Annotation of OpenXM_contrib/pari/src/basemath/bibli2.c, Revision 1.1.1.1
1.1 maekawa 1: /********************************************************************/
2: /** **/
3: /** BIBLIOTHEQUE MATHEMATIQUE **/
4: /** (deuxieme partie) **/
5: /** **/
6: /********************************************************************/
7: /* $Id: bibli2.c,v 1.1.1.1 1999/09/16 13:47:24 karim Exp $ */
8: #include "pari.h"
9:
10: /********************************************************************/
11: /** **/
12: /** DEVELOPPEMENTS LIMITES **/
13: /** **/
14: /********************************************************************/
15:
16: GEN
17: tayl(GEN x, long v, long precdl)
18: {
19: long tetpil,i,vx = gvar9(x), av=avma;
20: GEN p1,y;
21:
22: if (v <= vx)
23: {
24: long p1[] = { evaltyp(t_SER)|m_evallg(2), 0 };
25: p1[1] = evalvalp(precdl) | evalvarn(v);
26: return gadd(p1,x);
27: }
28: p1=cgetg(v+2,t_VEC);
29: for (i=0; i<v; i++) p1[i+1]=lpolx[i];
30: p1[vx+1]=lpolx[v]; p1[v+1]=lpolx[vx];
31: y = tayl(changevar(x,p1), vx,precdl); tetpil=avma;
32: return gerepile(av,tetpil, changevar(y,p1));
33: }
34:
35: GEN
36: grando0(GEN x, long n, long do_clone)
37: {
38: long m, v, tx=typ(x);
39: GEN y;
40:
41: if (gcmp0(x)) err(talker,"zero argument in O()");
42: if (tx == t_INT)
43: {
44: if (!gcmp1(x)) /* bug 3 + O(1). We suppose x is a truc() */
45: {
46: y=cgetg(5,t_PADIC);
47: y[1] = evalvalp(n) | evalprecp(0);
48: y[2] = do_clone? lclone(x): licopy(x);
49: y[3] = un; y[4] = zero; return y;
50: }
51: v=0; m=0; /* 1 = x^0 */
52: }
53: else
54: {
55: if (tx != t_POL && ! is_rfrac_t(tx))
56: err(talker,"incorrect argument in O()");
57: v=gvar(x); m=n*gval(x,v);
58: }
59: return zeroser(v,m);
60: }
61:
62: /*******************************************************************/
63: /** **/
64: /** SPECIAL POLYNOMIALS **/
65: /** **/
66: /*******************************************************************/
67: GEN addshiftw(GEN x, GEN y, long d);
68:
69: /* Tchebichev polynomial */
70: /* T0=1; T1=X; T(n)=2*X*T(n-1)-T(n-2) */
71: GEN
72: tchebi(long n, long v)
73: {
74: long av,tetpil,lim,m;
75: GEN p0,p1,p2,q,z;
76:
77: if (v<0) v = 0;
78: if (n==0) return polun[v];
79: if (n==1) return polx[v];
80:
81: p0=gneg(polun[v]); p1=polx[v]; z = zeropol(v);
82: av=avma; lim=stack_lim(av,2);
83: for (m=2; m<n; m++)
84: {
85: q = addshiftw(gmul2n(p1,1), z, 1); tetpil=avma;
86: setvarn(q,v);
87: p2 = gadd(q, p0); p0=gneg(p1); p1=p2;
88: if (low_stack(lim, stack_lim(av,2)))
89: {
90: GEN *gptr[2]; gptr[0]=&p0; gptr[1]=&p1;
91: if(DEBUGMEM>1) err(warnmem,"tchebi");
92: gerepilemanysp(av,tetpil,gptr,2);
93: }
94: }
95: q = addshiftw(gmul2n(p1,1), z, 1); tetpil=avma;
96: setvarn(q,v);
97: return gerepile(av,tetpil,gadd(q,p0));
98: }
99:
100: /* Legendre polynomial */
101: /* L0=1; L1=X; (n+1)*L(n+1)=(2*n+1)*X*L(n)-n*L(n-1) */
102: GEN
103: legendre(long n, long v)
104: {
105: long av,tetpil,m,lim;
106: GEN p0,p1,p2;
107:
108: if (v<0) v = 0;
109: if (n==0) return polun[v];
110: if (n==1) return polx[v];
111:
112: p0=polun[v]; av=avma; lim=stack_lim(av,2);
113: p1=gmul2n(polx[v],1);
114: for (m=1; m<n; m++)
115: {
116: p2 = addshiftw(gmulsg(4*m+2,p1), gmulsg(-4*m,p0), 1);
117: setvarn(p2,v);
118: p0 = p1; tetpil=avma; p1 = gdivgs(p2,m+1);
119: if (low_stack(lim, stack_lim(av,2)))
120: {
121: GEN *gptr[2];
122: if(DEBUGMEM>1) err(warnmem,"legendre");
123: p0=gcopy(p0); gptr[0]=&p0; gptr[1]=&p1;
124: gerepilemanysp(av,tetpil,gptr,2);
125: }
126: }
127: tetpil=avma; return gerepile(av,tetpil,gmul2n(p1,-n));
128: }
129:
130: /* cyclotomic polynomial */
131: GEN
132: cyclo(long n, long v)
133: {
134: long av=avma,tetpil,d,q,m;
135: GEN yn,yd;
136:
137: if (n<=0) err(arither2);
138: if (v<0) v = 0;
139: yn = yd = polun[0];
140: for (d=1; d*d<=n; d++)
141: {
142: if (n%d) continue;
143: q=n/d;
144: m = mu(stoi(q));
145: if (m)
146: { /* y *= (x^d - 1) */
147: if (m>0) yn = addshiftw(yn, gneg(yn), d);
148: else yd = addshiftw(yd, gneg(yd), d);
149: }
150: if (q==d) break;
151: m = mu(stoi(d));
152: if (m)
153: { /* y *= (x^q - 1) */
154: if (m>0) yn = addshiftw(yn, gneg(yn), q);
155: else yd = addshiftw(yd, gneg(yd), q);
156: }
157: }
