
872: for (i=1; i<=d; i++)
873: {
874: p1 = (GEN)vker[i];
875: for (j=2; j<lg(p1); j++) p1[j] = itos((GEN)p1[j]);
876: }
877: pol = cgetg(N+3,t_POL);
878:
879: for (kk=1; kk<d; )
880: {
881: GEN polt;
882: if (p)
883: {
884: if (p==2)
885: {
886: pol[2] = ((mymyrand() & 0x1000) == 0);
887: pol[1] = evallgef(pol[2]? 3: 2);
888: for (i=2; i<=d; i++)
889: Fp_pol_addmul(pol, (GEN)vker[i], ((mymyrand() & 0x1000) == 0), p);
890: }
891: else
892: {
893: pol[2] = mymyrand()%p; /* vker[1] = 1 */
894: pol[1] = evallgef(pol[2]? 3: 2);
895: for (i=2; i<=d; i++)
896: Fp_pol_addmul(pol, (GEN)vker[i], mymyrand()%p, p);
897: }
898: polt = small_to_pol(pol+2, lgef(pol), p);
899: setvarn(polt,vu);
900: }
901: else
902: {
903: pol[2] = (long)genrand(pp);
904: pol[1] = evallgef(signe(pol[2])? 3: 2) | evalvarn(vu);
905: for (i=2; i<=d; i++)
906: pol = gadd(pol, gmul((GEN)vker[i], genrand(pp)));
907: polt = Fp_pol_red(pol,pp);
908: }
909: for (i=1; i<=kk && kk<d; i++)
910: {
911: p1=t[i-1]; l1=lgef(p1)-3;
912: if (l1>1)
913: {
914: av = avma; p2 = Fp_res(polt, p1, pp);
915: if (lgef(p2) <= 3) { avma=av; continue; }
916: p2 = Fp_pow_mod_pol(p2,pps2, p1,pp);
917: /* set p2 = p2 - 1 */
918: p2[2] = laddis((GEN)p2[2], -1);
919: p2 = Fp_pol_gcd(p1,p2, pp); l2=lgef(p2)-3;
920: if (l2>0 && l2<l1)
921: {
922: p2 = normalize_mod_p(p2, pp);
923: t[i-1] = p2; kk++;
924: t[kk-1] = Fp_deuc(p1,p2,pp);
925: if (DEBUGLEVEL > 7) msgtimer("new factor");
926: }
927: else avma = av;
928: }
929: }
930: }
931: return d;
932: }
933:
934: GEN
935: factmod(GEN f, GEN pp)
936: {
937: long i,j,k,e,p,N,nbfact,av = avma,tetpil,d;
938: GEN pps2,ex,y,f2,p1,g1,Q,u,v, *t;
939:
940: if (!(d = factmod_init(&f, pp, &p))) { avma=av; return trivfact(); }
941: /* to hold factors and exponents */
942: t = (GEN*)cgetg(d+1,t_VEC); ex = new_chunk(d+1);
943: e = nbfact = 1;
944: pps2 = shifti(pp,-1);
945:
946: Q = cgetg(d+1,t_MAT);
947: for (i=1; i<=d; i++) Q[i] = lgetg(d+1, t_COL);
948: p1 = (GEN)Q[1];
949: for (i=1; i<=d; i++) p1[i] = zero;
950: for(;;)
951: {
952: f2 = Fp_pol_gcd(f,derivpol(f), pp);
953: g1 = gcmp1(f2)? f: Fp_deuc(f,f2,pp);
954: k = 0;
955: while (lgef(g1)>3)
956: {
957: k++; if (p && !(k%p)) { k++; f2 = Fp_deuc(f2,g1,pp); }
958: p1 = Fp_pol_gcd(f2,g1, pp); u = g1; g1 = p1;
959: if (!gcmp1(p1))
960: {
961: u = Fp_deuc( u,p1,pp);
962: f2= Fp_deuc(f2,p1,pp);
963: }
964: N = lgef(u)-3;
965: if (N)
966: {
967: /* here u is square-free (product of irred. of multiplicity e * k) */
968: t[nbfact] = normalize_mod_p(u,pp);
969: d = (N==1)? 1: split_berlekamp(Q, t+nbfact, pp, pps2);
970: for (j=0; j<d; j++) ex[nbfact+j] = e*k;
971: nbfact += d;
972: }
973: }
974: if (!p) break;
975: j=(lgef(f2)-3)/p+3; if (j==3) break;
976:
977: e*=p; setlg(f,j); setlgef(f,j);
978: for (i=2; i<j; i++) f[i] = f2[p*(i-2)+2];
979: }
980: tetpil=avma; y=cgetg(3,t_MAT);
981: y[1]=(long)t; setlg(t, nbfact);
982: y[2]=(long)ex; (void)sort_factor(y,cmpii);
983: u=cgetg(nbfact,t_COL); y[1]=(long)u;
984: v=cgetg(nbfact,t_COL); y[2]=(long)v;
985: for (j=1; j<nbfact; j++)
986: {
987: u[j] = (long)Fp_pol(t[j], pp);
988: v[j] = lstoi(ex[j]);
989: }
990: return gerepile(av,tetpil,y);
991: }
992:
993: GEN
994: factormod0(GEN f, GEN p, long flag)
995: {
996: switch(flag)
997: {
998: case 0: return factmod(f,p);
999: case 1: return simplefactmod(f,p);
1000: default: err(flagerr,"factormod");
1001: }
1002: return NULL; /* not reached */
1003: }
1004:
1005: /*******************************************************************/
1006: /* */
1007: /* Recherche de racines p-adiques */
1008: /* */
1009: /*******************************************************************/
1010: /* make f suitable for [root|factor]padic */
1011: static GEN
1012: padic_pol_to_int(GEN f)
1013: {
1014: long i, l = lgef(f);
1015: f = gdiv(f,content(f));
1016: for (i=2; i<l; i++)
1017: switch(typ(f[i]))
1018: {
1019: case t_INT: break;
1020: case t_PADIC: f[i] = ltrunc((GEN)f[i]); break;
1021: default: err(talker,"incorrect coeffs in padic_pol_to_int");
1022: }
1023: return f;
1024: }
1025:
1026: /* return x + O(pr), x in Z, pr = p^r */
1027: static GEN
1028: int_to_padic(GEN x, GEN p, GEN pr, long r)
1029: {
1030: GEN p1,y;
1031: long v;
1032: if (typ(x) == t_PADIC) return gcopy(x);
1033: y = cgetg(5,t_PADIC);
1034: if (signe(x) && (v = pvaluation(x,p,&p1)) < r)
1035: {
1036: y[4] = lmodii(p1,pr); r -= v;
1037: }
1038: else
1039: {
1040: y[4] = zero; v = r; r = 0;
1041: }
1042: y[3] = (long)pr;
1043: y[2] = (long)p;
1044: y[1] = evalprecp(r)|evalvalp(v); return y;
1045: }
1046:
1047: /* return x + O(p^r), x in Z[X] */
1048: static GEN
1049: pol_to_padic(GEN x, GEN pr, GEN p, long r)
1050: {
1051: long i,lx = lgef(x);
1052: GEN z = cgetg(lx,t_POL);
1053: for (i=2; i<lx; i++)
1054: z[i] = (long)int_to_padic((GEN)x[i],p,pr,r);
1055: z[1] = x[1]; return z;
1056: }
1057:
1058: static GEN
1059: padic_trivfact(GEN x, GEN p, long r)
1060: {
1061: GEN p1, y = cgetg(3,t_MAT);
1062: p1=cgetg(2,t_COL); y[1]=(long)p1; p = icopy(p);
1063: p1[1]=(long)pol_to_padic(x,gpowgs(p,r),p,r);
1064: p1=cgetg(2,t_COL); y[2]=(long)p1;
1065: p1[1]=un; return y;
1066: }
1067:
1068: /* a etant un p-adique, retourne le vecteur des racines p-adiques de f
1069: * congrues a a modulo p dans le cas ou on suppose f(a) congru a 0 modulo p
1070: * (ou a 4 si p=2).
