Line data Source code
1 : /* Copyright (C) 2000, 2012 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /********************************************************************/
16 : /** **/
17 : /** LINEAR ALGEBRA **/
18 : /** (first part) **/
19 : /** **/
20 : /********************************************************************/
21 : #include "pari.h"
22 : #include "paripriv.h"
23 :
24 : #define DEBUGLEVEL DEBUGLEVEL_mat
25 :
26 : /*******************************************************************/
27 : /* */
28 : /* GEREPILE */
29 : /* */
30 : /*******************************************************************/
31 :
32 : static void
33 0 : gerepile_mat(pari_sp av, pari_sp tetpil, GEN x, long k, long m, long n, long t)
34 : {
35 0 : pari_sp A, bot = pari_mainstack->bot;
36 : long u, i;
37 : size_t dec;
38 :
39 0 : (void)gerepile(av,tetpil,NULL); dec = av-tetpil;
40 :
41 0 : for (u=t+1; u<=m; u++)
42 : {
43 0 : A = (pari_sp)coeff(x,u,k);
44 0 : if (A < av && A >= bot) coeff(x,u,k) += dec;
45 : }
46 0 : for (i=k+1; i<=n; i++)
47 0 : for (u=1; u<=m; u++)
48 : {
49 0 : A = (pari_sp)coeff(x,u,i);
50 0 : if (A < av && A >= bot) coeff(x,u,i) += dec;
51 : }
52 0 : }
53 :
54 : static void
55 0 : gen_gerepile_gauss_ker(GEN x, long k, long t, pari_sp av, void *E, GEN (*copy)(void*, GEN))
56 : {
57 0 : pari_sp tetpil = avma;
58 0 : long u,i, n = lg(x)-1, m = n? nbrows(x): 0;
59 :
60 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"gauss_pivot_ker. k=%ld, n=%ld",k,n);
61 0 : for (u=t+1; u<=m; u++) gcoeff(x,u,k) = copy(E,gcoeff(x,u,k));
62 0 : for (i=k+1; i<=n; i++)
63 0 : for (u=1; u<=m; u++) gcoeff(x,u,i) = copy(E,gcoeff(x,u,i));
64 0 : gerepile_mat(av,tetpil,x,k,m,n,t);
65 0 : }
66 :
67 : /* special gerepile for huge matrices */
68 :
69 : #define COPY(x) {\
70 : GEN _t = (x); if (!is_universal_constant(_t)) x = gcopy(_t); \
71 : }
72 :
73 : INLINE GEN
74 0 : _copy(void *E, GEN x)
75 : {
76 0 : (void) E; COPY(x);
77 0 : return x;
78 : }
79 :
80 : static void
81 0 : gerepile_gauss_ker(GEN x, long k, long t, pari_sp av)
82 : {
83 0 : gen_gerepile_gauss_ker(x, k, t, av, NULL, &_copy);
84 0 : }
85 :
86 : static void
87 0 : gerepile_gauss(GEN x,long k,long t,pari_sp av, long j, GEN c)
88 : {
89 0 : pari_sp tetpil = avma, A, bot;
90 0 : long u,i, n = lg(x)-1, m = n? nbrows(x): 0;
91 : size_t dec;
92 :
93 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"gauss_pivot. k=%ld, n=%ld",k,n);
94 0 : for (u=t+1; u<=m; u++)
95 0 : if (u==j || !c[u]) COPY(gcoeff(x,u,k));
96 0 : for (u=1; u<=m; u++)
97 0 : if (u==j || !c[u])
98 0 : for (i=k+1; i<=n; i++) COPY(gcoeff(x,u,i));
99 :
100 0 : (void)gerepile(av,tetpil,NULL); dec = av-tetpil;
101 0 : bot = pari_mainstack->bot;
102 0 : for (u=t+1; u<=m; u++)
103 0 : if (u==j || !c[u])
104 : {
105 0 : A=(pari_sp)coeff(x,u,k);
106 0 : if (A<av && A>=bot) coeff(x,u,k)+=dec;
107 : }
108 0 : for (u=1; u<=m; u++)
109 0 : if (u==j || !c[u])
110 0 : for (i=k+1; i<=n; i++)
111 : {
112 0 : A=(pari_sp)coeff(x,u,i);
113 0 : if (A<av && A>=bot) coeff(x,u,i)+=dec;
114 : }
115 0 : }
116 :
117 : /*******************************************************************/
118 : /* */
119 : /* GENERIC */
120 : /* */
121 : /*******************************************************************/
122 : GEN
123 1571 : gen_ker(GEN x, long deplin, void *E, const struct bb_field *ff)
124 : {
125 1571 : pari_sp av0 = avma, av, tetpil;
126 : GEN y, c, d;
127 : long i, j, k, r, t, n, m;
128 :
129 1571 : n=lg(x)-1; if (!n) return cgetg(1,t_MAT);
130 1571 : m=nbrows(x); r=0;
131 1571 : x = RgM_shallowcopy(x);
132 1571 : c = zero_zv(m);
133 1571 : d=new_chunk(n+1);
134 1571 : av=avma;
135 5625 : for (k=1; k<=n; k++)
136 : {
137 11652 : for (j=1; j<=m; j++)
138 9906 : if (!c[j])
139 : {
140 6829 : gcoeff(x,j,k) = ff->red(E, gcoeff(x,j,k));
141 6829 : if (!ff->equal0(gcoeff(x,j,k))) break;
142 : }
143 4082 : if (j>m)
144 : {
145 1746 : if (deplin)
146 : {
147 28 : GEN c = cgetg(n+1, t_COL), g0 = ff->s(E,0), g1=ff->s(E,1);
148 63 : for (i=1; i<k; i++) gel(c,i) = ff->red(E, gcoeff(x,d[i],k));
149 28 : gel(c,k) = g1; for (i=k+1; i<=n; i++) gel(c,i) = g0;
150 28 : return gerepileupto(av0, c);
151 : }
152 1718 : r++; d[k]=0;
153 4239 : for(j=1; j<k; j++)