158: tetpil=avma; yn = gerepile(av,tetpil,gdeuc(yn,yd));
159: setvarn(yn,v); return yn;
160: }
161:
162: /* compute prod (L*x ± a[i]) */
163: GEN
164: roots_to_pol_intern(GEN L, GEN a, long v, int plus)
165: {
166: long i,k,lx = lg(a), code;
167: GEN p1,p2;
168: if (lx == 1) return polun[v];
169: p1 = cgetg(lx, t_VEC);
170: code = evalsigne(1)|evalvarn(v)|evallgef(5);
171: for (k=1,i=1; i<lx-1; i+=2)
172: {
173: p2 = cgetg(5,t_POL); p1[k++] = (long)p2;
174: p2[2] = lmul((GEN)a[i],(GEN)a[i+1]);
175: p2[3] = ladd((GEN)a[i],(GEN)a[i+1]);
176: if (plus == 0) p2[3] = lneg((GEN)p2[3]);
177: p2[4] = (long)L; p2[1] = code;
178: }
179: if (i < lx)
180: {
181: p2 = cgetg(4,t_POL); p1[k++] = (long)p2;
182: p2[1] = code = evalsigne(1)|evalvarn(v)|evallgef(4);
183: p2[2] = plus? a[i]: lneg((GEN)a[i]);
184: p2[3] = (long)L;
185: }
186: setlg(p1, k); return divide_conquer_prod(p1, gmul);
187: }
188:
189: GEN
190: roots_to_pol(GEN a, long v)
191: {
192: return roots_to_pol_intern(gun,a,v,0);
193: }
194:
195: /* prod_{i=1..r1} (x - a[i]) prod_{i=1..r2} (x - a[i])(x - conj(a[i]))*/
196: GEN
197: roots_to_pol_r1r2(GEN a, long r1, long v)
198: {
199: long i,k,lx = lg(a), code;
200: GEN p1;
201: if (lx == 1) return polun[v];
202: p1 = cgetg(lx, t_VEC);
203: code = evalsigne(1)|evalvarn(v)|evallgef(5);
204: for (k=1,i=1; i<r1; i+=2)
205: {
206: GEN p2 = cgetg(5,t_POL); p1[k++] = (long)p2;
207: p2[2] = lmul((GEN)a[i],(GEN)a[i+1]);
208: p2[3] = lneg(gadd((GEN)a[i],(GEN)a[i+1]));
209: p2[4] = un; p2[1] = code;
210: }
211: if (i < r1+1)
212: p1[k++] = ladd(polx[v], gneg((GEN)a[i]));
213: for (i=r1+1; i<lx; i++)
214: {
215: GEN p2 = cgetg(5,t_POL); p1[k++] = (long)p2;
216: p2[2] = lnorm((GEN)a[i]);
217: p2[3] = lneg(gtrace((GEN)a[i]));
218: p2[4] = un; p2[1] = code;
219: }
220: setlg(p1, k); return divide_conquer_prod(p1, gmul);
221: }
222:
223: /* finds an equation for the d-th degree subfield of Q(zeta_n).
224: * (Z/nZ)* must be cyclic.
225: */
226: GEN
227: subcyclo(GEN nn, GEN dd, int v)
228: {
229: long av=avma,tetpil,i,j,k,prec,q,d,p,pp,al,n,ex0,ex,aad,aa;
230: GEN a,z,pol,fa,powz,alpha;
231:
232: if (typ(dd)!=t_INT || signe(dd)<=0) err(typeer,"subcyclo");
233: if (is_bigint(dd)) err(talker,"degree too large in subcyclo");
234: if (typ(nn)!=t_INT || signe(nn)<=0) err(typeer,"subcyclo");
235: if (v<0) v = 0;
236: d=itos(dd); if (d==1) return polx[v];
237: if (is_bigint(nn)) err(impl,"subcyclo for huge cyclotomic fields");
238: n = nn[2]; if ((n & 3) == 2) n >>= 1;
239: if (n == 1) err(talker,"degree does not divide phi(n) in subcyclo");
240: fa = factor(stoi(n));
241: p = itos(gmael(fa,1,1));
242: al= itos(gmael(fa,2,1));
243: if (lg((GEN)fa[1]) > 2 || (p==2 && al>2))
244: err(impl,"subcyclo in non-cyclic case");
245: if (d < n)
246: {
247: k = 1 + svaluation(d,p,&i);
248: if (k<al) { al = k; nn = gpowgs(stoi(p),al); n = nn[2]; }
249: }
250: avma=av; q = (n/p)*(p-1); /* = phi(n) */
251: if (q == d) return cyclo(n,v);
252: if (q % d) err(talker,"degree does not divide phi(n) in subcyclo");
253: q /= d;
254: if (p==2)
255: {
256: pol = powgi(polx[v],gdeux); pol[2]=un; /* replace gzero */
257: return pol; /* = x^2 + 1 */
258: }
259: a=gener(stoi(n)); aa = mael(a,2,2);
260: a=gpowgs(a,d); aad = mael(a,2,2);
261: #if 1
262: prec = expi(binome(stoi(d*q-1),d)) + expi(stoi(n));
263: prec = 2 + (prec>>TWOPOTBITS_IN_LONG);
264: if (prec<DEFAULTPREC) prec = DEFAULTPREC;
265: if (DEBUGLEVEL) fprintferr("subcyclo prec = %ld\n",prec);
266: z = cgetg(3,t_COMPLEX); a=mppi(prec); setexpo(a,2); /* a = 2\pi */
267: gsincos(divrs(a,n),(GEN*)(z+2),(GEN*)(z+1),prec); /* z = e_n(1) */
268: powz = cgetg(n,t_VEC); powz[1] = (long)z;
269: k = (n+3)>>1;
270: for (i=2; i<k; i++) powz[i] = lmul(z,(GEN)powz[i-1]);
271: if ((q&1) == 0) /* totally real field, take real part */
272: {
273: for (i=1; i<k; i++) powz[i] = mael(powz,i,1);
274: for ( ; i<n; i++) powz[i] = powz[n-i];
275: }
276: else
277: for ( ; i<n; i++) powz[i] = lconj((GEN)powz[n-i]);
278:
279: alpha = cgetg(d+1,t_VEC) + 1; pol=gun;
280: for (ex0=1,k=0; k<d; k++, ex0=(ex0*aa)%n)
281: {
282: GEN p1 = gzero;
283: long av1 = avma; (void)new_chunk(2*prec + 3);
284: for (ex=ex0,i=0; i<q; i++)
285: {