1071: */
1072: GEN
1073: apprgen(GEN f, GEN a)
1074: {
1075: GEN fp,p1,p,pro,x,x2,u,ip,quatre;
1076: long av=avma,tetpil,v,ps,i,j,k,lu,n,fl2;
1077:
1078: if (typ(f)!=t_POL) err(notpoler,"apprgen");
1079: if (gcmp0(f)) err(zeropoler,"apprgen");
1080: if (typ(a) != t_PADIC) err(rootper1);
1081: f = padic_pol_to_int(f);
1082: fp=derivpol(f); p1=ggcd(f,fp);
1083: if (lgef(p1)>3) { f=gdeuc(f,p1); fp=derivpol(f); }
1084: p=(GEN)a[2]; p1=poleval(f,a);
1085: v=ggval(p1,p); if (v <= 0) err(rootper2);
1086: fl2=egalii(p,gdeux);
1087: if (fl2)
1088: {
1089: if (v==1) err(rootper2);
1090: quatre=stoi(4);
1091: }
1092: v=ggval(poleval(fp,a),p);
1093: if (!v) /* simple zero */
1094: {
1095: while (!gcmp0(p1))
1096: {
1097: a = gsub(a,gdiv(p1,poleval(fp,a)));
1098: p1 = poleval(f,a);
1099: }
1100: tetpil=avma; pro=cgetg(2,t_VEC); pro[1]=lcopy(a);
1101: return gerepile(av,tetpil,pro);
1102: }
1103: n=lgef(f)-3; pro=cgetg(n+1,t_VEC);
1104: p1=poleval(f,gadd(a,gmul(fl2?quatre:p,polx[varn(f)])));
1105: if (!gcmp0(p1)) p1=gdiv(p1,gpuigs(p,ggval(p1,p)));
1106:
1107: if (is_bigint(p)) err(impl,"apprgen for p>=2^31");
1108: x = ggrandocp(p, valp(a) | precp(a));
1109: if (fl2)
1110: {
1111: ps=4; x2=ggrandocp(p,2); p=quatre;
1112: }
1113: else
1114: {
1115: ps=itos(p); x2=ggrandocp(p,1);
1116: }
1117: for (j=0,i=0; i<ps; i++)
1118: {
1119: ip=stoi(i);
1120: if (gcmp0(poleval(p1,gadd(ip,x2))))
1121: {
1122: u=apprgen(p1,gadd(x,ip)); lu=lg(u);
1123: for (k=1; k<lu; k++)
1124: {
1125: j++; pro[j]=ladd(a,gmul(p,(GEN)u[k]));
1126: }
1127: }
1128: }
1129: tetpil=avma; setlg(pro,j+1);
1130: return gerepile(av,tetpil,gcopy(pro));
1131: }
1132:
1133: /* Retourne le vecteur des racines p-adiques de f en precision r */
1134: GEN
1135: rootpadic(GEN f, GEN p, long r)
1136: {
1137: GEN fp,y,z,p1,pr,rac;
1138: long lx,i,j,k,n,av=avma,tetpil,fl2;
1139:
1140: if (typ(f)!=t_POL) err(notpoler,"rootpadic");
1141: if (gcmp0(f)) err(zeropoler,"rootpadic");
1142: if (r<=0) err(rootper4);
1143: f = padic_pol_to_int(f);
1144: fp=derivpol(f); p1=ggcd(f,fp);
1145: if (lgef(p1)>3) { f=gdeuc(f,p1); fp=derivpol(f); }
1146: fl2=egalii(p,gdeux); rac=(fl2 && r>=2)? rootmod(f,stoi(4)): rootmod(f,p);
1147: lx=lg(rac); p=gclone(p);
1148: if (r==1)
1149: {
1150: tetpil=avma; y=cgetg(lx,t_COL);
1151: for (i=1; i<lx; i++)
1152: {
1153: z=cgetg(5,t_PADIC); y[i]=(long)z;
1154: z[1] = evalprecp(1)|evalvalp(0);
1155: z[2] = z[3] = (long)p;
1156: z[4] = lcopy(gmael(rac,i,2));
1157: }
1158: return gerepile(av,tetpil,y);
1159: }
1160: n=lgef(f)-3; y=cgetg(n+1,t_COL);
1161: j=0; pr = NULL;
1162: z = cgetg(5,t_PADIC);
1163: z[2] = (long)p;
1164: for (i=1; i<lx; i++)
1165: {
1166: p1 = gmael(rac,i,2);
1167: if (signe(p1))
1168: {
1169: if (!fl2 || mod2(p1))
1170: {
1171: z[1] = evalvalp(0)|evalprecp(r);
1172: z[4] = (long)p1;
1173: }
1174: else
1175: {
1176: z[1] = evalvalp(1)|evalprecp(r);
1177: z[4] = un;
1178: }
1179: if (!pr) pr=gpuigs(p,r);
1180: z[3] = (long)pr;
1181: }
1182: else
1183: {
1184: z[1] = evalvalp(r);
1185: z[3] = un;
1186: z[4] = (long)p1;
1187: }
1188: p1 = apprgen(f,z);
1189: for (k=1; k<lg(p1); k++) y[++j]=p1[k];
1190: }
1191: tetpil=avma; setlg(y,j+1);
1192: return gerepile(av,tetpil,gcopy(y));
1193: }
1194:
1195: /* a usage interne. Pas de verifs ni de gestion de pile. On suppose que f est
1196: * un polynome a coeffs dans Z de degre n ayant n racines distinctes mod p, et
1197: * p>2, r>=2. On rend les n racines p-adiques en precision r si flall>0,
1198: * 1 seule si flall=0
1199: */
1200: GEN
1201: rootpadicfast(GEN f, GEN p, long r, long flall)
1202: {
1203: GEN rac,fp,y,z,p1,p2;
1204: long i,e,n;
1205:
1206: rac=rootmod(f,p); n=flall? lgef(f)-3: 1;
1207: fp=derivpol(f); y=cgetg(n+1,t_COL);
1208: p=gclone(p); p2=NULL;
1209: z=cgetg(5,t_PADIC); z[2]=(long)p;
1210: for (i=1; i<=n; i++)
1211: {
1212: p1=gmael(rac,i,2);
1213: if (signe(p1))
1214: {
1215: if (!p2) p2=sqri(p);
1216: z[1] = evalvalp(0)|evalprecp(2);
1217: z[3] = (long)p2;
1218: }
1219: else
1220: {
1221: z[1] = evalvalp(2);
1222: z[3] = un;
1223: }
1224: z[4] = (long)p1; p1 = z;
1225: for(e=2;;)
1226: {
1227: p1 = gsub(p1, gdiv(poleval(f,p1),poleval(fp,p1)));
1228: if (e==r) break;
1229: e<<=1; if (e>r) e=r;
1230: p1 = gprec(p1,e);
1231: }
1232: y[i] = (long)p1;
1233: }
1234: return y;
1235: }
1236:
1237: static long
1238: getprec(GEN x, long prec, GEN *p)
1239: {
1240: long i,e;
1241: GEN p1;
1242:
1243: for (i = lgef(x)-1; i>1; i--)
1244: {
1245: p1=(GEN)x[i];
1246: if (typ(p1)==t_PADIC)
1247: {
1248: e=valp(p1); if (signe(p1[4])) e += precp(p1);
1249: if (e<prec) prec = e; *p = (GEN)p1[2];
1250: }
1251: }
1252: return prec;
1253: }
1254:
1255: /* a appartenant a une extension finie de Q_p, retourne le vecteur des
1256: * racines de f congrues a a modulo p dans le cas ou on suppose f(a) congru a
1257: * 0 modulo p (ou a 4 si p=2).
1258: */
1259: GEN
1260: apprgen9(GEN f, GEN a)
1261: {
1262: GEN fp,p1,p,pro,x,x2,u,ip,t,vecg,quatre;
1263: long av=avma,tetpil,v,ps_1,i,j,k,lu,n,prec,d,va,fl2;
1264:
1265: if (typ(f)!=t_POL) err(notpoler,"apprgen9");
1266: if (gcmp0(f)) err(zeropoler,"apprgen9");
1267: if (typ(a)==t_PADIC) return apprgen(f,a);
1268: if (typ(a)!=t_POLMOD || typ(a[2])!=t_POL) err(rootper1);
1269: fp=derivpol(f); p1=ggcd(f,fp);
1270: if (lgef(p1)>3) { f=gdeuc(f,p1); fp=derivpol(f); }
1271: t=(GEN)a[1];
1272: prec = getprec((GEN)a[2], BIGINT, &p);
1273: prec = getprec(t, prec, &p);
1274: if (prec==BIGINT) err(rootper1);
1275:
1276: p1=poleval(f,a); v=ggval(lift_intern(p1),p); if (v<=0) err(rootper2);
1277: fl2=egalii(p,gdeux);
1278: if (fl2)
1279: {
1280: if (v==1) err(rootper2);
1281: quatre=stoi(4);
1282: }
1283: v=ggval(lift_intern(poleval(fp,a)), p);
1284: if (!v)
1285: {
1286: while (!gcmp0(p1))
1287: {
1288: a = gsub(a, gdiv(p1,poleval(fp,a)));
1289: p1 = poleval(f,a);
1290: }
1291: tetpil=avma; pro=cgetg(2,t_COL); pro[1]=lcopy(a);
1292: return gerepile(av,tetpil,pro);
1293: }
1294: n=lgef(f)-3; pro=cgetg(n+1,t_COL); j=0;
1295: p1=poleval(f,gadd(a,gmul(fl2?quatre:p,polx[varn(f)])));
1296: if (!gcmp0(p1)) p1=gdiv(p1,gpuigs(p,ggval(p1,p)));
1297:
1298: if (is_bigint(p)) err(impl,"apprgen9 for p>=2^31");
1299: x=gmodulcp(ggrandocp(p,prec), t);
1300: if (fl2)
1301: {
1302: ps_1=3; x2=ggrandocp(p,2); p=quatre;
1303: }
1304: else
1305: {
1306: ps_1=itos(p)-1; x2=ggrandocp(p,1);
1307: }
1308: d=lgef(t)-3; vecg=cgetg(d+1,t_COL);
1309: for (i=1; i<=d; i++)
1310: vecg[i] = (long)setloop(gzero);
1311: va=varn(t);
1312: for(;;) /* loop through F_q */
1313: {
1314: ip=gmodulcp(gtopoly(vecg,va),t);
1315: if (gcmp0(poleval(p1,gadd(ip,x2))))
1316: {
1317: u=apprgen9(p1,gadd(ip,x)); lu=lg(u);
1318: for (k=1; k<lu; k++)
1319: {
1320: j++; pro[j]=ladd(a,gmul(p,(GEN)u[k]));
1321: }
1322: }
1323: for (i=d; i; i--)
1324: {
1325: p1 = (GEN)vecg[i];
1326: if (p1[2] != ps_1) { (void)incloop(p1); break; }
1327: affsi(0, p1);
1328: }
1329: if (!i) break;
1330: }
1331: tetpil=avma; setlg(pro,j+1);
1332: return gerepile(av,tetpil,gcopy(pro));
1333: }
1334:
1335: /*****************************************/
1336: /* Factorisation p-adique d'un polynome */
1337: /*****************************************/
1338:
1339: /* factorise le polynome T=nf[1] dans Zp avec la precision pr */
1340: static GEN
1341: padicff2(GEN nf,GEN p,long pr)
1342: {
1343: long N=lgef(nf[1])-3,i,j,d,l;
1344: GEN mat,V,D,fa,p1,pk,dec_p,pke,a,theta;
1345:
1346: pk=gpuigs(p,pr); dec_p=primedec(nf,p);
1347: l=lg(dec_p); fa=cgetg(l,t_COL);
1348: for (i=1; i<l; i++)
1349: {
1350: p1 = (GEN)dec_p[i];
1351: pke = idealpows(nf,p1, pr * itos((GEN)p1[3]));
1352: p1=smith2(pke); V=(GEN)p1[3]; D=(GEN)p1[1];
1353: for (d=1; d<=N; d++)