154 2521 : if (d[j]) gcoeff(x,d[j],k) = gclone(gcoeff(x,d[j],k));
155 : }
156 : else
157 : {
158 2336 : GEN piv = ff->neg(E,ff->inv(E,gcoeff(x,j,k)));
159 2336 : c[j] = k; d[k] = j;
160 2336 : gcoeff(x,j,k) = ff->s(E,-1);
161 5453 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = ff->red(E,ff->mul(E,piv,gcoeff(x,j,i)));
162 11662 : for (t=1; t<=m; t++)
163 : {
164 9326 : if (t==j) continue;
165 :
166 6990 : piv = ff->red(E,gcoeff(x,t,k));
167 6990 : if (ff->equal0(piv)) continue;
168 :
169 2012 : gcoeff(x,t,k) = ff->s(E,0);
170 5019 : for (i=k+1; i<=n; i++)
171 3007 : gcoeff(x,t,i) = ff->red(E, ff->add(E, gcoeff(x,t,i),
172 3007 : ff->mul(E,piv,gcoeff(x,j,i))));
173 2012 : if (gc_needed(av,1))
174 0 : gen_gerepile_gauss_ker(x,k,t,av,E,ff->red);
175 : }
176 : }
177 : }
178 1543 : if (deplin) return gc_NULL(av0);
179 :
180 1515 : tetpil=avma; y=cgetg(r+1,t_MAT);
181 3233 : for (j=k=1; j<=r; j++,k++)
182 : {
183 1718 : GEN C = cgetg(n+1,t_COL);
184 1718 : GEN g0 = ff->s(E,0), g1 = ff->s(E,1);
185 3391 : gel(y,j) = C; while (d[k]) k++;
186 4239 : for (i=1; i<k; i++)
187 2521 : if (d[i])
188 : {
189 2111 : GEN p1=gcoeff(x,d[i],k);
190 2111 : gel(C,i) = ff->red(E,p1); gunclone(p1);
191 : }
192 : else
193 410 : gel(C,i) = g0;
194 2332 : gel(C,k) = g1; for (i=k+1; i<=n; i++) gel(C,i) = g0;
195 : }
196 1515 : return gerepile(av0,tetpil,y);
197 : }
198 :
199 : GEN
200 1556 : gen_Gauss_pivot(GEN x, long *rr, void *E, const struct bb_field *ff)
201 : {
202 : pari_sp av;
203 : GEN c, d;
204 1556 : long i, j, k, r, t, m, n = lg(x)-1;
205 :
206 1556 : if (!n) { *rr = 0; return NULL; }
207 :
208 1556 : m=nbrows(x); r=0;
209 1556 : d = cgetg(n+1, t_VECSMALL);
210 1556 : x = RgM_shallowcopy(x);
211 1556 : c = zero_zv(m);
212 1556 : av=avma;
213 5747 : for (k=1; k<=n; k++)
214 : {
215 10941 : for (j=1; j<=m; j++)
216 10648 : if (!c[j])
217 : {
218 7172 : gcoeff(x,j,k) = ff->red(E,gcoeff(x,j,k));
219 7172 : if (!ff->equal0(gcoeff(x,j,k))) break;
220 : }
221 4191 : if (j>m) { r++; d[k]=0; }
222 : else
223 : {
224 3898 : GEN piv = ff->neg(E,ff->inv(E,gcoeff(x,j,k)));
225 3898 : GEN g0 = ff->s(E,0);
226 3898 : c[j] = k; d[k] = j;
227 8116 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = ff->red(E,ff->mul(E,piv,gcoeff(x,j,i)));
228 24208 : for (t=1; t<=m; t++)
229 : {
230 20310 : if (c[t]) continue; /* already a pivot on that line */
231 :
232 12507 : piv = ff->red(E,gcoeff(x,t,k));
233 12507 : if (ff->equal0(piv)) continue;
234 4947 : gcoeff(x,t,k) = g0;
235 8952 : for (i=k+1; i<=n; i++)
236 4005 : gcoeff(x,t,i) = ff->red(E, ff->add(E,gcoeff(x,t,i), ff->mul(E,piv,gcoeff(x,j,i))));
237 4947 : if (gc_needed(av,1))
238 0 : gerepile_gauss(x,k,t,av,j,c);
239 : }
240 12014 : for (i=k; i<=n; i++) gcoeff(x,j,i) = g0; /* dummy */
241 : }
242 : }
243 1556 : *rr = r; return gc_const((pari_sp)d, d);
244 : }
245 :
246 : GEN
247 294 : gen_det(GEN a, void *E, const struct bb_field *ff)
248 : {
249 294 : pari_sp av = avma;
250 294 : long i,j,k, s = 1, nbco = lg(a)-1;
251 294 : GEN x = ff->s(E,1);
252 294 : if (!nbco) return x;
253 287 : a = RgM_shallowcopy(a);
254 1064 : for (i=1; i<nbco; i++)
255 : {
256 : GEN q;
257 1029 : for(k=i; k<=nbco; k++)
258 : {
259 994 : gcoeff(a,k,i) = ff->red(E,gcoeff(a,k,i));
260 994 : if (!ff->equal0(gcoeff(a,k,i))) break;
261 : }
262 812 : if (k > nbco) return gerepileupto(av, gcoeff(a,i,i));
263 777 : if (k != i)
264 : { /* exchange the lines s.t. k = i */
265 413 : for (j=i; j<=nbco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
266 105 : s = -s;
267 : }
268 777 : q = gcoeff(a,i,i);
269 777 : x = ff->red(E,ff->mul(E,x,q));
270 777 : q = ff->inv(E,q);
271 2324 : for (k=i+1; k<=nbco; k++)
272 : {
273 1547 : GEN m = ff->red(E,gcoeff(a,i,k));
274 1547 : if (ff->equal0(m)) continue;
275 1092 : m = ff->neg(E, ff->red(E,ff->mul(E,m, q)));
276 3528 : for (j=i+1; j<=nbco; j++)
277 2436 : gcoeff(a,j,k) = ff->red(E, ff->add(E, gcoeff(a,j,k),
278 2436 : ff->mul(E, m, gcoeff(a,j,i))));
279 : }
280 777 : if (gc_needed(av,2))
281 : {
282 0 : if(DEBUGMEM>1) pari_warn(warnmem,"det. col = %ld",i);
283 0 : gerepileall(av,2, &a,&x);
284 : }
285 : }
286 252 : if (s < 0) x = ff->neg(E,x);
287 252 : return gerepileupto(av, ff->red(E,ff->mul(E, x, gcoeff(a,nbco,nbco))));