286: for (pp=ex,j=0; j<al; j++)
287: {
288: p1 = gadd(p1,(GEN)powz[pp]);
289: pp = mulssmod(pp,p, n);
290: }
291: ex = mulssmod(ex,aad, n);
292: }
293: /* p1 = sum z^{p^k*h}, k = 0..al-1, h runs through the subgroup of order
294: * q = phi(n)/d of (Z/nZ)^* */
295: avma = av1; alpha[k] = lneg(p1);
296: }
297: pol = roots_to_pol_intern(gun,alpha-1,v, 1);
298: if (q&1) pol=greal(pol); /* already done otherwise */
299: tetpil=avma; return gerepile(av,tetpil,ground(pol));
300: #else
301: {
302: /* exact computation (much slower) */
303: GEN p1 = cgetg(n+2,t_POL)+2; for (i=0; i<n; i++) p1[i]=0;
304: for (ex=1,i=0; i<q; i++, ex=(ex*aad)%n)
305: for (pp=ex,j=0; j<al; j++, pp=(pp*p)%n) p1[pp]++;
306: for (i=0; i<n; i++) p1[i] = lstoi(p1[i]);
307: p1 = normalizepol_i(p1-2,n+2); setvarn(p1,v);
308: z = cyclo(n,v); a = caract2(z,gres(p1,z),v);
309: a = gdeuc(a, modulargcd(a,derivpol(a)));
310: return gerepileupto(av, a);
311: }
312: #endif
313: }
314:
315: /********************************************************************/
316: /** **/
317: /** HILBERT & PASCAL MATRICES **/
318: /** **/
319: /********************************************************************/
320: GEN addshiftpol(GEN x, GEN y, long d);
321:
322: GEN
323: mathilbert(long n) /* Hilbert matrix of order n */
324: {
325: long i,j;
326: GEN a,p;
327:
328: if (n<0) n = 0;
329: p = cgetg(n+1,t_MAT);
330: for (j=1; j<=n; j++)
331: {
332: p[j]=lgetg(n+1,t_COL);
333: for (i=1; i<=n; i++)
334: {
335: a=cgetg(3,t_FRAC); a[1]=un; a[2]=lstoi(i+j-1);
336: coeff(p,i,j)=(long)a;
337: }
338: }
339: return p;
340: }
341:
342: /* q-Pascal triangle = (choose(i,j)_q) (ordinary binomial if q = NULL) */
343: GEN
344: matqpascal(long n, GEN q)
345: {
346: long i,j,I, av = avma;
347: GEN m, *qpow;
348:
349: if (n<0) n = -1;
350: n++; m = cgetg(n+1,t_MAT);
351: for (j=1; j<=n; j++) m[j] = lgetg(n+1,t_COL);
352: if (q)
353: {
354: I = (n+1)/2;
355: if (I > 1) { qpow = (GEN*)new_chunk(I+1); qpow[2]=q; }
356: for (j=3; j<=I; j++) qpow[j] = gmul(q, qpow[j-1]);
357: }
358: for (i=1; i<=n; i++)
359: {
360: I = (i+1)/2; coeff(m,i,1)=un;
361: if (q)
362: {
363: for (j=2; j<=I; j++)
364: coeff(m,i,j) = ladd(gmul(qpow[j],gcoeff(m,i-1,j)), gcoeff(m,i-1,j-1));
365: }
366: else
367: {
368: for (j=2; j<=I; j++)
369: coeff(m,i,j) = laddii(gcoeff(m,i-1,j), gcoeff(m,i-1,j-1));
370: }
371: for ( ; j<=i; j++) coeff(m,i,j) = coeff(m,i,i+1-j);
372: for ( ; j<=n; j++) coeff(m,i,j) = zero;
373: }
374: return gerepileupto(av, gcopy(m));
375: }
376:
377: /********************************************************************/
378: /** **/
379: /** LAPLACE TRANSFORM (OF A SERIES) **/
380: /** **/
381: /********************************************************************/
382:
383: GEN
384: laplace(GEN x)
385: {
386: long i,l,ec,av,tetpil;
387: GEN y,p1;
388:
389: if (typ(x)!=t_SER) err(talker,"not a series in laplace");
390: if (gcmp0(x)) return gcopy(x);
391:
392: av=avma; ec=valp(x);
393: if (ec<0) err(talker,"negative valuation in laplace");
394: l=lg(x); y=cgetg(l,t_SER);
395: p1=mpfact(ec); y[1]=x[1];
396: for (i=2; i<l; i++)
397: {
398: y[i]=lmul(p1,(GEN)x[i]);
399: ec++; p1=mulsi(ec,p1);
400: }
401: tetpil=avma; return gerepile(av,tetpil,gcopy(y));
402: }
403:
404: /********************************************************************/
405: /** **/
406: /** CONVOLUTION PRODUCT (OF TWO SERIES) **/
407: /** **/
408: /********************************************************************/
409:
410: GEN
411: convol(GEN x, GEN y)
412: {
413: long l,i,j,v, vx=varn(x), lx=lg(x), ly=lg(y), ex=valp(x), ey=valp(y);
414: GEN z;
415:
416: if (typ(x) != t_SER || typ(y) != t_SER)
417: err(talker,"not a series in convol");
418: if (gcmp0(x) || gcmp0(y))
419: err(talker,"zero series in convol");
420: if (varn(y) != vx)
421: err(talker,"different variables in convol");
422: v=ex; if (ey>v) v=ey;
423: l=ex+lx; i=ey+ly; if (i<l) l=i;
424: l -= v; if (l<3) err(talker,"non significant result in convol");
425: for (i=v+2; i < v+l; i++)
426: if (!gcmp0((GEN)x[i-ex]) && !gcmp0((GEN)y[i-ey])) { i++; break; }
427: if (i == l+v) return zeroser(vx, v+l-2);
428:
429: z = cgetg(l-i+3+v,t_SER);