1354: if (! egalii(gcoeff(V,d,d),pk)) break;
1355: a=ginv(D); theta=gmael(nf,8,2); mat=cgetg(d,t_MAT);
1356: for (j=1; j<d; j++)
1357: {
1358: p1 = gmul(D, element_mul(nf,theta,(GEN)a[j]));
1359: setlg(p1,d); mat[j]=(long)p1;
1360: }
1361: fa[i]=(long)caradj(mat,0,NULL);
1362: }
1363: a = cgetg(l,t_COL); pk = icopy(pk);
1364: for (i=1; i<l; i++)
1365: a[i] = (long)pol_to_padic((GEN)fa[i],pk,p,pr);
1366: return a;
1367: }
1368:
1369: static GEN
1370: padicff(GEN x,GEN p,long pr)
1371: {
1372: GEN q,bas,mul,dx,nf,mat;
1373: long n=lgef(x)-3,av=avma;
1374:
1375: nf=cgetg(10,t_VEC); nf[1]=(long)x; dx=discsr(x);
1376: mat=cgetg(3,t_MAT); mat[1]=lgetg(3,t_COL); mat[2]=lgetg(3,t_COL);
1377: coeff(mat,1,1)=(long)p; coeff(mat,1,2)=lstoi(pvaluation(dx,p,&q));
1378: coeff(mat,2,1)=(long)q; coeff(mat,2,2)=un;
1379: bas=allbase4(x,(long)mat,(GEN*)(nf+3),NULL);
1380: if (!carrecomplet(divii(dx,(GEN)nf[3]),(GEN*)(nf+4)))
1381: err(bugparier,"factorpadic2 (incorrect discriminant)");
1382: mul = get_mul_table(x,bas,NULL);
1383: nf[7]=(long)bas;
1384: nf[8]=linv(vecpol_to_mat(bas,n));
1385: nf[9]=lmul((GEN)nf[8],mul); nf[2]=nf[5]=nf[6]=zero;
1386: return gerepileupto(av,padicff2(nf,p,pr));
1387: }
1388:
1389: GEN
1390: factorpadic2(GEN x, GEN p, long r)
1391: {
1392: long av=avma,av2,k,i,j,i1,f,nbfac;
1393: GEN res,p1,p2,y,d,a,ap,t,v,w;
1394: GEN *fa;
1395:
1396: if (typ(x)!=t_POL) err(notpoler,"factorpadic");
1397: if (gcmp0(x)) err(zeropoler,"factorpadic");
1398: if (r<=0) err(rootper4);
1399:
1400: if (lgef(x)==3) return trivfact();
1401: if (lgef(x)==4) return padic_trivfact(x,p,r);
1402: y=cgetg(3,t_MAT);
1403: fa = (GEN*)new_chunk(lgef(x)-2);
1404: d=content(x); a=gdiv(x,d);
1405: ap=derivpol(a); t=ggcd(a,ap); v=gdeuc(a,t);
1406: w=gdeuc(ap,t); j=0; f=1; nbfac=0;
1407: while (f)
1408: {
1409: j++; w=gsub(w,derivpol(v)); f=signe(w);
1410: if (f) { res=ggcd(v,w); v=gdeuc(v,res); w=gdeuc(w,res); }
1411: else res=v;
1412: fa[j]=(lgef(res)>3) ? padicff(res,p,r) : cgetg(1,t_COL);
1413: nbfac += (lg(fa[j])-1);
1414: }
1415: av2=avma; y=cgetg(3,t_MAT);
1416: p1=cgetg(nbfac+1,t_COL); y[1]=(long)p1;
1417: p2=cgetg(nbfac+1,t_COL); y[2]=(long)p2;
1418: for (i=1,k=0; i<=j; i++)
1419: for (i1=1; i1<lg(fa[i]); i1++)
1420: {
1421: p1[++k]=lcopy((GEN)fa[i][i1]); p2[k]=lstoi(i);
1422: }
1423: return gerepile(av,av2,y);
1424: }
1425:
1426: /*******************************************************************/
1427: /* */
1428: /* FACTORISATION P-adique avec ROUND 4 */
1429: /* */
1430: /*******************************************************************/
1431: GEN polmodi(GEN x, GEN y);
1432: GEN polmodi_keep(GEN x, GEN y);
1433:
1434: static GEN
1435: Decomppadic(GEN p,long r,GEN f,long mf,GEN theta,GEN chi,GEN nu)
1436: {
1437: GEN pk,ph,pdr,unmodp;
1438: GEN pr,res,b1,b2,b3,a1,e,f1,f2;
1439: long valk;
1440:
1441: if (DEBUGLEVEL>=3)
1442: {
1443: fprintferr(" entering Decomp_padic ");
1444: if (DEBUGLEVEL>=4)
1445: {
1446: fprintferr("with params: p=%Z, exponent=%ld, prec=%ld\n", p,mf,r);
1447: fprintferr(" f=%Z",f);
1448: }
1449: fprintferr("\n");
1450: }
1451: unmodp=gmodulsg(1,p);
1452:
1453: pdr = respm(f, derivpol(f), gpuigs(p,mf));
1454:
1455: b1=lift_intern(gmul(chi,unmodp)); a1=gun;
1456: b2=gun; b3=lift_intern(gmul(nu,unmodp));
1457: while (lgef(b3) > 3)
1458: {
1459: GEN p1;
1460: b1 = Fp_deuc(b1,b3, p);
1461: b2 = Fp_pol_red(gmul(b2,b3), p);
1462: b3 = Fp_pol_extgcd(b2,b1, p, &a1,&p1);
1463: p1 = leading_term(b3);
1464: if (!gcmp1(p1))
1465: {
1466: p1 = mpinvmod(p1,p);
1467: b3 = gmul(b3,p1);
1468: a1 = gmul(a1,p1);
1469: }
1470: }
1471: e=eleval(f,Fp_pol_red(gmul(a1,b2), p),theta);
1472: if (padicprec(e,p) > 0)
1473: e=gdiv(polmodi(gmul(pdr,e), mulii(pdr,p)),pdr);
1474:
1475: pk=p; pr=gpuigs(p,r); ph=mulii(pdr,pr); valk = 1;
1476: /* E(t)-e(t) belongs to p^k Op, which is contained in p^(k-df)*Zp[xi] */
1477: while (cmpii(pk,ph) < 0)
1478: {
1479: e = gmul(e,gmul(e,gsubsg(3,gmul2n(e,1))));
1480: e = gres(e,f); pk=sqri(pk); valk <<= 1;
1481: if (valk<=padicprec(e,p))
1482: e = gdiv(polmodi(gmul(pdr,e), mulii(pk,pdr)), pdr);
1483: }
1484: f1=gcdpm(f,gmul(pdr,gsubsg(1,e)), ph);
1485: f1 = Fp_res(f1,f, pr);
1486: f2 = Fp_res(Fp_deuc(f,f1, pr), f, pr);
1487:
1488: if (DEBUGLEVEL>=4)
1489: {
1490: fprintferr(" leaving Decomp_padic with parameters: ");
1491: fprintferr("f1=%Z, f2=%Z\n",f1,f2);
1492: }
1493: b1=factorpadic4(f1,p,r);
1494: b2=factorpadic4(f2,p,r); res=cgetg(3,t_MAT);
1495: res[1]=lconcat((GEN)b1[1],(GEN)b2[1]);
1496: res[2]=lconcat((GEN)b1[2],(GEN)b2[2]); return res;
1497: }
1498:
1499: static GEN
1500: nilordpadic(GEN p,long r,GEN fx,long mf,GEN gx)
1501: {
1502: long Da,Na,La,Ma,first=1,n,v=varn(fx);
1503: GEN alpha,chi,nu,eta,w,phi,pmf,Dchi;
1504:
1505: if (DEBUGLEVEL>=3)
1506: {
1507: fprintferr(" entering Nilord_padic ");
1508: if (DEBUGLEVEL>=4)
1509: {
1510: fprintferr("with parameters: p=%Z, expo=%ld\n",p,mf);
1511: fprintferr(" fx=%Z, gx=%Z",fx,gx);
1512: }
1513: fprintferr("\n");
1514: }
1515: pmf=gpuigs(p,mf+1); alpha=polx[v]; n=lgef(fx)-3;
1516: for(;;)
1517: {
1518: if (first) { chi=fx; nu=gx; first=0; }
1519: else
1520: {
1521: w=factcp(p,fx,alpha);
1522: chi=(GEN)w[1]; nu=(GEN)w[2];
1523: if (cmpis((GEN)w[3],1)==1)
1524: return Decomppadic(p,r,fx,mf,alpha,chi,nu);
1525: }
1526: Da=lgef(nu)-3; Na=n/Da;
1527:
1528: if (mf+1<=padicprec(chi,p))
1529: {
1530: Dchi=modii(discsr(polmodi_keep(chi,pmf)), pmf);
1531: if (gcmp0(Dchi)) Dchi=discsr(chi);
1532: }
1533: else
1534: Dchi=discsr(chi);
1535:
1536: if (gcmp0(Dchi))
1537: alpha=gadd(alpha,gmul(p,polx[v]));
1538: else
1539: {
1540: if (vstar(p,chi)[0] > 0)
1541: alpha=gadd(alpha,gun);
1542: else
1543: {
1544: eta=setup(p,chi,polx[v],nu, &La, &Ma);
1545: if (La>1)
1546: alpha=gadd(alpha,eleval(fx,eta,alpha));
1547: else
1548: {
1549: if (Ma==Na) return NULL; /* correspond to [fx; 1] */
1550: w=bsrch(p,chi,ggval(Dchi,p),eta,Ma);
1551: phi=eleval(fx,(GEN)w[2],alpha);
1552: if (gcmp1((GEN)w[1]))
1553: return Decomppadic(p,r,fx,mf,phi,(GEN)w[3],(GEN)w[4]);
1554: alpha=phi;
1555: }
1556: }
1557: }
1558: }
1559: }
1560:
1561: static GEN
1562: squarefree(GEN f, GEN *ex)
1563: {
1564: GEN T,V,W,A,B;
1565: long n,i,k;
1566:
1567: T=ggcd(derivpol(f),f); V=gdeuc(f,T);
1568: n=lgef(f)-2; A=cgetg(n,t_COL); B=cgetg(n,t_COL);
1569: k=1; i=1;
1570: do
1571: {
1572: W=ggcd(T,V); T=gdeuc(T,W);
1573: if (lgef(V) != lgef(W))
1574: {
1575: A[i]=ldeuc(V,W); B[i]=k; i++;
1576: }
1577: k++; V=W;
1578: }
1579: while (lgef(V)>3);
1580: setlg(A,i); *ex=B; return A;
1581: }
1582:
1583: GEN
1584: factorpadic4(GEN f,GEN p,long prec)
1585: {
1586: GEN w,g,poly,fx,y,p1,p2,ex,pols,exps,ppow,lead;
1587: long v=varn(f),n=lgef(f)-3,av,tetpil,mfx,i,k,j,m,r,pr;
1588:
1589: if (typ(f)!=t_POL) err(notpoler,"factorpadic");
1590: if (gcmp0(f)) err(zeropoler,"factorpadic");
1591: if (prec<=0) err(rootper4);
1592:
1593: if (n==0) return trivfact();
1594: av=avma; f = padic_pol_to_int(f);
1595: if (n==1) return gerepileupto(av, padic_trivfact(f,p,prec));
1596: f = pol_to_monic(f, &lead);
1597: pr = lead? prec + (n-1) * ggval(lead,p): prec;
1598: poly=squarefree(f,&ex);
1599: pols=cgetg(n+1,t_COL);
1600: exps=cgetg(n+1,t_COL); n = lg(poly);
1601: for (j=1,i=1; i<n; i++)
1602: {
1603: long av1 = avma;
1604: fx=(GEN)poly[i]; mfx=ggval(discsr(fx),p);
1605: m = (pr<=mfx)?mfx+1:pr;
1606: w = (GEN)factmod(fx,p)[1]; r = lg(w)-1;
1607: g = lift_intern((GEN)w[r]);
1608: p2 = (r == 1)? nilordpadic(p,m,fx,mfx,g)
1609: : Decomppadic(p,m,fx,mfx,polx[v],fx,g);
1610: if (p2)
1611: {
1612: p2 = gerepileupto(av1,p2);
1613: p1 = (GEN)p2[1];
1614: p2 = (GEN)p2[2];
1615: for (k=1; k<lg(p1); k++,j++)
1616: {