288 : }
289 :
290 : INLINE void
291 143984 : _gen_addmul(GEN b, long k, long i, GEN m, void *E, const struct bb_field *ff)
292 : {
293 143984 : gel(b,i) = ff->red(E,gel(b,i));
294 143984 : gel(b,k) = ff->add(E,gel(b,k), ff->mul(E,m, gel(b,i)));
295 143984 : }
296 :
297 : static GEN
298 54854 : _gen_get_col(GEN a, GEN b, long li, void *E, const struct bb_field *ff)
299 : {
300 54854 : GEN u = cgetg(li+1,t_COL);
301 54854 : pari_sp av = avma;
302 : long i, j;
303 :
304 54854 : gel(u,li) = gerepileupto(av, ff->red(E,ff->mul(E,gel(b,li), gcoeff(a,li,li))));
305 285179 : for (i=li-1; i>0; i--)
306 : {
307 230325 : pari_sp av = avma;
308 230325 : GEN m = gel(b,i);
309 921848 : for (j=i+1; j<=li; j++) m = ff->add(E,m, ff->neg(E,ff->mul(E,gcoeff(a,i,j), gel(u,j))));
310 230325 : m = ff->red(E, m);
311 230325 : gel(u,i) = gerepileupto(av, ff->red(E,ff->mul(E,m, gcoeff(a,i,i))));
312 : }
313 54854 : return u;
314 : }
315 :
316 : GEN
317 12193 : gen_Gauss(GEN a, GEN b, void *E, const struct bb_field *ff)
318 : {
319 : long i, j, k, li, bco, aco;
320 12193 : GEN u, g0 = ff->s(E,0);
321 12193 : pari_sp av = avma;
322 12193 : a = RgM_shallowcopy(a);
323 12193 : b = RgM_shallowcopy(b);
324 12193 : aco = lg(a)-1; bco = lg(b)-1; li = nbrows(a);
325 53713 : for (i=1; i<=aco; i++)
326 : {
327 : GEN invpiv;
328 64675 : for (k = i; k <= li; k++)
329 : {
330 64619 : GEN piv = ff->red(E,gcoeff(a,k,i));
331 64619 : if (!ff->equal0(piv)) { gcoeff(a,k,i) = ff->inv(E,piv); break; }
332 10962 : gcoeff(a,k,i) = g0;
333 : }
334 : /* found a pivot on line k */
335 53713 : if (k > li) return NULL;
336 53657 : if (k != i)
337 : { /* swap lines so that k = i */
338 44212 : for (j=i; j<=aco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
339 59304 : for (j=1; j<=bco; j++) swap(gcoeff(b,i,j), gcoeff(b,k,j));
340 : }
341 53657 : if (i == aco) break;
342 :
343 41520 : invpiv = gcoeff(a,i,i); /* 1/piv mod p */
344 157435 : for (k=i+1; k<=li; k++)
345 : {
346 115915 : GEN m = ff->red(E,gcoeff(a,k,i)); gcoeff(a,k,i) = g0;
347 115915 : if (ff->equal0(m)) continue;
348 :
349 17697 : m = ff->red(E,ff->neg(E,ff->mul(E,m, invpiv)));
350 70921 : for (j=i+1; j<=aco; j++) _gen_addmul(gel(a,j),k,i,m,E,ff);
351 108457 : for (j=1 ; j<=bco; j++) _gen_addmul(gel(b,j),k,i,m,E,ff);
352 : }
353 41520 : if (gc_needed(av,1))
354 : {
355 0 : if(DEBUGMEM>1) pari_warn(warnmem,"gen_Gauss. i=%ld",i);
356 0 : gerepileall(av,2, &a,&b);
357 : }
358 : }
359 :
360 12137 : if(DEBUGLEVEL>4) err_printf("Solving the triangular system\n");
361 12137 : u = cgetg(bco+1,t_MAT);
362 66991 : for (j=1; j<=bco; j++) gel(u,j) = _gen_get_col(a, gel(b,j), aco, E, ff);
363 12137 : return u;
364 : }
365 :
366 : /* compatible t_MAT * t_COL, lgA = lg(A) = lg(B) > 1, l = lgcols(A) */
367 : static GEN
368 576823 : gen_matcolmul_i(GEN A, GEN B, ulong lgA, ulong l,
369 : void *E, const struct bb_field *ff)
370 : {
371 576823 : GEN C = cgetg(l, t_COL);
372 : ulong i;
373 3855180 : for (i = 1; i < l; i++) {
374 3278357 : pari_sp av = avma;
375 3278357 : GEN e = ff->mul(E, gcoeff(A, i, 1), gel(B, 1));
376 : ulong k;
377 14093105 : for(k = 2; k < lgA; k++)
378 10814748 : e = ff->add(E, e, ff->mul(E, gcoeff(A, i, k), gel(B, k)));
379 3278357 : gel(C, i) = gerepileupto(av, ff->red(E, e));
380 : }
381 576823 : return C;
382 : }
383 :
384 : GEN
385 180173 : gen_matcolmul(GEN A, GEN B, void *E, const struct bb_field *ff)
386 : {
387 180173 : ulong lgA = lg(A);
388 180173 : if (lgA != (ulong)lg(B))
389 0 : pari_err_OP("operation 'gen_matcolmul'", A, B);
390 180173 : if (lgA == 1)
391 0 : return cgetg(1, t_COL);
392 180173 : return gen_matcolmul_i(A, B, lgA, lgcols(A), E, ff);
393 : }
394 :
395 : static GEN
396 76357 : gen_matmul_classical(GEN A, GEN B, long l, long la, long lb,
397 : void *E, const struct bb_field *ff)
398 : {
399 : long j;
400 76357 : GEN C = cgetg(lb, t_MAT);
401 473007 : for(j = 1; j < lb; j++)
402 396650 : gel(C, j) = gen_matcolmul_i(A, gel(B, j), la, l, E, ff);
403 76357 : return C;
404 : }
405 :
406 : /* Strassen-Winograd algorithm */
407 :
408 : /*
409 : Return A[ma+1..ma+da, na+1..na+ea] - B[mb+1..mb+db, nb+1..nb+eb]
410 : as an (m x n)-matrix, padding the input with zeroes as necessary.