430: z[1] = evalsigne(1) | evalvalp(i-3) | evalvarn(vx);
431: for (j=i-1; j<l+v; j++) z[j-i+3]=lmul((GEN)x[j-ex],(GEN)y[j-ey]);
432: return z;
433: }
434:
435: /******************************************************************/
436: /** **/
437: /** PRECISION CHANGES **/
438: /** **/
439: /******************************************************************/
440:
441: GEN
442: gprec(GEN x, long l)
443: {
444: long tx=typ(x),lx=lg(x),i,pr;
445: GEN y;
446:
447: if (l<=0) err(talker,"precision<=0 in gprec");
448: switch(tx)
449: {
450: case t_REAL:
451: pr = (long) (l*pariK1+3); y=cgetr(pr); affrr(x,y); break;
452:
453: case t_PADIC:
454: y=cgetg(lx,tx); copyifstack(x[2], y[2]);
455: if (!signe(x[4]))
456: {
457: y[1]=evalvalp(l+precp(x)) | evalprecp(0);
458: y[3]=un; y[4]=zero; return y;
459: }
460: y[1]=x[1]; setprecp(y,l);
461: y[3]=lpuigs((GEN)x[2],l);
462: y[4]=lmodii((GEN)x[4],(GEN)y[3]);
463: break;
464:
465: case t_SER:
466: if (gcmp0(x)) return zeroser(varn(x), l);
467: y=cgetg(l+2,t_SER); y[1]=x[1]; l++; i=l;
468: if (l>=lx)
469: for ( ; i>=lx; i--) y[i]=zero;
470: for ( ; i>=2; i--) y[i]=lcopy((GEN)x[i]);
471: break;
472:
473: case t_POL:
474: lx=lgef(x); y=cgetg(lx,tx); y[1]=x[1];
475: for (i=2; i<lx; i++) y[i]=lprec((GEN)x[i],l);
476: break;
477:
478: case t_COMPLEX: case t_POLMOD: case t_RFRAC: case t_RFRACN:
479: case t_VEC: case t_COL: case t_MAT:
480: y=cgetg(lx,tx);
481: for (i=1; i<lx; i++) y[i]=lprec((GEN)x[i],l);
482: break;
483: default: y=gcopy(x);
484: }
485: return y;
486: }
487:
488: /* internal: precision given in word length (including codewords) */
489: GEN
490: gprec_w(GEN x, long pr)
491: {
492: long tx=typ(x),lx=lg(x),i;
493: GEN y;
494:
495: switch(tx)
496: {
497: case t_REAL:
498: y=cgetr(pr); affrr(x,y); break;
499:
500: case t_POL:
501: lx=lgef(x); y=cgetg(lx,tx); y[1]=x[1];
502: for (i=2; i<lx; i++) y[i]=(long)gprec_w((GEN)x[i],pr);
503: break;
504:
505: case t_COMPLEX: case t_POLMOD: case t_RFRAC: case t_RFRACN:
506: case t_VEC: case t_COL: case t_MAT:
507: y=cgetg(lx,tx);
508: for (i=1; i<lx; i++) y[i]=(long)gprec_w((GEN)x[i],pr);
509: break;
510: default: y=gprec(x,pr);
511: }
512: return y;
513: }
514:
515: /*******************************************************************/
516: /** **/
517: /** RECIPROCAL POLYNOMIAL **/
518: /** **/
519: /*******************************************************************/
520:
521: GEN
522: polrecip(GEN x)
523: {
524: long lx=lgef(x),i,j;
525: GEN y;
526:
527: if (typ(x) != t_POL) err(typeer,"polrecip");
528: y=cgetg(lx,t_POL); y[1]=x[1];
529: for (i=2,j=lx-1; i<lx; i++,j--) y[i]=lcopy((GEN)x[j]);
530: return normalizepol_i(y,lx);
531: }
532:
533: /* as above. Internal (don't copy or normalize) */
534: GEN
535: polrecip_i(GEN x)
536: {
537: long lx=lgef(x),i,j;
538: GEN y;
539:
540: y=cgetg(lx,t_POL); y[1]=x[1];
541: for (i=2,j=lx-1; i<lx; i++,j--) y[i]=x[j];
542: return y;
543: }
544:
545: /*******************************************************************/
546: /** **/
547: /** BINOMIAL COEFFICIENTS **/
548: /** **/
549: /*******************************************************************/
550:
551: GEN
552: binome(GEN n, long k)
553: {
554: long av,i;
555: GEN y;
556:
557: if (k <= 1)
558: {
559: if (k < 0) return gzero;
560: if (k == 0) return gun;
561: return gcopy(n);
562: }
563: av = avma; y = n;
564: if (typ(n) == t_INT)
565: {
566: if (signe(n) > 0)
567: {
568: GEN z = subis(n,k);
569: if (cmpis(z,k) < 0) k = itos(z);
570: avma = av;
571: if (k <= 1)
572: {
573: if (k < 0) return gzero;
574: if (k == 0) return gun;
575: return gcopy(n);
576: }
577: }
578: for (i=2; i<=k; i++)
579: y = gdivgs(gmul(y,addis(n,i-1-k)), i);
580: }
581: else
582: {
583: for (i=2; i<=k; i++)
584: y = gdivgs(gmul(y,gaddgs(n,i-1-k)), i);
585: }
586: return gerepileupto(av, y);
587: }
588:
589: /********************************************************************/
590: /** **/
591: /** POLYNOMIAL INTERPOLATION **/
592: /** **/
593: /********************************************************************/
594:
595: GEN
596: polint_i(GEN xa, GEN ya, GEN x, long n, GEN *ptdy)
597: {
598: long av = avma,tetpil,i,m, ns=0, tx=typ(x);
599: GEN den,ho,hp,w,y,c,d,dy;
600:
601: if (is_scalar_t(tx) && tx != t_INTMOD && tx != t_PADIC && tx != t_POLMOD)
602: {
603: GEN dif = NULL, dift;
604: for (i=0; i<n; i++)
605: {
606: dift = gabs(gsub(x,(GEN)xa[i]), MEDDEFAULTPREC);
607: if (!dif || gcmp(dift,dif)<0) { ns=i; dif=dift; }
608: }
609: }
610: c=new_chunk(n);
611: d=new_chunk(n); for (i=0; i<n; i++) c[i] = d[i] = ya[i];
612: y=(GEN)d[ns--];
613: for (m=1; m<n; m++)
614: {
615: for (i=0; i<n-m; i++)
616: {
617: ho = gsub((GEN)xa[i],x);
618: hp = gsub((GEN)xa[i+m],x); den = gsub(ho,hp);
619: if (gcmp0(den)) err(talker,"two abcissas are equal in polint");
620: w=gsub((GEN)c[i+1],(GEN)d[i]); den = gdiv(w,den);
621: c[i]=lmul(ho,den);
622: d[i]=lmul(hp,den);
623: }
624: dy = (2*(ns+1) < n-m)? (GEN)c[ns+1]: (GEN)d[ns--];
625: tetpil=avma; y=gadd(y,dy);
626: }
627: if (!ptdy) y = gerepile(av,tetpil,y);
628: else
629: {
630: GEN *gptr[2];
631: *ptdy=gcopy(dy); gptr[0]=&y; gptr[1]=ptdy;
632: gerepilemanysp(av,tetpil,gptr,2);
633: }
634: return y;
635: }
636:
637: GEN
638: polint(GEN xa, GEN ya, GEN x, GEN *ptdy)
639: {
640: long tx=typ(xa), ty=typ(ya), lx=lg(xa);
641:
642: if (! is_vec_t(tx) || ! is_vec_t(ty))
643: err(talker,"not vectors in polinterpolate");
644: if (lx != lg(ya))
645: err(talker,"different lengths in polinterpolate");
646: if (lx <= 2)
647: {
648: if (lx == 1) err(talker,"no data in polinterpolate");
649: ya=gcopy((GEN)ya[1]); if (ptdy) *ptdy = ya;
650: return ya;
651: }
652: if (!x) x = polx[0];
653: return polint_i(xa+1,ya+1,x,lx-1,ptdy);
654: }
655:
656: /***********************************************************************/
657: /* */
658: /* SET OPERATIONS */
659: /* */
660: /***********************************************************************/
661:
662: static GEN
663: gtostr(GEN x)
664: {
665: char *s=GENtostr(x);
666: x = strtoGENstr(s,0); free(s); return x;
667: }
668:
669: GEN
670: gtoset(GEN x)
671: {
672: long i,c,av,tetpil,tx,lx;
673: GEN y;
674:
675: if (!x) return cgetg(1, t_VEC);
676: tx = typ(x); lx = lg(x);
677: if (!is_vec_t(tx))
678: {
679: if (tx != t_LIST)
680: { y=cgetg(2,t_VEC); y[1]=(long)gtostr(x); return y; }
681: lx = lgef(x)-1; x++;
682: }
683: if (lx==1) return cgetg(1,t_VEC);
684: av=avma; y=cgetg(lx,t_VEC);
685: for (i=1; i<lx; i++) y[i]=(long)gtostr((GEN)x[i]);
686: y = sort(y);
687: c=1;
688: for (i=2; i<lx; i++)
689: if (!gegal((GEN)y[i], (GEN)y[c])) y[++c] = y[i];
690: tetpil=avma; setlg(y,c+1);
691: return gerepile(av,tetpil,gcopy(y));
692: }
693:
694: long
695: setisset(GEN x)
696: {
697: long lx,i;
698:
699: if (typ(x)!=t_VEC) return 0;
700: lx=lg(x)-1; if (!lx) return 1;
701: for (i=1; i<lx; i++)
702: if (typ(x[i]) != t_STR || gcmp((GEN)x[i+1],(GEN)x[i])<=0) return 0;
703: return typ(x[i]) == t_STR;
704: }
705:
706: /* looks if y belongs to the set x and returns the index if yes, 0 if no */
707: long
708: setsearch(GEN x, GEN y, long flag)
709: {
710: long av = avma,lx,j,li,ri,fl, tx = typ(x);
711:
712: if (tx==t_VEC) lx = lg(x);
713: else
714: {
715: if (tx!=t_LIST) err(talker,"not a set in setsearch");
716: lx=lgef(x)-1; x++;
717: }
718: if (lx==1) return flag? 1: 0;
719:
720: li=1; ri=lx-1;
721: if (typ(y) != t_STR) y = gtostr(y);
722: while (ri>=li)
723: {
724: j = (ri+li)>>1; fl = gcmp((GEN)x[j],y);
725: if (!fl) { avma=av; return flag? 0: j; }
726: if (fl<0) li=j+1; else ri=j-1;
727: }
728: avma=av; if (!flag) return 0;
729: return (fl<0)? j+1: j;
730: }
731:
732: GEN
733: setunion(GEN x, GEN y)
734: {
735: long av=avma,tetpil;
736: GEN z;
737:
738: if (typ(x) != t_VEC || typ(y) != t_VEC) err(talker,"not a set in setunion");
739: z=concatsp(x,y); tetpil=avma; return gerepile(av,tetpil,gtoset(z));
740: }
741:
742: GEN
743: setintersect(GEN x, GEN y)
744: {
745: long av=avma,tetpil,i,lx,c;
746: GEN z;
747:
748: if (!setisset(x) || !setisset(y)) err(talker,"not a set in setintersect");
749: lx=lg(x); z=cgetg(lx,t_VEC); c=1;
750: for (i=1; i<lx; i++)
751: if (setsearch(y, (GEN)x[i], 0)) z[c++] = x[i];
752: tetpil=avma; setlg(z,c);
753: return gerepile(av,tetpil,gcopy(z));
754: }
755:
756: GEN
757: setminus(GEN x, GEN y)
758: {
759: long av=avma,tetpil,i,lx,c;
760: GEN z;
761:
762: if (!setisset(x) || !setisset(y)) err(talker,"not a set in setminus");