1617: pols[j]=p1[k];
1618: exps[j]=lmulis((GEN)p2[k],ex[i]);
1619: }
1620: }
1621: else
1622: {
1623: avma=av1;
1624: pols[j]=(long)fx;
1625: exps[j]=lstoi(ex[i]); j++;
1626: }
1627: }
1628: if (lead)
1629: {
1630: p1 = gmul(polx[v],lead);
1631: for (i=1; i<j; i++)
1632: {
1633: p2 = poleval((GEN)pols[i], p1);
1634: pols[i] = ldiv(p2, content(p2));
1635: }
1636: }
1637:
1638: tetpil=avma; y=cgetg(3,t_MAT);
1639: p1 = cgetg(j,t_COL); ppow = gpowgs(p,prec); p = icopy(p);
1640: for (i=1; i<j; i++)
1641: p1[i] = (long)pol_to_padic((GEN)pols[i],ppow,p,prec);
1642: y[1]=(long)p1; setlg(exps,j);
1643: y[2]=lcopy(exps); return gerepile(av,tetpil,y);
1644: }
1645:
1646: GEN
1647: factorpadic0(GEN f,GEN p,long r,long flag)
1648: {
1649: switch(flag)
1650: {
1651: case 0: return factorpadic4(f,p,r);
1652: case 1: return factorpadic2(f,p,r);
1653: default: err(flagerr,"factorpadic");
1654: }
1655: return NULL; /* not reached */
1656: }
1657:
1658: /*******************************************************************/
1659: /* */
1660: /* FACTORISATION DANS F_q */
1661: /* */
1662: /*******************************************************************/
1663: static GEN
1664: to_fq(GEN x, GEN a, GEN p)
1665: {
1666: long i,lx = lgef(x);
1667: GEN p1, z = cgetg(3,t_POLMOD), pol = cgetg(lx,t_POL);
1668: pol[1] = x[1];
1669: for (i=2; i<lx; i++)
1670: {
1671: p1 = cgetg(3,t_INTMOD);
1672: p1[1] = (long)p;
1673: p1[2] = x[i]; pol[i] = (long)p1;
1674: }
1675: /* assume deg(pol) < deg(a) */
1676: z[1] = (long)a;
1677: z[2] = (long)pol; return z;
1678: }
1679:
1680: /* pol. in v whose coeff are the digits of m in base qq */
1681: static GEN
1682: stopoly9(GEN pg, GEN mm, GEN qq, long v, GEN a)
1683: {
1684: long q,l,m,l1,i,va, smll = !is_bigint(mm), p = pg[2];
1685: GEN y,p1,r;
1686:
1687: y = cgetg(bit_accuracy(lgefint(mm)) + 2, t_POL);
1688: va = varn(a);
1689: p1 = cgetg(bit_accuracy(lgefint(qq)) + 2,t_POL);
1690: p1[1] = evalsigne(1) | evalvarn(va);
1691: l = 2;
1692: if (smll)
1693: {
1694: q = itos(qq); m = itos(mm);
1695: do { y[l++] = m % q; m /= q; } while (m);
1696: }
1697: else
1698: do { mm=dvmdii(mm,qq,&r); y[l++]=(long)r; } while (signe(mm));
1699: if (smll)
1700: for (i=2; i<l; i++)
1701: {
1702: m=y[i]; l1=2;
1703: do { p1[l1++] = lstoi(m % p); m /= p; } while (m);
1704: setlgef(p1,l1); y[i]=(long)to_fq(p1,a,pg);
1705: }
1706: else
1707: for (i=2; i<l; i++)
1708: {
1709: mm=(GEN)y[i]; l1=2;
1710: do { mm=dvmdis(mm,p,&r); p1[l1++]=(long)r; } while (signe(mm));
1711: setlgef(p1,l1); y[i]=(long)to_fq(p1,a,pg);
1712: }
1713: y[1] = evalsigne(1) | evalvarn(v) | evallgef(l);
1714: return y;
1715: }
1716:
1717: /* renvoie un polynome aleatoire de la variable v
1718: de degre inferieur ou egal a 2*d1-1 */
1719: static GEN
1720: stopoly92(GEN pg, long d1, long v, GEN a, GEN *ptres)
1721: {
1722: GEN y,p1;
1723: long m,l1,i,d2,l,va=varn(a),k=lgef(a)-3,nsh;
1724:
1725: d2=2*d1+1; y=cgetg(d2+1,t_POL); y[1]=1;
1726: nsh=BITS_IN_RANDOM-1-k; if (nsh<=0) nsh=1;
1727: do
1728: {
1729: for (l=2; l<=d2; l++) y[l] = mymyrand() >> nsh;
1730: l=d2; while (!y[l]) l--;
1731: }
1732: while (l<=2);
1733: l++; y[1] = mymyrand() >> nsh;
1734: p1 = cgetg(BITS_IN_LONG+2,t_POL);
1735: p1[1] = evalsigne(1) | evalvarn(va);
1736: for (i=1; i<l; i++)
1737: {
1738: m=y[i]; l1=2;
1739: do { p1[l1++] = m&1? un: zero; m>>=1; } while (m);
1740: setlgef(p1,l1); y[i]=(long)to_fq(p1,a,pg);
1741: }
1742: *ptres = (GEN)y[1];
1743: y[1] = evalsigne(1) | evalvarn(v) | evallgef(l);
1744: return y;
1745: }
1746:
1747: static void
1748: split9(GEN m, GEN *t, long d, GEN pg, GEN q, GEN munfq, GEN qq, GEN a)
1749: {
1750: long l,dv,v,av0,av,tetpil,p;
1751: GEN w,res,polmod;
1752:
1753: dv=lgef(*t)-3; if (dv==d) return;
1754: v=varn(*t); m=setloop(m);
1755: av0=avma; p = pg[2];
1756: polmod=cgetg(3,t_POLMOD);
1757: polmod[1]=(long)dummyclone(*t);
1758: for(av=avma;;avma=av)
1759: {
1760: if (p==2)
1761: {
1762: polmod[2] = lres(stopoly92(pg,d,v,a,&res), *t);
1763: w = polmod; for (l=1; l<d; l++) w = gadd(polmod, powgi(w,qq));
1764: w = gadd((GEN)w[2], res); /* - res = res ! */
1765: }
1766: else
1767: {
1768: polmod[2] = lres(stopoly9(pg,m,qq,v,a), *t);
1769: m = incpos(m);
1770: w = powgi(polmod,q);
1771: w = gadd((GEN)w[2], munfq);
1772: }
1773: tetpil=avma; w=ggcd(*t,w); l=lgef(w)-3;
1774: if (l && l!=dv) break;
1775: }
1776: free((GEN)polmod[1]); w = gerepile(av0,tetpil,w);
1777: l /= d; t[l]=gdeuc(*t,w); *t=w;
1778: split9(m,t+l,d,pg,q,munfq,qq,a);
1779: split9(m,t ,d,pg,q,munfq,qq,a);
1780: }
1781:
1782: /* to "compare" (real) scalars and t_INTMODs */
1783: static int
1784: cmp_coeff(GEN x, GEN y)
1785: {
1786: if (typ(x) == t_INTMOD) x = (GEN)x[2];
1787: if (typ(y) == t_INTMOD) y = (GEN)y[2];
1788: return gcmp(x,y);
1789: }
1790:
1791: int
1792: cmp_pol(GEN x, GEN y)
1793: {
1794: long fx[3], fy[3];
1795: long i,lx,ly;
1796: int fl;
1797: if (typ(x) == t_POLMOD) x = (GEN)x[2];
1798: if (typ(y) == t_POLMOD) y = (GEN)y[2];
1799: if (typ(x) == t_POL) lx = lgef(x); else { lx = 3; fx[2] = (long)x; x = fx; }
1800: if (typ(y) == t_POL) ly = lgef(y); else { ly = 3; fy[2] = (long)y; y = fy; }
1801: if (lx > ly) return 1;
1802: if (lx < ly) return -1;
1803: for (i=lx-1; i>1; i--)
1804: if ((fl = cmp_coeff((GEN)x[i], (GEN)y[i]))) return fl;
1805: return 0;
1806: }
1807:
1808: GEN
1809: factmod9(GEN f, GEN pp, GEN a)
1810: {
1811: long av = avma, tetpil,p,i,j,k,d,e,vf,va,nbfact,nbf,pk;
1812: GEN ex,y,f2,f3,df1,df2,g,g1,xmod,u,v,pd,q,qq,unfp,unfq,munfq,tokill, *t;
1813: GEN frobinv = gpowgs(pp, lgef(a)-4);
1814:
1815: if (typ(a)!=t_POL || typ(f)!=t_POL || gcmp0(a)) err(typeer,"factmod9");
1816: vf=varn(f); va=varn(a);
1817: if (va<=vf) err(talker,"polynomial variable must be of higher priority than finite field\nvariable in factorff");
1818: p=itos(pp); unfp=gmodulss(1,p); a=gmul(unfp,a);
1819: unfq=gmodulo(gmul(unfp,polun[va]), a); tokill = (GEN)unfq[1];
1820: f = gmul(unfq,f); if (!signe(f)) err(zeropoler,"factmod9");
1821: d = lgef(f)-3; if (!d) { avma=av; gunclone(tokill); return trivfact(); }
1822:
1823: pp = gmael(a,2,1); /* out of the stack */
1824: t = (GEN*)cgetg(d+1,t_VEC); ex = new_chunk(d+1);
1825:
1826: xmod = cgetg(3,t_POLMOD);
1827: xmod[2] = lmul(polx[vf],unfq);
1828: munfq = gneg(unfq);
1829: qq=gpuigs(pp,lgef(a)-3);
1830: e = nbfact = 1;
1831: pk=1; df1=derivpol(f); f3=NULL;
1832: for(;;)
1833: {
1834: while (gcmp0(df1))
1835: {
1836: pk *= p; e=pk;
1837: j=(lgef(f)-3)/p+3; setlg(f,j); setlgef(f,j);
1838: for (i=2; i<j; i++) f[i] = (long)powgi((GEN)f[p*(i-2)+2], frobinv);
1839: df1=derivpol(f); f3=NULL;
1840: }
1841: f2 = f3? f3: ggcd(f,df1);
1842: if (lgef(f2)==3) u=f;
1843: else
1844: {
1845: g1=gdeuc(f,f2); df2=derivpol(f2);
1846: if (gcmp0(df2)) { u=g1; f3=f2; }
1847: else
1848: {
1849: f3=ggcd(f2,df2);
1850: if (lgef(f3)==3) u=gdeuc(g1,f2);
1851: else
1852: u=gdeuc(g1,gdeuc(f2,f3));
1853: }
1854: }
1855: /* Ici u est square-free (produit des facteurs premiers de meme
1856: * multiplicite e). On cherche le produit des facteurs de meme degre d
1857: */
1858: pd=gun; xmod[1]=(long)u; v=xmod;
1859: for (d=1; d <= (lgef(u)-3)>>1; d++)
1860: {
1861: pd=mulii(pd,qq); v=powgi(v,qq);
1862: g=ggcd((GEN)gsub(v,xmod)[2],u);
1863:
1864: if (lgef(g) > 3)
1865: {
1866: /* Ici g est produit de pol irred ayant tous le meme degre d; */
1867: j = nbfact+(lgef(g)-3)/d;
1868:
1869: t[nbfact]=g;
1870: q=shifti(subis(pd,1),-1);
1871: /* le premier parametre est un entier variable m qui sera
1872: * converti en un polynome w dont les coeff sont ses digits en
1873: * base p (initialement m = p --> X) pour faire pgcd de g avec
1874: * w^(p^d-1)/2 jusqu'a casser.