411 : */
412 : static GEN
413 0 : add_slices(long m, long n,
414 : GEN A, long ma, long da, long na, long ea,
415 : GEN B, long mb, long db, long nb, long eb,
416 : void *E, const struct bb_field *ff)
417 : {
418 0 : long min_d = minss(da, db), min_e = minss(ea, eb), i, j;
419 0 : GEN M = cgetg(n + 1, t_MAT), C;
420 :
421 0 : for (j = 1; j <= min_e; j++) {
422 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
423 0 : for (i = 1; i <= min_d; i++)
424 0 : gel(C, i) = ff->add(E, gcoeff(A, ma + i, na + j),
425 0 : gcoeff(B, mb + i, nb + j));
426 0 : for (; i <= da; i++)
427 0 : gel(C, i) = gcoeff(A, ma + i, na + j);
428 0 : for (; i <= db; i++)
429 0 : gel(C, i) = gcoeff(B, mb + i, nb + j);
430 0 : for (; i <= m; i++)
431 0 : gel(C, i) = ff->s(E, 0);
432 : }
433 0 : for (; j <= ea; j++) {
434 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
435 0 : for (i = 1; i <= da; i++)
436 0 : gel(C, i) = gcoeff(A, ma + i, na + j);
437 0 : for (; i <= m; i++)
438 0 : gel(C, i) = ff->s(E, 0);
439 : }
440 0 : for (; j <= eb; j++) {
441 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
442 0 : for (i = 1; i <= db; i++)
443 0 : gel(C, i) = gcoeff(B, mb + i, nb + j);
444 0 : for (; i <= m; i++)
445 0 : gel(C, i) = ff->s(E, 0);
446 : }
447 0 : for (; j <= n; j++) {
448 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
449 0 : for (i = 1; i <= m; i++)
450 0 : gel(C, i) = ff->s(E, 0);
451 : }
452 0 : return M;
453 : }
454 :
455 : /*
456 : Return A[ma+1..ma+da, na+1..na+ea] - B[mb+1..mb+db, nb+1..nb+eb]
457 : as an (m x n)-matrix, padding the input with zeroes as necessary.
458 : */
459 : static GEN
460 0 : subtract_slices(long m, long n,
461 : GEN A, long ma, long da, long na, long ea,
462 : GEN B, long mb, long db, long nb, long eb,
463 : void *E, const struct bb_field *ff)
464 : {
465 0 : long min_d = minss(da, db), min_e = minss(ea, eb), i, j;
466 0 : GEN M = cgetg(n + 1, t_MAT), C;
467 :
468 0 : for (j = 1; j <= min_e; j++) {
469 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
470 0 : for (i = 1; i <= min_d; i++)
471 0 : gel(C, i) = ff->add(E, gcoeff(A, ma + i, na + j),
472 0 : ff->neg(E, gcoeff(B, mb + i, nb + j)));
473 0 : for (; i <= da; i++)
474 0 : gel(C, i) = gcoeff(A, ma + i, na + j);
475 0 : for (; i <= db; i++)
476 0 : gel(C, i) = ff->neg(E, gcoeff(B, mb + i, nb + j));
477 0 : for (; i <= m; i++)
478 0 : gel(C, i) = ff->s(E, 0);
479 : }
480 0 : for (; j <= ea; j++) {
481 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
482 0 : for (i = 1; i <= da; i++)
483 0 : gel(C, i) = gcoeff(A, ma + i, na + j);
484 0 : for (; i <= m; i++)
485 0 : gel(C, i) = ff->s(E, 0);
486 : }
487 0 : for (; j <= eb; j++) {
488 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
489 0 : for (i = 1; i <= db; i++)
490 0 : gel(C, i) = ff->neg(E, gcoeff(B, mb + i, nb + j));
491 0 : for (; i <= m; i++)
492 0 : gel(C, i) = ff->s(E, 0);
493 : }
494 0 : for (; j <= n; j++) {
495 0 : gel(M, j) = C = cgetg(m + 1, t_COL);
496 0 : for (i = 1; i <= m; i++)
497 0 : gel(C, i) = ff->s(E, 0);
498 : }
499 0 : return M;
500 : }
501 :
502 : static GEN gen_matmul_i(GEN A, GEN B, long l, long la, long lb,
503 : void *E, const struct bb_field *ff);
504 :
505 : static GEN
506 0 : gen_matmul_sw(GEN A, GEN B, long m, long n, long p,
507 : void *E, const struct bb_field *ff)
508 : {
509 0 : pari_sp av = avma;
510 0 : long m1 = (m + 1)/2, m2 = m/2,
511 0 : n1 = (n + 1)/2, n2 = n/2,
512 0 : p1 = (p + 1)/2, p2 = p/2;
513 : GEN A11, A12, A22, B11, B21, B22,
514 : S1, S2, S3, S4, T1, T2, T3, T4,
515 : M1, M2, M3, M4, M5, M6, M7,
516 : V1, V2, V3, C11, C12, C21, C22, C;
517 :
518 0 : T2 = subtract_slices(n1, p2, B, 0, n1, p1, p2, B, n1, n2, p1, p2, E, ff);
519 0 : S1 = subtract_slices(m2, n1, A, m1, m2, 0, n1, A, 0, m2, 0, n1, E, ff);
520 0 : M2 = gen_matmul_i(S1, T2, m2 + 1, n1 + 1, p2 + 1, E, ff);
521 0 : if (gc_needed(av, 1))
522 0 : gerepileall(av, 2, &T2, &M2); /* destroy S1 */
523 0 : T3 = subtract_slices(n1, p1, T2, 0, n1, 0, p2, B, 0, n1, 0, p1, E, ff);
524 0 : if (gc_needed(av, 1))
525 0 : gerepileall(av, 2, &M2, &T3); /* destroy T2 */
526 0 : S2 = add_slices(m2, n1, A, m1, m2, 0, n1, A, m1, m2, n1, n2, E, ff);
527 0 : T1 = subtract_slices(n1, p1, B, 0, n1, p1, p2, B, 0, n1, 0, p2, E, ff);
528 0 : M3 = gen_matmul_i(S2, T1, m2 + 1, n1 + 1, p2 + 1, E, ff);
529 0 : if (gc_needed(av, 1))