763: lx=lg(x); z=cgetg(lx,t_VEC); c=1;
764: for (i=1; i<lx; i++)
765: if (setsearch(y, (GEN)x[i], 1)) z[c++] = x[i];
766: tetpil=avma; setlg(z,c);
767: return gerepile(av,tetpil,gcopy(z));
768: }
769:
770: /***********************************************************************/
771: /* */
772: /* OPERATIONS ON DIRICHLET SERIES */
773: /* */
774: /***********************************************************************/
775:
776: /* Addition, subtraction and scalar multiplication of Dirichlet series
777: are done on the corresponding vectors */
778:
779: static long
780: dirval(GEN x)
781: {
782: long i=1,lx=lg(x);
783: while (i<lx && gcmp0((GEN)x[i])) i++;
784: return i;
785: }
786:
787: GEN
788: dirmul(GEN x, GEN y)
789: {
790: long lx,ly,lz,dx,dy,av,tetpil,i,j;
791: GEN z,p1;
792:
793: if (typ(x)!=t_VEC || typ(y)!=t_VEC) err(talker,"not a dirseries in dirmul");
794: av=avma; dx=dirval(x); dy=dirval(y); lx=lg(x); ly=lg(y);
795: if (ly-dy<lx-dx) { z=y; y=x; x=z; lz=ly; ly=lx; lx=lz; lz=dy; dy=dx; dx=lz; }
796: lz=min(lx*dy,ly*dx);
797: z=cgetg(lz,t_VEC); for (i=1; i<lz; i++) z[i]=zero;
798: for (j=dx; j<lx; j++)
799: {
800: p1=(GEN)x[j];
801: if (!gcmp0(p1))
802: {
803: if (gcmp1(p1))
804: for (i=j*dy; i<lz; i+=j) z[i]=ladd((GEN)z[i],(GEN)y[i/j]);
805: else
806: {
807: if (gcmp_1(p1))
808: for (i=j*dy; i<lz; i+=j) z[i]=lsub((GEN)z[i],(GEN)y[i/j]);
809: else
810: for (i=j*dy; i<lz; i+=j) z[i]=ladd((GEN)z[i],gmul(p1,(GEN)y[i/j]));
811: }
812: }
813: }
814: tetpil=avma; return gerepile(av,tetpil,gcopy(z));
815: }
816:
817: GEN
818: dirdiv(GEN x, GEN y)
819: {
820: long lx,ly,lz,dx,dy,av,tetpil,i,j;
821: GEN z,p1;
822:
823: if (typ(x)!=t_VEC || typ(y)!=t_VEC) err(talker,"not a dirseries in dirmul");
824: av=avma; dx=dirval(x); dy=dirval(y); lx=lg(x); ly=lg(y);
825: if (dy!=1) err(talker,"not an invertible dirseries in dirdiv");
826: lz=min(lx,ly*dx); p1=(GEN)y[1];
827: if (!gcmp1(p1)) { y=gdiv(y,p1); x=gdiv(x,p1); }
828: else x=gcopy(x);
829: z=cgetg(lz,t_VEC); for (i=1; i<dx; i++) z[i]=zero;
830: for (j=dx; j<lz; j++)
831: {
832: p1=(GEN)x[j]; z[j]=(long)p1;
833: if (!gcmp0(p1))
834: {
835: if (gcmp1(p1))
836: for (i=j+j; i<lz; i+=j) x[i]=lsub((GEN)x[i],(GEN)y[i/j]);
837: else
838: {
839: if (gcmp_1(p1))
840: for (i=j+j; i<lz; i+=j) x[i]=ladd((GEN)x[i],(GEN)y[i/j]);
841: else
842: for (i=j+j; i<lz; i+=j) x[i]=lsub((GEN)x[i],gmul(p1,(GEN)y[i/j]));
843: }
844: }
845: }
846: tetpil=avma; return gerepile(av,tetpil,gcopy(z));
847: }
848:
849: /*************************************************************************/
850: /** **/
851: /** RANDOM **/
852: /** **/
853: /*************************************************************************/
854: static long pari_randseed = 1;
855:
856: /* BSD rand gives this: seed = 1103515245*seed + 12345 */
857: long
858: mymyrand()
859: {
860: #if BITS_IN_RANDOM == 64
861: pari_randseed = (1000000000000654397*pari_randseed + 12347) & ~HIGHBIT;
862: #else
863: pari_randseed = (1000276549*pari_randseed + 12347) & 0x7fffffff;
864: #endif
865: return pari_randseed;
866: }
867:
868: GEN muluu(ulong x, ulong y);
869:
870: static ulong
871: gp_rand()
872: {
873: #define GLUE2(hi, lo) (((hi) << BITS_IN_HALFULONG) | (lo))
874: #if !defined(LONG_IS_64BIT) || BITS_IN_RANDOM == 64
875: return GLUE2((mymyrand()>>12)&LOWMASK,
876: (mymyrand()>>12)&LOWMASK);
877: #else
878: #define GLUE4(hi1,hi2, lo1,lo2) GLUE2(((hi1)<<16)|(hi2), ((lo1)<<16)|(lo2))
879: # define LOWMASK2 0xffffUL
880: return GLUE4((mymyrand()>>12)&LOWMASK2,
881: (mymyrand()>>12)&LOWMASK2,
882: (mymyrand()>>12)&LOWMASK2,
883: (mymyrand()>>12)&LOWMASK2);
884: #endif
885: }
886:
887: GEN
888: genrand(GEN N)
889: {
890: long lx,i;
891: GEN x;
892:
893: if (!N) return stoi(mymyrand());
894: if (typ(N)!=t_INT || signe(N)<=0) err(talker,"invalid bound in random");
895:
896: lx = lgefint(N); x = new_chunk(lx);
897: for (i=2; i<lx; i++)
898: {
899: long av = avma, n = N[i];
900: ulong r = gp_rand();
901: if (i < lx-1) n++; else if (!n) r = 0;
902: if (n) { GEN p1 = muluu(n,r); r = (lgefint(p1)<=3)? 0: p1[2]; }
903: avma = av; x[i] = r;
904: if (r < (ulong)N[i]) break;
905: }