1875: */
1876: split9(addis(qq,1),t+nbfact,d,pp,q,munfq,qq,a);
1877: for (; nbfact<j; nbfact++) ex[nbfact]=e;
1878: u=gdeuc(u,g); v=gmodulcp((GEN)v[2],u);
1879: }
1880: }
1881: if (lgef(u)>3) { t[nbfact]=u; ex[nbfact++]=e; }
1882: if (lgef(f2) == 3) break;
1883:
1884: f=f2; df1=df2; e += pk;
1885: }
1886:
1887: nbf=nbfact; tetpil=avma; y=cgetg(3,t_MAT);
1888: for (j=1; j<nbfact; j++)
1889: {
1890: t[j]=gdiv((GEN)t[j],leading_term(t[j]));
1891: for (k=1; k<j; k++)
1892: if (ex[k] && gegal(t[j],t[k]))
1893: {
1894: ex[k] += ex[j]; ex[j]=0;
1895: nbf--; break;
1896: }
1897: }
1898: u=cgetg(nbf,t_COL); y[1]=(long)u;
1899: v=cgetg(nbf,t_COL); y[2]=(long)v;
1900: for (j=1,k=0; j<nbfact; j++)
1901: if (ex[j])
1902: {
1903: k++;
1904: u[k]=(long)t[j];
1905: v[k]=lstoi(ex[j]);
1906: }
1907: y = gerepile(av,tetpil,y);
1908: u=(GEN)y[1]; a = forcecopy(tokill);
1909: for (j=1; j<nbf; j++) fqunclone((GEN)u[j], a);
1910: (void)sort_factor(y, cmp_pol);
1911: gunclone(tokill); return y;
1912: }
1913:
1914: /*******************************************************************/
1915: /* */
1916: /* RACINES COMPLEXES */
1917: /* l represente la longueur voulue pour les parties */
1918: /* reelles et imaginaires des racines de x */
1919: /* */
1920: /*******************************************************************/
1921: GEN square_free_factorization(GEN pol);
1922: static GEN gnorml1(GEN x, long PREC);
1923: static GEN laguer(GEN pol,long N,GEN y0,GEN EPS,long PREC);
1924: static GEN zrhqr(GEN a,long PREC);
1925:
1926: GEN
1927: rootsold(GEN x, long l)
1928: {
1929: long av1=avma,i,j,f,g,gg,fr,deg,l0,l1,l2,l3,l4,ln;
1930: long exc,expmin,m,deg0,k,ti,h,ii,e,e1,emax,v;
1931: GEN y,xc,xd0,xd,xdabs,p1,p2,p3,p4,p5,p6,p7,p8;
1932: GEN p9,p10,p11,p12,p14,p15,pa,pax,pb,pp,pq,ps;
1933:
1934: if (typ(x)!=t_POL) err(typeer,"rootsold");
1935: v=varn(x); deg0=lgef(x)-3; expmin=12 - bit_accuracy(l);
1936: if (!signe(x)) err(zeropoler,"rootsold");
1937: y=cgetg(deg0+1,t_COL); if (!deg0) return y;
1938: for (i=1; i<=deg0; i++)
1939: {
1940: p1=cgetg(3,t_COMPLEX); p1[1]=lgetr(l); p1[2]=lgetr(l); y[i]=(long)p1;
1941: for (j=3; j<l; j++) ((GEN)p1[2])[j]=((GEN)p1[1])[j]=0;
1942: }
1943: g=1; gg=1; f=1;
1944: for (i=2; i<=deg0+2; i++)
1945: {
1946: ti=typ(x[i]);
1947: if (ti==t_REAL) gg=0;
1948: else if (ti==t_QUAD)
1949: {
1950: p2=gmael3(x,i,1,2);
1951: if (gsigne(p2)>0) g=0;
1952: } else if (ti != t_INT && ti != t_INTMOD && !is_frac_t(ti)) g=0;
1953: }
1954: l1=avma; p2=cgetg(3,t_COMPLEX);
1955: p2[1]=lmppi(DEFAULTPREC); p2[2]=ldivrs((GEN)p2[1],10);
1956: p11=cgetg(4,t_POL); p11[1]=evalsigne(1)+evallgef(4);
1957: setvarn(p11,v); p11[3]=un;
1958:
1959: p12=cgetg(5,t_POL); p12[1]=evalsigne(1)+evallgef(5);
1960: setvarn(p12,v); p12[4]=un;
1961: for (i=2; i<=deg0+2 && gcmp0((GEN)x[i]); i++) gaffsg(0,(GEN)y[i-1]);
1962: k=i-2;
1963: if (k!=deg0)
1964: {
1965: if (k)
1966: {
1967: j=deg0+3-k; pax=cgetg(j,t_POL);
1968: pax[1]=evalsigne(1)+evallgef(j); setvarn(pax,v);
1969: for (i=2; i<j; i++) pax[i]=x[i+k];
1970: }
1971: else pax=x;
1972: xd0=deriv(pax,v); m=1; pa=pax;
1973: if (gg) { pp=ggcd(pax,xd0); h=isnonscalar(pp); if (h) pq=gdeuc(pax,pp); }
1974: else{ pp=gun; h=0; }
1975: do
1976: {
1977: if (h)
1978: {
1979: pa=pp; pb=pq; pp=ggcd(pa,deriv(pa,v)); h=isnonscalar(pp);
1980: if (h) pq=gdeuc(pa,pp); else pq=pa; ps=gdeuc(pb,pq);
1981: }
1982: else ps=pa;
1983: /* calcul des racines d'ordre exactement m */
1984: deg=lgef(ps)-3;
1985: if (deg)
1986: {
1987: l3=avma; e=gexpo((GEN)ps[deg+2]); emax=e;
1988: for (i=2; i<deg+2; i++)
1989: {
1990: p3=(GEN)(ps[i]);
1991: e1=gexpo(p3); if (e1>emax) emax=e1;
1992: }
1993: e=emax-e; if (e<0) e=0; avma=l3; if (ps!=pax) xd0=deriv(ps,v);
1994: xdabs=cgetg(deg+2,t_POL); xdabs[1]=xd0[1];
1995: for (i=2; i<deg+2; i++)
1996: {
1997: l3=avma; p3=(GEN)xd0[i];
1998: p4=gabs(greal(p3),l);
1999: p5=gabs(gimag(p3),l); l4=avma;
2000: xdabs[i]=lpile(l3,l4,gadd(p4,p5));
2001: }
2002: l0=avma; xc=gcopy(ps); xd=gcopy(xd0); l2=avma;
2003: for (i=1; i<=deg; i++)
2004: {
2005: if (i==deg)
2006: {
2007: p1=(GEN)y[k+m*i]; gdivz(gneg_i((GEN)xc[2]),(GEN)xc[3],p1);
2008: p14=(GEN)(p1[1]); p15=(GEN)(p1[2]);
2009: }
2010: else
2011: {
2012: p3=gshift(p2,e); p4=poleval(xc,p3); p5=gnorm(p4); exc=0;
2013: while (exc>= -20)
2014: {
2015: p6=poleval(xd,p3); p7=gneg_i(gdiv(p4,p6)); f=1;
2016: l3=avma;
2017: if (gcmp0(p5)) exc= -32;
2018: else exc=expo(gnorm(p7))-expo(gnorm(p3));
2019: avma=l3;
2020: for (j=1; j<=10 && f; j++)
2021: {
2022: p8=gadd(p3,p7); p9=poleval(xc,p8); p10=gnorm(p9);
2023: f=(cmprr(p10,p5)>=0)&&(exc>= -20);
2024: if (f){ gshiftz(p7,-2,p7); avma=l3; }
2025: }
2026: if (f)
2027: {
2028: avma=av1;
2029: if (DEBUGLEVEL)
2030: {
2031: fprintferr("too many iterations in rootsold(): ");
2032: fprintferr("using roots2()\n"); flusherr();
2033: }
2034: return roots2(x,l);
2035: }
2036: else
2037: {
2038: GEN *gptr[3];
2039: p3=p8; p4=p9; p5=p10;
2040: gptr[0]=&p3; gptr[1]=&p4; gptr[2]=&p5;
2041: gerepilemanysp(l2,l3,gptr,3);
2042: }
2043: }
2044: p1=(GEN)y[k+m*i]; setlg(p1[1],3); setlg(p1[2],3); gaffect(p3,p1);
2045: avma=l2; p14=(GEN)(p1[1]); p15=(GEN)(p1[2]);
2046: for (ln=4; ln<=l; ln=(ln<<1)-2)
2047: {
2048: setlg(p14,ln); setlg(p15,ln);
2049: if (gcmp0(p14)) { settyp(p14,t_INT); p14[1]=2; }
2050: if (gcmp0(p15)) { settyp(p15,t_INT); p15[1]=2; }
2051: p4=poleval(xc,p1);
2052: p5=poleval(xd,p1); p6=gneg_i(gdiv(p4,p5));
2053: settyp(p14,t_REAL); settyp(p15,t_REAL);
2054: gaffect(gadd(p1,p6),p1); avma=l2;
2055: }
2056: }
2057: setlg(p14,l); setlg(p15,l);
2058: p7=gcopy(p1); p14=(GEN)(p7[1]); p15=(GEN)(p7[2]);
2059: setlg(p14,l+1); setlg(p15,l+1);
2060: if (gcmp0(p14)) { settyp(p14,t_INT); p14[1]=2; }
2061: if (gcmp0(p15)) { settyp(p15,t_INT); p15[1]=2; }
2062: for (ii=1; ii<=5; ii++)
2063: {
2064: p4=poleval(ps,p7); p5=poleval(xd0,p7);
2065: p6=gneg_i(gdiv(p4,p5)); p7=gadd(p7,p6);
2066: p14=(GEN)(p7[1]); p15=(GEN)(p7[2]);
2067: if (gcmp0(p14)) { settyp(p14,t_INT); p14[1]=2; }
2068: if (gcmp0(p15)) { settyp(p15,t_INT); p15[1]=2; }
2069: }
2070: gaffect(p7,p1); p4=poleval(ps,p7);
2071: p6=gdiv(p4,poleval(xdabs,gabs(p7,l)));
2072: if (gexpo(p6)>=expmin)
2073: {
2074: avma=av1;
2075: if (DEBUGLEVEL)
2076: {
2077: fprintferr("internal error in rootsold(): using roots2()\n");
2078: flusherr();
2079: }
2080: return roots2(x,l);
2081: }
2082: avma=l2;
2083: if (expo(p1[2])<expmin && g)
2084: {
2085: gaffect(gzero,(GEN)p1[2]);
2086: for (j=1; j<m; j++) gaffect(p1,(GEN)y[k+(i-1)*m+j]);
2087: p11[2]=lneg((GEN)p1[1]);
2088: l4=avma; xc=gerepile(l0,l4,gdeuc(xc,p11));
2089: }
2090: else
2091: {
2092: for (j=1; j<m; j++) gaffect(p1,(GEN)y[k+(i-1)*m+j]);
2093: if (g)
2094: {
2095: p1=gconj(p1);
2096: for (j=1; j<=m; j++) gaffect(p1,(GEN)y[k+i*m+j]);
2097: i++;
2098: p12[2]=lnorm(p1); p12[3]=lmulsg(-2,(GEN)p1[1]); l4=avma;
2099: xc=gerepile(l0,l4,gdeuc(xc,p12));
2100: }
2101: else
2102: {
2103: p11[2]=lneg(p1); l4=avma;
2104: xc=gerepile(l0,l4,gdeuc(xc,p11));
2105: }
2106: }