530 0 : gerepileall(av, 4, &M2, &T3, &S2, &M3); /* destroy T1 */
531 0 : S3 = subtract_slices(m1, n1, S2, 0, m2, 0, n1, A, 0, m1, 0, n1, E, ff);
532 0 : if (gc_needed(av, 1))
533 0 : gerepileall(av, 4, &M2, &T3, &M3, &S3); /* destroy S2 */
534 0 : A11 = matslice(A, 1, m1, 1, n1);
535 0 : B11 = matslice(B, 1, n1, 1, p1);
536 0 : M1 = gen_matmul_i(A11, B11, m1 + 1, n1 + 1, p1 + 1, E, ff);
537 0 : if (gc_needed(av, 1))
538 0 : gerepileall(av, 5, &M2, &T3, &M3, &S3, &M1); /* destroy A11, B11 */
539 0 : A12 = matslice(A, 1, m1, n1 + 1, n);
540 0 : B21 = matslice(B, n1 + 1, n, 1, p1);
541 0 : M4 = gen_matmul_i(A12, B21, m1 + 1, n2 + 1, p1 + 1, E, ff);
542 0 : if (gc_needed(av, 1))
543 0 : gerepileall(av, 6, &M2, &T3, &M3, &S3, &M1, &M4); /* destroy A12, B21 */
544 0 : C11 = add_slices(m1, p1, M1, 0, m1, 0, p1, M4, 0, m1, 0, p1, E, ff);
545 0 : if (gc_needed(av, 1))
546 0 : gerepileall(av, 6, &M2, &T3, &M3, &S3, &M1, &C11); /* destroy M4 */
547 0 : M5 = gen_matmul_i(S3, T3, m1 + 1, n1 + 1, p1 + 1, E, ff);
548 0 : S4 = subtract_slices(m1, n2, A, 0, m1, n1, n2, S3, 0, m1, 0, n2, E, ff);
549 0 : if (gc_needed(av, 1))
550 0 : gerepileall(av, 7, &M2, &T3, &M3, &M1, &C11, &M5, &S4); /* destroy S3 */
551 0 : T4 = add_slices(n2, p1, B, n1, n2, 0, p1, T3, 0, n2, 0, p1, E, ff);
552 0 : if (gc_needed(av, 1))
553 0 : gerepileall(av, 7, &M2, &M3, &M1, &C11, &M5, &S4, &T4); /* destroy T3 */
554 0 : V1 = subtract_slices(m1, p1, M1, 0, m1, 0, p1, M5, 0, m1, 0, p1, E, ff);
555 0 : if (gc_needed(av, 1))
556 0 : gerepileall(av, 6, &M2, &M3, &S4, &T4, &C11, &V1); /* destroy M1, M5 */
557 0 : B22 = matslice(B, n1 + 1, n, p1 + 1, p);
558 0 : M6 = gen_matmul_i(S4, B22, m1 + 1, n2 + 1, p2 + 1, E, ff);
559 0 : if (gc_needed(av, 1))
560 0 : gerepileall(av, 6, &M2, &M3, &T4, &C11, &V1, &M6); /* destroy S4, B22 */
561 0 : A22 = matslice(A, m1 + 1, m, n1 + 1, n);
562 0 : M7 = gen_matmul_i(A22, T4, m2 + 1, n2 + 1, p1 + 1, E, ff);
563 0 : if (gc_needed(av, 1))
564 0 : gerepileall(av, 6, &M2, &M3, &C11, &V1, &M6, &M7); /* destroy A22, T4 */
565 0 : V3 = add_slices(m1, p2, V1, 0, m1, 0, p2, M3, 0, m2, 0, p2, E, ff);
566 0 : C12 = add_slices(m1, p2, V3, 0, m1, 0, p2, M6, 0, m1, 0, p2, E, ff);
567 0 : if (gc_needed(av, 1))
568 0 : gerepileall(av, 6, &M2, &M3, &C11, &V1, &M7, &C12); /* destroy V3, M6 */
569 0 : V2 = add_slices(m2, p1, V1, 0, m2, 0, p1, M2, 0, m2, 0, p2, E, ff);
570 0 : if (gc_needed(av, 1))
571 0 : gerepileall(av, 5, &M3, &C11, &M7, &C12, &V2); /* destroy V1, M2 */
572 0 : C21 = add_slices(m2, p1, V2, 0, m2, 0, p1, M7, 0, m2, 0, p1, E, ff);
573 0 : if (gc_needed(av, 1))
574 0 : gerepileall(av, 5, &M3, &C11, &C12, &V2, &C21); /* destroy M7 */
575 0 : C22 = add_slices(m2, p2, V2, 0, m2, 0, p2, M3, 0, m2, 0, p2, E, ff);
576 0 : if (gc_needed(av, 1))
577 0 : gerepileall(av, 4, &C11, &C12, &C21, &C22); /* destroy V2, M3 */
578 0 : C = mkmat2(mkcol2(C11, C21), mkcol2(C12, C22));
579 0 : return gerepileupto(av, matconcat(C));
580 : }
581 :
582 : /* Strassen-Winograd used for dim >= gen_matmul_sw_bound */
583 : static const long gen_matmul_sw_bound = 24;
584 :
585 : static GEN
586 76357 : gen_matmul_i(GEN A, GEN B, long l, long la, long lb,
587 : void *E, const struct bb_field *ff)
588 : {
589 76357 : if (l <= gen_matmul_sw_bound
590 7 : || la <= gen_matmul_sw_bound
591 0 : || lb <= gen_matmul_sw_bound)
592 76357 : return gen_matmul_classical(A, B, l, la, lb, E, ff);
593 : else
594 0 : return gen_matmul_sw(A, B, l - 1, la - 1, lb - 1, E, ff);
595 : }
596 :
597 : GEN
598 76357 : gen_matmul(GEN A, GEN B, void *E, const struct bb_field *ff)
599 : {
600 76357 : ulong lgA, lgB = lg(B);
601 76357 : if (lgB == 1)
602 0 : return cgetg(1, t_MAT);
603 76357 : lgA = lg(A);
604 76357 : if (lgA != (ulong)lgcols(B))
605 0 : pari_err_OP("operation 'gen_matmul'", A, B);
606 76357 : if (lgA == 1)
607 0 : return zeromat(0, lgB - 1);
608 76357 : return gen_matmul_i(A, B, lgcols(A), lgA, lgB, E, ff);
609 : }
610 :
611 : static GEN
612 18632 : gen_colneg(GEN A, void *E, const struct bb_field *ff)
613 : {
614 : long i, l;
615 18632 : GEN B = cgetg_copy(A, &l);
616 72593 : for (i = 1; i < l; i++)
617 53961 : gel(B, i) = ff->neg(E, gel(A, i));
618 18632 : return B;
619 : }
620 :
621 : static GEN
622 3958 : gen_matneg(GEN A, void *E, const struct bb_field *ff)
623 : {
624 : long i, l;