906: for (i++; i<lx; i++) x[i] = gp_rand();
907: i=2; while (i<lx && !x[i]) i++;
908: i -= 2; x += i; lx -= i;
909: x[1] = evalsigne(lx>2) | evallgefint(lx);
910: x[0] = evaltyp(t_INT) | evallg(lx);
911: avma = (long)x; return x;
912: }
913:
914: long
915: setrand(long seed) { return (pari_randseed = seed); }
916:
917: long
918: getrand() { return pari_randseed; }
919:
920: long
921: getstack() { return top-avma; }
922:
923: long
924: gettime() { return timer2(); }
925:
926: /***********************************************************************/
927: /** **/
928: /** PERMUTATIONS **/
929: /** **/
930: /***********************************************************************/
931:
932: GEN
933: permute(long n, GEN x)
934: {
935: long av=avma,i,a,r;
936: GEN v,w,y;
937:
938: v=(GEN)gpmalloc((n+1)*sizeof(long)); v[1]=1;
939: for (r=2; r<=n; r++)
940: {
941: x=dvmdis(x,r,&w); a=itos(w);
942: for (i=r; i>=a+2; i--) v[i]=v[i-1];
943: v[i]=r;
944: }
945: avma=av; y=cgetg(n+1,t_VEC);
946: for (i=1; i<=n; i++) y[i]=lstoi(v[i]);
947: free(v); return y;
948: }
949:
950: GEN
951: permuteInv(GEN x)
952: {
953: long av=avma,tetpil, len=lg(x)-1, ini=len, last, ind;
954: GEN ary,res;
955:
956: if (typ(x)!=t_VEC && typ(x)!=t_COL) err(talker,"not a vector in permuteInv");
957: res=gzero; ary=cgetg(len+1,t_VEC);
958: for (ind=1; ind<=len; ind++) ary[ind]=*++x;
959: ary++;
960: for (last=len; last>0; last--)
961: {
962: len--; ind=len;
963: while (ind>0 && itos((GEN)ary[ind])!=last) ind--;
964: res=mulis(res,last); tetpil=avma; res=addis(res,ind);
965: while (ind++<len) ary[ind-1]=ary[ind];
966: }
967: if (!signe(res)) { tetpil=avma; res=mpfact(ini); }
968: return gerepile(av,tetpil,res);
969: }
970:
971: /********************************************************************/
972: /** **/
973: /** MODREVERSE **/
974: /** **/
975: /********************************************************************/
976:
977: GEN
978: polymodrecip(GEN x)
979: {
980: long v,i,j,n,av,tetpil,lx;
981: GEN p1,p2,p3,p,phi,y,col;
982:
983: if (typ(x)!=t_POLMOD) err(talker,"not a polymod in polymodrecip");
984: p=(GEN)x[1]; phi=(GEN)x[2];
985: v=varn(p); n=lgef(p)-3; if (n<=0) return gcopy(x);
986: if (n==1)
987: {
988: y=cgetg(3,t_POLMOD);
989: if (typ(phi)==t_POL) phi = (GEN)phi[2];
990: p1=cgetg(4,t_POL); p1[1]=p[1]; p1[2]=lneg(phi); p1[3]=un;
991: y[1]=(long)p1;
992: if (gcmp0((GEN)p[2])) p1 = zeropol(v);
993: else
994: {
995: p1=cgetg(3,t_POL); av=avma;
996: p1[1] = evalsigne(1) | evalvarn(n) | evallgef(3);
997: p2=gdiv((GEN)p[2],(GEN)p[3]); tetpil=avma;
998: p1[2] = lpile(av,tetpil,gneg(p2));
999: }
1000: y[2]=(long)p1; return y;
1001: }
1002: if (gcmp0(phi) || typ(phi) != t_POL)
1003: err(talker,"reverse polymod does not exist");
1004: av=avma; y=cgetg(n+1,t_MAT);
1005: y[1]=(long)gscalcol_i(gun,n);
1006: p2=phi;
1007: for (j=2; j<=n; j++)
1008: {
1009: lx=lgef(p2); p1=cgetg(n+1,t_COL); y[j]=(long)p1;
1010: for (i=1; i<=lx-2; i++) p1[i]=p2[i+1];
1011: for ( ; i<=n; i++) p1[i]=zero;
1012: if (j<n) p2 = gmod(gmul(p2,phi), p);
1013: }
1014: col=cgetg(n+1,t_COL); col[1]=zero; col[2]=un;
1015: for (i=3; i<=n; i++) col[i]=zero;
1016: p1=gauss(y,col); p2=gtopolyrev(p1,v); p3=caract(x,v);
1017: tetpil=avma; return gerepile(av,tetpil,gmodulcp(p2,p3));
1018: }
1019:
1020: /********************************************************************/
1021: /** **/
1022: /** HEAPSORT **/
1023: /** **/
1024: /********************************************************************/
1025: static GEN vcmp_k;
1026: static int vcmp_lk;
1027: static int (*vcmp_cmp)(GEN,GEN);
1028:
1029: int
1030: pari_compare_int(int *a,int *b)
1031: {
1032: return *a - *b;
1033: }
1034:
1035: int
1036: pari_compare_long(long *a,long *b)
1037: {
1038: return *a - *b;
1039: }
1040:
1041: static int
1042: veccmp(GEN x, GEN y)
1043: {
1044: int i,s;
1045:
1046: for (i=1; i<vcmp_lk; i++)
1047: {
1048: s = vcmp_cmp((GEN) x[vcmp_k[i]], (GEN) y[vcmp_k[i]]);
1049: if (s) return s;
1050: }
1051: return 0;
1052: }
1053:
1054: static int
1055: longcmp(GEN x, GEN y)
1056: {
1057: return ((long)x > (long)y)? 1: ((x == y)? 0: -1);
1058: }
1059:
1060: /* Sort x = vector of elts, using cmp to compare them.