2107: xd=deriv(xc,v); l2=avma;
2108: }
2109: k=k+deg*m;
2110: }
2111: m++;
2112: }
2113: while (k!=deg0);
2114: }
2115: avma=l1;
2116: if (deg0>1)
2117: {
2118: for (j=2; j<=deg0; j++)
2119: {
2120: p1=(GEN)y[j]; if (gcmp0((GEN)p1[2])) fr=0; else fr=1;
2121: for (k=j-1; k>=1; k--)
2122: {
2123: if (gcmp0((GEN)((GEN)y[k])[2])) f=0; else f=1;
2124: if (f<fr) break;
2125: if (f==fr && gcmp(gmael(y,k,1),(GEN)p1[1])<=0) break;
2126: y[k+1]=y[k];
2127: }
2128: y[k+1]=(long)p1;
2129: }
2130: }
2131: return y;
2132: }
2133:
2134: #if 0
2135: GEN
2136: rootslong(GEN x, long l)
2137: {
2138: long av1=avma,i,j,f,g,fr,deg,l0,l1,l2,l3,l4,ln;
2139: long exc,expmin,m,deg0,k,ti,h,ii,e,e1,emax,v;
2140: GEN y,xc,xd0,xd,xdabs,p1,p2,p3,p4,p5,p6,p7,p8;
2141: GEN p9,p10,p11,p12,p14,p15,pa,pax,pb,pp,pq,ps;
2142:
2143: if (typ(x)!=t_POL) err(typeer,"rootslong");
2144: v=varn(x); deg0=lgef(x)-3; expmin = -bit_accuracy(l)+12;
2145: if (!signe(x)) err(zeropoler,"rootslong");
2146: y=cgetg(deg0+1,t_COL); if (!deg0) return y;
2147: for (i=1; i<=deg0; i++)
2148: {
2149: p1=cgetg(3,t_COMPLEX); y[i]=(long)p1;
2150: p1[1]=lgetr(l); p1[2]=lgetr(l);
2151: for (j=3; j<l; j++) mael(p1,2,j)=mael(p1,1,j)=0;
2152: }
2153: g=1; f=1;
2154: for (i=2; i<=deg0+2; i++)
2155: {
2156: ti=typ(x[i]);
2157: if (ti==t_QUAD)
2158: {
2159: p2=gmael3(x,i,1,2);
2160: if (gcmpgs(p2,0)>0) g=0;
2161: }
2162: else
2163: if (!is_const_t(ti) || ti==t_PADIC || ti==t_COMPLEX) g=0;
2164: }
2165: l1=avma; p2=cgetg(3,t_COMPLEX);
2166: p2[1]=lmppi(l);
2167: p2[2]=ldivrs((GEN)p2[1],10);
2168: p11=cgetg(4,t_POL); p11[1]=evalsigne(1)+evallgef(4); setvarn(p11,v); p11[3]=un;
2169: p12=cgetg(5,t_POL); p12[1]=evalsigne(1)+evallgef(5); setvarn(p12,v); p12[4]=un;
2170: for (i=2; (i<=deg0+2)&&(gcmp0((GEN)x[i])); i++)
2171: gaffsg(0,(GEN)y[i-1]); k=i-2;
2172: if (k!=deg0)
2173: {
2174: if (k)
2175: {
2176: j=deg0+3-k; pax=cgetg(j,t_POL); pax[1]=evalsigne(1)+evallgef(j);
2177: setvarn(pax,v);
2178: for (i=2; i<j; i++)
2179: pax[i]=x[i+k];
2180: }
2181: else pax=x;
2182: xd0=deriv(pax,v); pp=ggcd(pax,xd0); m=1; pa=pax;
2183: h=isnonscalar(pp); if (h) pq=gdeuc(pax,pp);
2184: do
2185: {
2186: if (h)
2187: {
2188: pa=pp; pb=pq;
2189: pp=ggcd(pa,deriv(pa,v)); h=isnonscalar(pp);
2190: if (h) pq=gdeuc(pa,pp); else pq=pa;
2191: ps=gdeuc(pb,pq);
2192: }
2193: else ps=pa;
2194: /* calcul des racines d'ordre exactement m */
2195: deg=lgef(ps)-3;
2196: if (deg)
2197: {
2198: l3=avma; e=gexpo((GEN)ps[deg+2]); emax=e;
2199: for (i=2; i<deg+2; i++)
2200: {
2201: p3=(GEN)(ps[i]);
2202: if (!gcmp0(p3))
2203: {
2204: e1=gexpo(p3);
2205: if (e1>emax) emax=e1;
2206: }
2207: }
2208: e=emax-e; if (e<0) e=0; avma=l3;
2209: if (ps!=pax) xd0=deriv(ps,v);
2210: xdabs=cgetg(deg+2,t_POL); xdabs[1]=xd0[1];
2211: for (i=2; i<deg+2; i++)
2212: {
2213: l3=avma; p3=(GEN)xd0[i]; p4=gabs(greal(p3),l);
2214: p5=gabs(gimag(p3),l); l4=avma;
2215: xdabs[i]=lpile(l3,l4,gadd(p4,p5));
2216: }
2217: l0=avma; xc=gcopy(ps); xd=gcopy(xd0); l2=avma;
2218: for (i=1; i<=deg; i++)
2219: {
2220: if (i==deg)
2221: {
2222: p1=(GEN)y[k+m*i];
2223: gdivz(gneg_i((GEN)xc[2]),(GEN)xc[3],p1);
2224: p14=(GEN)(p1[1]); p15=(GEN)(p1[2]);
2225: }
2226: else
2227: {
2228: p3=gshift(p2,e); p4=poleval(xc,p3);
2229: p5=gnorm(p4); exc=0;
2230: while (exc>= -20)
2231: {
2232: p6=poleval(xd,p3); p7=gneg_i(gdiv(p4,p6));
2233: f=1; l3=avma; if (gcmp0(p5)) exc= -32;
2234: else exc=expo(gnorm(p7))-expo(gnorm(p3));
2235: avma=l3;
2236: for (j=1; (j<=50)&&f; j++)
2237: {
2238: p8=gadd(p3,p7); p9=poleval(xc,p8);
2239: p10=gnorm(p9);
2240: f=(cmprr(p10,p5)>=0)&&(exc>= -20);
2241: if (f) { gshiftz(p7,-2,p7); avma=l3; }
2242: }
2243: if (f) err(poler9);
2244: else
2245: {
2246: GEN *gptr[3];
2247: p3=p8; p4=p9; p5=p10;
2248: gptr[0]=&p3; gptr[1]=&p4; gptr[2]=&p5;
2249: gerepilemanysp(l2,l3,gptr,3);
2250: }
2251: }
2252: p1=(GEN)y[k+m*i]; gaffect(p3,p1); avma=l2;
2253: p14=(GEN)(p1[1]); p15=(GEN)(p1[2]);
2254: for (ln=4; ln<=l; ln=(ln<<1)-2)
2255: {
2256: if (gcmp0(p14))
2257: { settyp(p14,t_INT); p14[1]=2; }
2258: if (gcmp0(p15))
2259: { settyp(p15,t_INT); p15[1]=2; }
2260: p4=poleval(xc,p1); p5=poleval(xd,p1);
2261: p6=gneg_i(gdiv(p4,p5));
2262: settyp(p14,t_REAL); settyp(p15,t_REAL);
2263: gaffect(gadd(p1,p6),p1); avma=l2;
2264: }
2265: }
2266: p7=gcopy(p1);
2267: p14=(GEN)(p7[1]); p15=(GEN)(p7[2]);
2268: setlg(p14,l+1); setlg(p15,l+1);
2269: if (gcmp0(p14))
2270: { settyp(p14,t_INT); p14[1]=2; }
2271: if (gcmp0(p15))
2272: { settyp(p15,t_INT); p15[1]=2; }
2273: for (ii=1; ii<=max(32,((e<<TWOPOTBITS_IN_LONG)+2)); ii<<=1)
2274: {
2275: p4=poleval(ps,p7); p5=poleval(xd0,p7);
2276: p6=gneg_i(gdiv(p4,p5)); p7=gadd(p7,p6);
2277: p14=(GEN)(p7[1]); p15=(GEN)(p7[2]);
2278: if (gcmp0(p14))
2279: { settyp(p14,t_INT); p14[1]=2; }
2280: if (gcmp0(p15))
2281: { settyp(p15,t_INT); p15[1]=2; }
2282: }
2283: gaffect(p7,p1); p4=poleval(ps,p7);
2284: p6=gdiv(p4,poleval(xdabs,gabs(p7,l)));
2285: if ((!gcmp0(p6))&&(gexpo(p6)>=expmin))
2286: {
2287: avma=av1;
2288: if (DEBUGLEVEL)
2289: {
2290: fprintferr("internal error in roots: using roots2\n"); flusherr();
2291: }
2292: return roots2(x,l);
2293: }
2294: avma=l2;
2295: if ((expo(p1[2])<expmin)&&g)
2296: {
2297: gaffect(gzero,(GEN)p1[2]);
2298: for (j=1; j<m; j++)
2299: gaffect(p1,(GEN)y[k+(i-1)*m+j]);
2300: p11[2]=lneg((GEN)p1[1]); l4=avma;
2301: xc=gerepile(l0,l4,gdeuc(xc,p11));
2302: }
2303: else
2304: {
2305: for (j=1; j<m; j++)
2306: gaffect(p1,(GEN)y[k+(i-1)*m+j]);
2307: if (g)
2308: {
2309: p1=gconj(p1);
2310: for (j=1; j<=m; j++)
2311: gaffect(p1,(GEN)y[k+i*m+j]); i++;
2312: p12[2]=lnorm(p1);
2313: p12[3]=lmulsg(-2,(GEN)p1[1]);
2314: l4=avma;
2315: xc=gerepile(l0,l4,gdeuc(xc,p12));
2316: }
2317: else
2318: {
2319: p11[2]=lneg(p1); l4=avma;
2320: xc=gerepile(l0,l4,gdeuc(xc,p11));
2321: }
2322: }
2323: xd=deriv(xc,v); l2=avma;
2324: }
2325: k=k+deg*m;
2326: }
2327: m++;
2328: }
2329: while (k!=deg0);
2330: }
2331: avma=l1;
2332: if (deg0>1)
2333: {
2334: for (j=2; j<=deg0; j++)
2335: {
2336: p1=(GEN)y[j]; if (gcmp0((GEN)p1[2])) fr=0; else fr=1;
2337: for (k=j-1; k>=1; k--)
2338: {
2339: if (gcmp0((GEN)((GEN)y[k])[2])) f=0; else f=1;
2340: if (f<fr) break;
2341: if ((f==fr)&&(gcmp((GEN)((GEN)y[k])[1],(GEN)p1[1])<=0)) break;
2342: y[k+1]=y[k];
2343: }
2344: y[k+1]=(long)p1;
2345: }
2346: }
2347: return y;
2348: }
2349: #endif
2350:
2351: GEN
2352: roots2(GEN pol,long PREC)
2353: {
2354: long av = avma,tetpil,N,flagexactpol,flagrealpol,flagrealrac,ti,i,j;
2355: long nbpol,k,av1,multiqol,deg,nbroot,fr,f;
2356: GEN p1,p2,rr,EPS,qol,qolbis,x,b,c,*ad,v,tabqol;
2357:
2358: if (typ(pol)!=t_POL) err(typeer,"roots2");
2359: if (!signe(pol)) err(zeropoler,"roots2");
2360: N=lgef(pol)-3;
2361: if (!N) return cgetg(1,t_COL);
2362: if (N==1)
2363: {
2364: p1=gmul(realun(PREC),(GEN)pol[3]);
2365: p2=gneg_i(gdiv((GEN)pol[2],p1));
2366: tetpil=avma; return gerepile(av,tetpil,gcopy(p2));
2367: }
2368: EPS=realun(3); setexpo(EPS, 12 - bit_accuracy(PREC));
2369: flagrealpol=1; flagexactpol=1;