625 3958 : GEN B = cgetg_copy(A, &l);
626 22534 : for (i = 1; i < l; i++)
627 18576 : gel(B, i) = gen_colneg(gel(A, i), E, ff);
628 3958 : return B;
629 : }
630 :
631 : static GEN
632 284291 : gen_colscalmul(GEN A, GEN b, void *E, const struct bb_field *ff)
633 : {
634 : long i, l;
635 284291 : GEN B = cgetg_copy(A, &l);
636 681214 : for (i = 1; i < l; i++)
637 396923 : gel(B, i) = ff->red(E, ff->mul(E, gel(A, i), b));
638 284291 : return B;
639 : }
640 :
641 : static GEN
642 50970 : gen_matscalmul(GEN A, GEN b, void *E, const struct bb_field *ff)
643 : {
644 : long i, l;
645 50970 : GEN B = cgetg_copy(A, &l);
646 335261 : for (i = 1; i < l; i++)
647 284291 : gel(B, i) = gen_colscalmul(gel(A, i), b, E, ff);
648 50970 : return B;
649 : }
650 :
651 : static GEN
652 534315 : gen_colsub(GEN A, GEN C, void *E, const struct bb_field *ff)
653 : {
654 : long i, l;
655 534315 : GEN B = cgetg_copy(A, &l);
656 1948132 : for (i = 1; i < l; i++)
657 1413817 : gel(B, i) = ff->add(E, gel(A, i), ff->neg(E, gel(C, i)));
658 534315 : return B;
659 : }
660 :
661 : static GEN
662 73095 : gen_matsub(GEN A, GEN C, void *E, const struct bb_field *ff)
663 : {
664 : long i, l;
665 73095 : GEN B = cgetg_copy(A, &l);
666 607410 : for (i = 1; i < l; i++)
667 534315 : gel(B, i) = gen_colsub(gel(A, i), gel(C, i), E, ff);
668 73095 : return B;
669 : }
670 :
671 : static GEN
672 44332 : gen_zerocol(long n, void* data, const struct bb_field *R)
673 : {
674 44332 : GEN C = cgetg(n+1,t_COL), zero = R->s(data, 0);
675 : long i;
676 266059 : for (i=1; i<=n; i++) gel(C,i) = zero;
677 44332 : return C;
678 : }
679 :
680 : static GEN
681 14126 : gen_zeromat(long m, long n, void* data, const struct bb_field *R)
682 : {
683 14126 : GEN M = cgetg(n+1,t_MAT);
684 : long i;
685 58458 : for (i=1; i<=n; i++) gel(M,i) = gen_zerocol(m, data, R);
686 14126 : return M;
687 : }
688 :
689 : static GEN
690 140 : gen_colei(long n, long i, void *E, const struct bb_field *S)
691 : {
692 140 : GEN y = cgetg(n+1,t_COL), _0, _1;
693 : long j;
694 140 : if (n < 0) pari_err_DOMAIN("gen_colei", "dimension","<",gen_0,stoi(n));
695 140 : _0 = S->s(E,0);
696 140 : _1 = S->s(E,1);
697 2268 : for (j=1; j<=n; j++)
698 2128 : gel(y, j) = i==j ? _1: _0;
699 140 : return y;
700 : }
701 :
702 : /* assume dim A >= 1, A invertible + upper triangular */
703 : static GEN
704 77 : gen_matinv_upper_ind(GEN A, long index, void *E, const struct bb_field *ff)
705 : {
706 77 : long n = lg(A) - 1, i, j;
707 77 : GEN u = cgetg(n + 1, t_COL);
708 147 : for (i = n; i > index; i--)
709 70 : gel(u, i) = ff->s(E, 0);
710 77 : gel(u, i) = ff->inv(E, gcoeff(A, i, i));
711 147 : for (i--; i > 0; i--) {
712 70 : pari_sp av = avma;
713 70 : GEN m = ff->neg(E, ff->mul(E, gcoeff(A, i, i + 1), gel(u, i + 1)));
714 112 : for (j = i + 2; j <= n; j++)
715 42 : m = ff->add(E, m, ff->neg(E, ff->mul(E, gcoeff(A, i, j), gel(u, j))));
716 70 : gel(u, i) = gerepileupto(av, ff->red(E, ff->mul(E, m, ff->inv(E, gcoeff(A, i, i)))));
717 : }
718 77 : return u;
719 : }
720 :
721 : static GEN
722 28 : gen_matinv_upper(GEN A, void *E, const struct bb_field *ff)
723 : {
724 : long i, l;
725 28 : GEN B = cgetg_copy(A, &l);
726 105 : for (i = 1; i < l; i++)
727 77 : gel(B,i) = gen_matinv_upper_ind(A, i, E, ff);
728 28 : return B;
729 : }
730 :
731 : /* find z such that A z = y. Return NULL if no solution */
732 : GEN
733 0 : gen_matcolinvimage(GEN A, GEN y, void *E, const struct bb_field *ff)
734 : {
735 0 : pari_sp av = avma;
736 0 : long i, l = lg(A);
737 : GEN M, x, t;
738 :
739 0 : M = gen_ker(shallowconcat(A, y), 0, E, ff);
740 0 : i = lg(M) - 1;
741 0 : if (!i) return gc_NULL(av);
742 :
743 0 : x = gel(M, i);
744 0 : t = gel(x, l);
745 0 : if (ff->equal0(t)) return gc_NULL(av);
746 :
747 0 : t = ff->neg(E, ff->inv(E, t));
748 0 : setlg(x, l);
749 0 : for (i = 1; i < l; i++)
750 0 : gel(x, i) = ff->red(E, ff->mul(E, t, gel(x, i)));
751 0 : return gerepilecopy(av, x);
752 : }
753 :
754 : /* find Z such that A Z = B. Return NULL if no solution */
755 : GEN
756 77 : gen_matinvimage(GEN A, GEN B, void *E, const struct bb_field *ff)
757 : {
758 77 : pari_sp av = avma;
759 : GEN d, x, X, Y;
760 : long i, j, nY, nA, nB;
761 77 : x = gen_ker(shallowconcat(gen_matneg(A, E, ff), B), 0, E, ff);
762 : /* AX = BY, Y in strict upper echelon form with pivots = 1.