1061: * flag & cmp_IND: indirect sort: return permutation that would sort x
1062: * For private use:
1063: * flag & cmp_C : as cmp_IND, but return permutation as vector of C-longs
1064: */
1065: GEN
1066: gen_sort(GEN x, int flag, int (*cmp)(GEN,GEN))
1067: {
1068: long i,j,indxt,ir,l,tx=typ(x),lx=lg(x);
1069: GEN q,y,indx;
1070:
1071: if (!is_matvec_t(tx) && tx != t_VECSMALL) err(typeer,"gen_sort");
1072: if (flag & cmp_C) tx = t_VECSMALL;
1073: else if (flag & cmp_IND) tx = t_VEC;
1074: y = cgetg(lx, tx);
1075: if (lx==1) return y;
1076: if (lx==2)
1077: {
1078: if (flag & cmp_C)
1079: y[1] = 1;
1080: else if (flag & cmp_IND)
1081: y[1] = un;
1082: else
1083: y[1] = lcopy((GEN)x[1]);
1084: return y;
1085: }
1086: if (!cmp) cmp = &longcmp;
1087: indx = (GEN) gpmalloc(lx*sizeof(long));
1088: for (j=1; j<lx; j++) indx[j]=j;
1089:
1090: ir=lx-1; l=(ir>>1)+1;
1091: for(;;)
1092: {
1093: if (l>1)
1094: { l--; indxt = indx[l]; }
1095: else
1096: {
1097: indxt = indx[ir]; indx[ir]=indx[1]; ir--;
1098: if (ir == 1)
1099: {
1100: indx[1] = indxt;
1101: if (flag & cmp_C)
1102: for (i=1; i<lx; i++) y[i]=indx[i];
1103: else if (flag & cmp_IND)
1104: for (i=1; i<lx; i++) y[i]=lstoi(indx[i]);
1105: else
1106: for (i=1; i<lx; i++) y[i]=lcopy((GEN)x[indx[i]]);
1107: free(indx); return y;
1108: }
1109: }
1110: q = (GEN)x[indxt]; i=l;
1111: for (j=i<<1; j<=ir; j<<=1)
1112: {
1113: if (j<ir && cmp((GEN)x[indx[j]],(GEN)x[indx[j+1]]) < 0) j++;
1114: if (cmp(q,(GEN)x[indx[j]]) >= 0) break;
1115:
1116: indx[i]=indx[j]; i=j;
1117: }
1118: indx[i]=indxt;
1119: }
1120: }
1121:
1122: #define sort_fun(flag) ((flag & cmp_LEX)? &lexcmp: &gcmp)
1123:
1124: GEN
1125: gen_vecsort(GEN x, GEN k, long flag)
1126: {
1127: long i,j,l,t, lx = lg(x), tmp[2];
1128:
1129: if (lx<=2) return gen_sort(x,flag,sort_fun(flag));
1130: t = typ(k); vcmp_cmp = sort_fun(flag);
1131: if (t==t_INT)
1132: {
1133: tmp[1] = (long)k; k = tmp;
1134: vcmp_lk = 2;
1135: }
1136: else
1137: {
1138: if (! is_vec_t(t)) err(talker,"incorrect lextype in vecsort");
1139: vcmp_lk = lg(k);
1140: }
1141: l = 0;
1142: vcmp_k = (GEN)gpmalloc(vcmp_lk * sizeof(long));
1143: for (i=1; i<vcmp_lk; i++)
1144: {
1145: j = itos((GEN)k[i]);
1146: if (j<=0) err(talker,"negative index in vecsort");
1147: vcmp_k[i]=j; if (j>l) l=j;
1148: }
1149: t = typ(x);
1150: if (! is_matvec_t(t)) err(typeer,"vecsort");
1151: for (j=1; j<lx; j++)
1152: {
1153: t = typ(x[j]);
1154: if (! is_vec_t(t)) err(typeer,"vecsort");
1155: if (lg((GEN)x[j]) <= l) err(talker,"index too large in vecsort");
1156: }
1157: x = gen_sort(x, flag, veccmp);
1158: free(vcmp_k); return x;
1159: }
1160:
1161: GEN
1162: vecsort0(GEN x, GEN k, long flag)
1163: {
1164: if (flag < 0 || flag >= cmp_C) err(flagerr,"vecsort");
1165: return k? gen_vecsort(x,k,flag): gen_sort(x,flag, sort_fun(flag));
1166: }
1167:
1168: GEN
1169: vecsort(GEN x, GEN k)
1170: {
1171: return gen_vecsort(x,k, 0);
1172: }
1173:
1174: GEN
1175: sindexsort(GEN x)
1176: {
1177: return gen_sort(x, cmp_IND | cmp_C, gcmp);
1178: }
1179:
1180: GEN
1181: sindexlexsort(GEN x)
1182: {
1183: return gen_sort(x, cmp_IND | cmp_C, lexcmp);
1184: }
1185:
1186: GEN
1187: indexsort(GEN x)
1188: {
1189: return gen_sort(x, cmp_IND, gcmp);
1190: }
1191:
1192: GEN
1193: indexlexsort(GEN x)
1194: {
1195: return gen_sort(x, cmp_IND, lexcmp);
1196: }
1197:
1198: GEN
1199: sort(GEN x)
1200: {
1201: return gen_sort(x, 0, gcmp);
1202: }
1203:
1204: GEN
1205: lexsort(GEN x)
1206: {
1207: return gen_sort(x, 0, lexcmp);
1208: }
1209:
1210: /* index of x in table T, 0 otherwise */
1211: long
1212: tablesearch(GEN T, GEN x, int (*cmp)(GEN,GEN))
1213: {
1214: long l=1,u=lg(T)-1,i,s;
1215:
1216: while (u>=l)
1217: {
1218: i = (l+u)>>1; s = cmp(x,(GEN)T[i]);
1219: if (!s) return i;
1220: if (s<0) u=i-1; else l=i+1;
1221: }
1222: return 0;
1223: }
1224:
1225: /* assume lg(x) = lg(y), x,y in Z^n */
1226: int
1227: cmp_vecint(GEN x, GEN y)
1228: {
1229: long fl,i, lx = lg(x);
1230: for (i=1; i<lx; i++)
1231: if (( fl = cmpii((GEN)x[i], (GEN)y[i]) )) return fl;
1232: return 0;
1233: }
1234:
1235: /* assume x and y in primedec format. */
1236: int
1237: cmp_prime_over_p(GEN x, GEN y)
1238: {
1239: int k = mael(x,4,2) - mael(y,4,2); /* diff. between residue degree */
1240: return k? ((k > 0)? 1: -1)
1241: : cmp_vecint((GEN)x[2], (GEN)y[2]);
1242: }
1243:
1244: int
1245: cmp_prime_ideal(GEN x, GEN y)
1246: {
1247: int k = cmpii((GEN)x[1], (GEN)y[1]);
1248: return k? k: cmp_prime_over_p(x,y);
1249: }
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