2370: for (i=2; i<=N+2; i++)
2371: {
2372: ti=typ(pol[i]);
2373: if (ti!=t_INT && ti!=t_INTMOD && !is_frac_t(ti))
2374: {
2375: flagexactpol=0;
2376: if (ti!=t_REAL) flagrealpol=0;
2377: }
2378: if (ti==t_QUAD)
2379: {
2380: p1=gmael3(pol,i,1,2);
2381: flagrealpol = (gsigne(p1)>0)? 0 : 1;
2382: }
2383: }
2384: rr=cgetg(N+1,t_COL);
2385: for (i=1; i<=N; i++)
2386: {
2387: rr[i]=lgetg(3,t_COMPLEX); p1=(GEN)rr[i];
2388: mael(rr,i,1)=lgetr(PREC); mael(rr,i,2)=lgetr(PREC);
2389: for (j=3; j<PREC; j++) mael(p1,2,j)=mael(p1,1,j)=0;
2390: }
2391: if (flagexactpol) tabqol=square_free_factorization(pol);
2392: else
2393: {
2394: tabqol=cgetg(3,t_MAT);
2395: tabqol[1]=lgetg(2,t_COL); mael(tabqol,1,1)=un;
2396: tabqol[2]=lgetg(2,t_COL); mael(tabqol,2,1)=lcopy(pol);
2397: }
2398: nbpol=lg(tabqol[1])-1; nbroot=0;
2399: for (k=1; k<=nbpol; k++)
2400: {
2401: av1=avma; qol=gmael(tabqol,2,k); qolbis=gcopy(qol);
2402: multiqol=itos(gmael(tabqol,1,k)); deg=lgef(qol)-3;
2403: for (j=deg; j>=1; j--)
2404: {
2405: x=gzero; flagrealrac=0;
2406: if (j==1) x=gneg_i(gdiv((GEN)qolbis[2],(GEN)qolbis[3]));
2407: else
2408: {
2409: x=laguer(qolbis,j,x,EPS,PREC);
2410: if (x == NULL) goto RLAB;
2411: }
2412: if (flagexactpol)
2413: {
2414: x=gprec(x,(long)((PREC-1)*pariK));
2415: x=laguer(qol,deg,x,gmul2n(EPS,-32),PREC+1);
2416: }
2417: else x=laguer(qol,deg,x,EPS,PREC);
2418: if (x == NULL) goto RLAB;
2419:
2420: if (typ(x)==t_COMPLEX &&
2421: gcmp(gabs(gimag(x),PREC),gmul2n(gmul(EPS,gabs(greal(x),PREC)),1))<=0)
2422: { x[2]=zero; flagrealrac=1; }
2423: else if (j==1 && flagrealpol)
2424: { x[2]=zero; flagrealrac=1; }
2425: else if (typ(x)!=t_COMPLEX) flagrealrac=1;
2426:
2427: for (i=1; i<=multiqol; i++) gaffect(x,(GEN)rr[nbroot+i]);
2428: nbroot+=multiqol;
2429: if (!flagrealpol || flagrealrac)
2430: {
2431: ad = (GEN*) new_chunk(j+1);
2432: for (i=0; i<=j; i++) ad[i]=(GEN)qolbis[i+2];
2433: b=(GEN)ad[j];
2434: for (i=j-1; i>=0; i--)
2435: {
2436: c=(GEN)ad[i]; ad[i]=b;
2437: b=gadd(gmul((GEN)rr[nbroot],b),c);
2438: }
2439: v=cgetg(j+1,t_VEC); for (i=1; i<=j; i++) v[i]=(long)ad[j-i];
2440: qolbis=gtopoly(v,varn(qolbis));
2441: if (flagrealpol)
2442: for (i=2; i<=j+1; i++)
2443: if (typ(qolbis[i])==t_COMPLEX) mael(qolbis,i,2)=zero;
2444: }
2445: else
2446: {
2447: ad = (GEN*) new_chunk(j-1); ad[j-2]=(GEN)qolbis[j+2];
2448: p1=gmulsg(2,greal((GEN)rr[nbroot])); p2=gnorm((GEN)rr[nbroot]);
2449: ad[j-3]=gadd((GEN)qolbis[j+1],gmul(p1,ad[j-2]));
2450: for (i=j-2; i>=2; i--)
2451: ad[i-2] = gadd((GEN)qolbis[i+2],gsub(gmul(p1,ad[i-1]),gmul(p2,ad[i])));
2452: v=cgetg(j,t_VEC); for (i=1; i<=j-1; i++) v[i]=(long)ad[j-1-i];
2453: qolbis=gtopoly(v,varn(qolbis));
2454: for (i=2; i<=j; i++)
2455: if (typ(qolbis[i])==t_COMPLEX) mael(qolbis,i,2)=zero;
2456: for (i=1; i<=multiqol; i++)
2457: gaffect(gconj((GEN)rr[nbroot]), (GEN)rr[nbroot+i]);
2458: nbroot+=multiqol; j--;
2459: }
2460: }
2461: avma=av1;
2462: }
2463: for (j=2; j<=N; j++)
2464: {
2465: x=(GEN)rr[j]; if (gcmp0((GEN)x[2])) fr=0; else fr=1;
2466: for (i=j-1; i>=1; i--)
2467: {
2468: if (gcmp0(gmael(rr,i,2))) f=0; else f=1;
2469: if (f<fr) break;
2470: if (f==fr && gcmp(greal((GEN)rr[i]),greal(x)) <= 0) break;
2471: rr[i+1]=rr[i];
2472: }
2473: rr[i+1]=(long)x;
2474: }
2475: tetpil=avma; return gerepile(av,tetpil,gcopy(rr));
2476:
2477: RLAB:
2478: avma=av;
2479: for(i=2;i<=N+2;i++)
2480: {
2481: ti=typ(pol[i]);
2482: if (is_intreal_t(ti) || ti==t_INTMOD) err(poler9);
2483: }
2484: if (DEBUGLEVEL)
2485: {
2486: fprintferr("too many iterations in roots2() ( laguer() ): \n");
2487: fprintferr(" real coefficients polynomial, using zrhqr()\n");
2488: flusherr();
2489: }
2490: return zrhqr(pol,PREC);
2491: }
2492:
2493: #define MR 8
2494: #define MT 10
2495:
2496: static GEN
2497: laguer(GEN pol,long N,GEN y0,GEN EPS,long PREC)
2498: {
2499: long av = avma, av1,MAXIT,iter,i,j;
2500: GEN rac,erre,I,x,abx,abp,abm,dx,x1,b,d,f,g,h,sq,gp,gm,g2,*ffrac;
2501:
2502: MAXIT=MR*MT; rac=cgetg(3,t_COMPLEX);
2503: rac[1]=lgetr(PREC); rac[2]=lgetr(PREC);
2504: av1 = avma;
2505: I=cgetg(3,t_COMPLEX); I[1]=un; I[2]=un;
2506: ffrac=(GEN*)new_chunk(MR+1); for (i=0; i<=MR; i++) ffrac[i]=cgetr(PREC);
2507: affrr(dbltor(0.0), ffrac[0]); affrr(dbltor(0.5), ffrac[1]);
2508: affrr(dbltor(0.25),ffrac[2]); affrr(dbltor(0.75),ffrac[3]);
2509: affrr(dbltor(0.13),ffrac[4]); affrr(dbltor(0.38),ffrac[5]);
2510: affrr(dbltor(0.62),ffrac[6]); affrr(dbltor(0.88),ffrac[7]);
2511: affrr(dbltor(1.0),ffrac[8]);
2512: x=y0;
2513: for (iter=1; iter<=MAXIT; iter++)
2514: {
2515: b=(GEN)pol[N+2]; erre=gnorml1(b,PREC);
2516: d=gzero; f=gzero; abx=gnorml1(x,PREC);
2517: for (j=N-1; j>=0; j--)
2518: {
2519: f=gadd(gmul(x,f),d); d=gadd(gmul(x,d),b);
2520: b=gadd(gmul(x,b),(GEN)pol[j+2]);
2521: erre=gadd(gnorml1(b,PREC),gmul(abx,erre));
2522: }
2523: erre=gmul(erre,EPS);
2524: if (gcmp(gnorml1(b,PREC),erre)<=0)
2525: {
2526: gaffect(x,rac); avma = av1; return rac;
2527: }
2528: g=gdiv(d,b); g2=gsqr(g); h=gsub(g2, gmul2n(gdiv(f,b),1));
2529: sq=gsqrt(gmulsg(N-1,gsub(gmulsg(N,h),g2)),PREC);
2530: gp=gadd(g,sq); gm=gsub(g,sq); abp=gnorm(gp); abm=gnorm(gm);
2531: if (gcmp(abp,abm)<0) gp=gcopy(gm);
2532: if (gsigne(gmax(abp,abm))==1)
2533: dx = gdivsg(N,gp);
2534: else
2535: dx = gmul(gadd(gun,abx),gexp(gmulgs(I,iter),PREC));
2536: x1=gsub(x,dx);
2537: if (gcmp(gnorml1(gsub(x,x1),PREC),EPS)<0)
2538: {
2539: gaffect(x,rac); avma = av1; return rac;
2540: }
2541: if (iter%MT) x=gcopy(x1); else x=gsub(x,gmul(ffrac[iter/MT],dx));
2542: }
2543: avma=av; return NULL;
2544: }
2545:
2546: #undef MR
2547: #undef MT
2548:
2549: static GEN
2550: gnorml1(GEN x,long PREC)
2551: {
2552: long av,tetpil,lx,i;
2553: GEN p1,p2,s;
2554: av=avma;
2555: switch(typ(x))
2556: {
2557: case t_INT: case t_REAL: case t_FRAC: case t_FRACN:
2558: return gabs(x,PREC);
2559:
2560: case t_INTMOD: case t_PADIC: case t_POLMOD: case t_POL:
2561: case t_SER: case t_RFRAC: case t_RFRACN: case t_QFR: case t_QFI:
2562: return gcopy(x);
2563:
2564: case t_COMPLEX:
2565: p1=gabs((GEN)x[1],PREC); p2=gabs((GEN)x[2],PREC); tetpil=avma;
2566: return gerepile(av,tetpil,gadd(p1,p2));
2567:
2568: case t_QUAD:
2569: p1=gabs((GEN)x[2],PREC); p2=gabs((GEN)x[3],PREC); tetpil=avma;
2570: return gerepile(av,tetpil,gadd(p1,p2));
2571:
2572: case t_VEC: case t_COL: case t_MAT:
2573: lx=lg(x); s=gzero;
2574: for (i=1; i<lx; i++) s=gadd(s,gnorml1((GEN)x[i],PREC)); tetpil=avma;
2575: return gerepile(av,tetpil,gcopy(s));
2576: }
2577: err(talker,"not a PARI object in gnorml1");
2578: return NULL; /* not reached */
2579: }
2580:
2581: /***********************************************************************/
2582: /** **/
2583: /** RACINES D'UN POLYNOME **/
2584: /** A COEFFICIENTS REELS **/
2585: /** **/
2586: /***********************************************************************/
2587:
2588: #define RADIX 1
2589: #define COF 0.95
2590:
2591: /* ONLY FOR REAL COEFFICIENTS MATRIX : replace the matrix x with