763 : * We must find T such that Y T = Id_nB then X T = Z. This exists
764 : * iff Y has at least nB columns and full rank. */
765 77 : nY = lg(x) - 1;
766 77 : nB = lg(B) - 1;
767 77 : if (nY < nB) return gc_NULL(av);
768 77 : nA = lg(A) - 1;
769 77 : Y = rowslice(x, nA + 1, nA + nB); /* nB rows */
770 77 : d = cgetg(nB + 1, t_VECSMALL);
771 182 : for (i = nB, j = nY; i >= 1; i--, j--) {
772 224 : for (; j >= 1; j--)
773 175 : if (!ff->equal0(gcoeff(Y, i, j))) { d[i] = j; break; }
774 154 : if (!j) return gc_NULL(av);
775 : }
776 : /* reduce to the case Y square, upper triangular with 1s on diagonal */
777 28 : Y = vecpermute(Y, d);
778 28 : x = vecpermute(x, d);
779 28 : X = rowslice(x, 1, nA);
780 28 : return gerepileupto(av, gen_matmul(X, gen_matinv_upper(Y, E, ff), E, ff));
781 : }
782 :
783 : static GEN
784 380826 : image_from_pivot(GEN x, GEN d, long r)
785 : {
786 : GEN y;
787 : long j, k;
788 :
789 380826 : if (!d) return gcopy(x);
790 : /* d left on stack for efficiency */
791 378054 : r = lg(x)-1 - r; /* = dim Im(x) */
792 378054 : y = cgetg(r+1,t_MAT);
793 2376748 : for (j=k=1; j<=r; k++)
794 1998694 : if (d[k]) gel(y,j++) = gcopy(gel(x,k));
795 378054 : return y;
796 : }
797 :
798 : /* r = dim Ker x, n = nbrows(x) */
799 : static GEN
800 272631 : get_suppl(GEN x, GEN d, long n, long r, GEN(*ei)(long,long))
801 : {
802 : pari_sp av;
803 : GEN y, c;
804 272631 : long j, k, rx = lg(x)-1; /* != 0 due to init_suppl() */
805 :
806 272631 : if (rx == n && r == 0) return gcopy(x);
807 201662 : y = cgetg(n+1, t_MAT);
808 201665 : av = avma; c = zero_zv(n);
809 : /* c = lines containing pivots (could get it from gauss_pivot, but cheap)
810 : * In theory r = 0 and d[j] > 0 for all j, but why take chances? */
811 916032 : for (k = j = 1; j<=rx; j++)
812 714364 : if (d[j]) { c[ d[j] ] = 1; gel(y,k++) = gel(x,j); }
813 1286816 : for (j=1; j<=n; j++)
814 1085148 : if (!c[j]) gel(y,k++) = (GEN)j; /* HACK */
815 201668 : set_avma(av);
816 :
817 201667 : rx -= r;
818 915959 : for (j=1; j<=rx; j++) gel(y,j) = gcopy(gel(y,j));
819 572521 : for ( ; j<=n; j++) gel(y,j) = ei(n, y[j]);
820 201666 : return y;
821 : }
822 :
823 : /* n = dim x, r = dim Ker(x), d from gauss_pivot */
824 : static GEN
825 189503 : indexrank0(long n, long r, GEN d)
826 : {
827 189503 : GEN p1, p2, res = cgetg(3,t_VEC);
828 : long i, j;
829 :
830 189503 : r = n - r; /* now r = dim Im(x) */
831 189503 : p1 = cgetg(r+1,t_VECSMALL); gel(res,1) = p1;
832 189503 : p2 = cgetg(r+1,t_VECSMALL); gel(res,2) = p2;
833 189503 : if (d)
834 : {
835 1058800 : for (i=0,j=1; j<=n; j++)
836 870662 : if (d[j]) { i++; p1[i] = d[j]; p2[i] = j; }
837 188138 : vecsmall_sort(p1);
838 : }
839 189503 : return res;
840 : }
841 :
842 : /*******************************************************************/
843 : /* */
844 : /* Echelon form and CUP decomposition */
845 : /* */
846 : /*******************************************************************/
847 :
848 : /* By Peter Bruin, based on
849 : C.-P. Jeannerod, C. Pernet and A. Storjohann, Rank-profile revealing
850 : Gaussian elimination and the CUP matrix decomposition. J. Symbolic
851 : Comput. 56 (2013), 46-68.
852 :
853 : Decompose an m x n-matrix A of rank r as C*U*P, with
854 : - C: m x r-matrix in column echelon form (not necessarily reduced)
855 : with all pivots equal to 1
856 : - U: upper-triangular r x n-matrix
857 : - P: permutation matrix
858 : The pivots of C and the known zeroes in C and U are not necessarily
859 : filled in; instead, we also return the vector R of pivot rows.
860 : Instead of the matrix P, we return the permutation p of [1..n]
861 : (t_VECSMALL) such that P[i,j] = 1 if and only if j = p[i].