2592: a symmetric matrix a with the same eigenvalues */
2593: static GEN
2594: balanc(GEN x)
2595: {
2596: long av,tetpil,n,last,j,i,sqrdx;
2597: GEN s,r,g,f,c,cofgen,a;
2598:
2599: av=avma; a=gcopy(x); n=lg(a)-1; sqrdx=RADIX+RADIX; last=0; cofgen=dbltor(COF);
2600: while (!last)
2601: {
2602: last=1;
2603: for (i=1; i<=n; i++)
2604: {
2605: r=c=gzero;
2606: for (j=1; j<=n; j++)
2607: if (j!=i){ c=gadd(gabs(gcoeff(a,j,i),0),c); r=gadd(gabs(gcoeff(a,i,j),0),r); }
2608: if ((!gcmp0(r))&&(!gcmp0(c)))
2609: {
2610: g=gmul2n(r,-RADIX); f=gun; s=gadd(c,r);
2611: while (gcmp(c,g)<0){ f=gmul2n(f,RADIX); c=gmul2n(c,sqrdx); }
2612: g=gmul2n(r,RADIX);
2613: while (gcmp(c,g)>0){ f=gmul2n(f,-RADIX); c=gmul2n(c,-sqrdx); }
2614: if (gcmp(gdiv(gadd(c,r),f),gmul(cofgen,s))<0)
2615: {
2616: last=0; g=ginv(f);
2617: for (j=1; j<=n; j++) coeff(a,i,j)=lmul(gcoeff(a,i,j),g);
2618: for (j=1; j<=n; j++) coeff(a,j,i)=lmul(gcoeff(a,j,i),f);
2619: }
2620: }
2621: }
2622: }
2623: tetpil=avma; return gerepile(av,tetpil,gcopy(a));
2624: }
2625:
2626: #define SIGN(a,b) ((b)>=0.0 ? fabs(a) : -fabs(a))
2627: static GEN
2628: hqr(GEN mat) /* find all the eigenvalues of the matrix mat */
2629: {
2630: long nn,n,m,l,k,j,its,i,mmin,flj,flk;
2631: double **a,p,q,r,s,t,u,v,w,x,y,z,anorm,*wr,*wi,eps;
2632: GEN eig;
2633:
2634: eps=0.000001;
2635: n=lg(mat)-1; a=(double**)gpmalloc(sizeof(double*)*(n+1));
2636: for (i=1; i<=n; i++) a[i]=(double*)gpmalloc(sizeof(double)*(n+1));
2637: for (j=1; j<=n; j++)
2638: for (i=1; i<=n; i++) a[i][j]=gtodouble((GEN)((GEN)mat[j])[i]);
2639: wr=(double*)gpmalloc(sizeof(double)*(n+1));
2640: wi=(double*)gpmalloc(sizeof(double)*(n+1));
2641:
2642: anorm=fabs(a[1][1]);
2643: for (i=2; i<=n; i++) for (j=(i-1); j<=n; j++) anorm+=fabs(a[i][j]);
2644: nn=n; t=0.0;
2645: if (DEBUGLEVEL>3)
2646: { fprintferr("* Finding eigenvalues\n"); flusherr(); }
2647: while (nn>=1)
2648: {
2649: its=0;
2650: do
2651: {
2652: for (l=nn; l>=2; l--)
2653: {
2654: s=fabs(a[l-1][l-1])+fabs(a[l][l]); if (s==0.0) s=anorm;
2655: if ((double)(fabs(a[l][l-1])+s)==s) break;
2656: }
2657: x=a[nn][nn];
2658: if (l==nn){ wr[nn]=x+t; wi[nn--]=0.0; }
2659: else
2660: {
2661: y=a[nn-1][nn-1]; w=a[nn][nn-1]*a[nn-1][nn];
2662: if (l==(nn-1))
2663: {
2664: p=0.5*(y-x); q=p*p+w; z=sqrt(fabs(q)); x+=t;
2665: if ((q>=0.0)||(fabs(q)<=eps))
2666: {
2667: z=p+SIGN(z,p); wr[nn-1]=wr[nn]=x+z;
2668: if (fabs(z)>eps) wr[nn]=x-w/z;
2669: wi[nn-1]=wi[nn]=0.0;
2670: }
2671: else{ wr[nn-1]=wr[nn]=x+p; wi[nn-1]=-(wi[nn]=z); }
2672: nn-=2;
2673: }
2674: else
2675: {
2676: if (its==30) err(talker,"too many iterations in hqr");
2677: if ((its==10)||(its==20))
2678: {
2679: t+=x; for (i=1; i<=nn; i++) a[i][i]-=x; s=fabs(a[nn][nn-1])+fabs(a[nn-1][nn-2]);
2680: y=x=0.75*s; w=-0.4375*s*s;
2681: }
2682: ++its;
2683: for (m=(nn-2); m>=l; m--)
2684: {
2685: z=a[m][m]; r=x-z; s=y-z; p=(r*s-w)/a[m+1][m]+a[m][m+1]; q=a[m+1][m+1]-z-r-s;
2686: r=a[m+2][m+1]; s=fabs(p)+fabs(q)+fabs(r); p/=s; q/=s; r/=s;
2687: if (m==l) break;
2688: u=fabs(a[m][m-1])*(fabs(q)+fabs(r));
2689: v=fabs(p)*(fabs(a[m-1][m-1])+fabs(z)+fabs(a[m+1][m+1]));
2690: if ((double)(u+v)==v) break;
2691: }
2692: for (i=m+2; i<=nn; i++){ a[i][i-2]=0.0; if (i!=(m+2)) a[i][i-3]=0.0; }
2693: for (k=m; k<=nn-1; k++)
2694: {
2695: if (k!=m)
2696: {
2697: p=a[k][k-1]; q=a[k+1][k-1]; r=0.0; if (k!=(nn-1)) r=a[k+2][k-1];
2698: if ((x=fabs(p)+fabs(q)+fabs(r))!=0.0){ p/=x; q/=x; r/=x; }
2699: }
2700: if ((s=SIGN(sqrt(p*p+q*q+r*r),p))!=0.0)
2701: {
2702: if (k==m){ if (l!=m) a[k][k-1]=-a[k][k-1]; }else a[k][k-1]=-s*x;
2703: p+=s; x=p/s; y=q/s; z=r/s; q/=p; r/=p;
2704: for (j=k; j<=nn; j++)
2705: {
2706: p=a[k][j]+q*a[k+1][j]; if (k!=(nn-1)){ p+=r*a[k+2][j]; a[k+2][j]-=p*z; }
2707: a[k+1][j]-=p*y; a[k][j]-=p*x;
2708: }
2709: mmin=(nn<k+3) ? nn : k+3;
2710: for (i=l; i<=mmin; i++)
2711: {
2712: p=x*a[i][k]+y*a[i][k+1]; if (k!=(nn-1)){ p+=z*a[i][k+2]; a[i][k+2]-=p*r; }
2713: a[i][k+1]-=p*q; a[i][k]-=p;
2714: }
2715: }
2716: }
2717: }
2718: }
2719: }
2720: while (l<nn-1);
2721: }
2722: for (j=2; j<=n; j++) /* ordering the roots */
2723: {
2724: x=wr[j]; y=wi[j]; if (y) flj=1; else flj=0;
2725: for (k=j-1; k>=1; k--)
2726: {
2727: if (wi[k]) flk=1; else flk=0;
2728: if (flk<flj) break;
2729: if ((!flk)&&(!flj)&&(wr[k]<=x)) break;
2730: if (flk&&flj&&(wr[k]<x)) break;
2731: if (flk&&flj&&(wr[k]==x)&&(wi[k]>0)) break;
2732: wr[k+1]=wr[k]; wi[k+1]=wi[k];
2733: }
2734: wr[k+1]=x; wi[k+1]=y;
2735: }
2736: if (DEBUGLEVEL>3)
2737: { fprintferr("* End of the computation of eigenvalues\n"); flusherr(); }
2738: for (i=1; i<=n; i++) free(a[i]); free(a); eig=cgetg(n+1,t_COL);
2739: for (i=1; i<=n; i++)
2740: {
2741: if (wi[i])
2742: {
2743: eig[i]=lgetg(3,t_COMPLEX);
2744: ((GEN)eig[i])[1]=(long)dbltor(wr[i]); ((GEN)eig[i])[2]=(long)dbltor(wi[i]);
2745: }
2746: else eig[i]=(long)dbltor(wr[i]);
2747: }
2748: free(wr); free(wi); return eig;
2749: }
2750:
2751: static GEN
2752: zrhqr(GEN a,long PREC)
2753: /* ONLY FOR POLYNOMIAL WITH REAL COEFFICIENTS : give the roots of
2754: * the polynomial a (first, the real roots, then the
2755: * non real roots) in increasing order of their real
2756: * parts MULTIPLE ROOTS ARE FORBIDDEN.
2757: */
2758: {
2759: long av,tetpil,n,i,j,k,ct,prec;
2760: GEN aa,b,p1,rt,rr,hess,x,dx,y,hessbis,eps,newval;
2761: GEN oldval = NULL; /* for lint */
2762:
2763: av=avma; n=lgef(a)-3; prec=PREC;
2764: hess=cgetg(n+1,t_MAT); for (k=1; k<=n; k++) hess[k]=lgetg(n+1,t_COL);
2765: for (k=1; k<=n; k++)
2766: {
2767: p1=(GEN)hess[k]; p1[1]=lneg(gdiv((GEN)a[n-k+2],(GEN)a[n+2]));
2768: for (j=2; j<=n; j++){ if (j==(k+1)) p1[j]=un; else p1[j]=zero; }
2769: }
2770: rr=cgetg(n+1,t_COL);
2771: for (i=1; i<=n; i++)
2772: {
2773: rr[i]=lgetg(3,t_COMPLEX);
2774: mael(rr,i,1)=lgetr(PREC);
2775: mael(rr,i,2)=lgetr(PREC);
2776: }
2777: hessbis=balanc(hess); rt=hqr(hessbis);
2778: eps=cgetr(prec);
2779: p1=gpuigs(gdeux,-bit_accuracy(prec)); gaffect(p1,eps);
2780: prec=2*PREC; /* polishing the roots */
2781: aa=cgetg(n+3,t_POL); aa[1]=a[1];
2782: for (i=2; i<=n+2; i++){ aa[i]=lgetr(prec); gaffect((GEN)a[i],(GEN)aa[i]); }
2783: b=deriv(aa,varn(aa));
2784: for (i=1; i<=n; i++)
2785: {
2786: ct=0;
2787: if (typ(rt[i])==t_REAL) { x=cgetr(prec); affrr((GEN)rt[i],x); }
2788: else
2789: {
2790: x=cgetg(3,t_COMPLEX);
2791: x[1]=lgetr(prec); affrr(gmael(rt,i,1),(GEN)x[1]);
2792: x[2]=lgetr(prec); affrr(gmael(rt,i,2),(GEN)x[2]);
2793: }
2794: LAB1:
2795: dx=poleval(b,x);
2796: if (gcmp(gabs(dx,prec),eps) <= 0)
2797: err(talker,"the polynomial has probably multiple roots in zrhqr");
2798: y=gsub(x,gdiv(poleval(aa,x),dx));
2799: newval=gabs(poleval(aa,y),prec);
2800: if (gcmp(newval,eps) > 0 && (!ct || gcmp(newval,oldval) < 0))
2801: {
2802: ct++; oldval=newval; x=y;
2803: goto LAB1;
2804: }
2805: gaffect(y,(GEN)rr[i]);
2806: }
2807: if (DEBUGLEVEL>3){ fprintferr("polished roots = %Z",rr); flusherr(); }
2808: tetpil=avma; return gerepile(av,tetpil,gcopy(rr));
2809: }
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