862 : */
863 :
864 : /* complement of a strictly increasing subsequence of (1, 2, ..., n) */
865 : static GEN
866 17049 : indexcompl(GEN v, long n)
867 : {
868 17049 : long i, j, k, m = lg(v) - 1;
869 17049 : GEN w = cgetg(n - m + 1, t_VECSMALL);
870 157070 : for (i = j = k = 1; i <= n; i++)
871 140021 : if (j <= m && v[j] == i) j++; else w[k++] = i;
872 17049 : return w;
873 : }
874 :
875 : static GEN
876 3754 : gen_solve_upper_1(GEN U, GEN B, void *E, const struct bb_field *ff)
877 3754 : { return gen_matscalmul(B, ff->inv(E, gcoeff(U, 1, 1)), E, ff); }
878 :
879 : static GEN
880 2074 : gen_rsolve_upper_2(GEN U, GEN B, void *E, const struct bb_field *ff)
881 : {
882 2074 : GEN a = gcoeff(U, 1, 1), b = gcoeff(U, 1, 2), d = gcoeff(U, 2, 2);
883 2074 : GEN D = ff->red(E, ff->mul(E, a, d)), Dinv = ff->inv(E, D);
884 2074 : GEN ainv = ff->red(E, ff->mul(E, d, Dinv));
885 2074 : GEN dinv = ff->red(E, ff->mul(E, a, Dinv));
886 2074 : GEN B1 = rowslice(B, 1, 1);
887 2074 : GEN B2 = rowslice(B, 2, 2);
888 2074 : GEN X2 = gen_matscalmul(B2, dinv, E, ff);
889 2074 : GEN X1 = gen_matscalmul(gen_matsub(B1, gen_matscalmul(X2, b, E, ff), E, ff),
890 : ainv, E, ff);
891 2074 : return vconcat(X1, X2);
892 : }
893 :
894 : /* solve U*X = B, U upper triangular and invertible */
895 : static GEN
896 5399 : gen_rsolve_upper(GEN U, GEN B, void *E, const struct bb_field *ff,
897 : GEN (*mul)(void *E, GEN a, GEN))
898 : {
899 5399 : long n = lg(U) - 1, n1;
900 : GEN U2, U11, U12, U22, B1, B2, X1, X2, X;
901 5399 : pari_sp av = avma;
902 :
903 5399 : if (n == 0) return B;
904 5399 : if (n == 1) return gen_solve_upper_1(U, B, E, ff);
905 4529 : if (n == 2) return gen_rsolve_upper_2(U, B, E, ff);
906 2455 : n1 = (n + 1)/2;
907 2455 : U2 = vecslice(U, n1 + 1, n);
908 2455 : U11 = matslice(U, 1,n1, 1,n1);
909 2455 : U12 = rowslice(U2, 1, n1);
910 2455 : U22 = rowslice(U2, n1 + 1, n);
911 2455 : B1 = rowslice(B, 1, n1);
912 2455 : B2 = rowslice(B, n1 + 1, n);
913 2455 : X2 = gen_rsolve_upper(U22, B2, E, ff, mul);
914 2455 : B1 = gen_matsub(B1, mul(E, U12, X2), E, ff);
915 2455 : if (gc_needed(av, 1)) gerepileall(av, 3, &B1, &U11, &X2);
916 2455 : X1 = gen_rsolve_upper(U11, B1, E, ff, mul);
917 2455 : X = vconcat(X1, X2);
918 2455 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
919 2455 : return X;
920 : }
921 :
922 : static GEN
923 5947 : gen_lsolve_upper_2(GEN U, GEN B, void *E, const struct bb_field *ff)
924 : {
925 5947 : GEN a = gcoeff(U, 1, 1), b = gcoeff(U, 1, 2), d = gcoeff(U, 2, 2);
926 5947 : GEN D = ff->red(E, ff->mul(E, a, d)), Dinv = ff->inv(E, D);
927 5947 : GEN ainv = ff->red(E, ff->mul(E, d, Dinv)), dinv = ff->red(E, ff->mul(E, a, Dinv));
928 5947 : GEN B1 = vecslice(B, 1, 1);
929 5947 : GEN B2 = vecslice(B, 2, 2);
930 5947 : GEN X1 = gen_matscalmul(B1, ainv, E, ff);
931 5947 : GEN X2 = gen_matscalmul(gen_matsub(B2, gen_matscalmul(X1, b, E, ff), E, ff), dinv, E, ff);
932 5947 : return shallowconcat(X1, X2);
933 : }
934 :
935 : /* solve X*U = B, U upper triangular and invertible */
936 : static GEN
937 13523 : gen_lsolve_upper(GEN U, GEN B, void *E, const struct bb_field *ff,
938 : GEN (*mul)(void *E, GEN a, GEN))
939 : {
940 13523 : long n = lg(U) - 1, n1;
941 : GEN U2, U11, U12, U22, B1, B2, X1, X2, X;
942 13523 : pari_sp av = avma;
943 :
944 13523 : if (n == 0) return B;
945 13523 : if (n == 1) return gen_solve_upper_1(U, B, E, ff);
946 10639 : if (n == 2) return gen_lsolve_upper_2(U, B, E, ff);
947 4692 : n1 = (n + 1)/2;
948 4692 : U2 = vecslice(U, n1 + 1, n);
949 4692 : U11 = matslice(U, 1,n1, 1,n1);
950 4692 : U12 = rowslice(U2, 1, n1);
951 4692 : U22 = rowslice(U2, n1 + 1, n);
952 4692 : B1 = vecslice(B, 1, n1);
953 4692 : B2 = vecslice(B, n1 + 1, n);
954 4692 : X1 = gen_lsolve_upper(U11, B1, E, ff, mul);
955 4692 : B2 = gen_matsub(B2, mul(E, X1, U12), E, ff);
956 4692 : if (gc_needed(av, 1)) gerepileall(av, 3, &B2, &U22, &X1);
957 4692 : X2 = gen_lsolve_upper(U22, B2, E, ff, mul);
958 4692 : X = shallowconcat(X1, X2);
959 4692 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
960 4692 : return X;
961 : }
962 :
963 : static GEN
964 15688 : gen_rsolve_lower_unit_2(GEN L, GEN A, void *E, const struct bb_field *ff)
965 : {
966 15688 : GEN X1 = rowslice(A, 1, 1);
967 15688 : GEN X2 = gen_matsub(rowslice(A, 2, 2), gen_matscalmul(X1, gcoeff(L, 2, 1), E, ff), E, ff);
968 15688 : return vconcat(X1, X2);
969 : }
970 :
971 : /* solve L*X = A, L lower triangular with ones on the diagonal
972 : * (at least as many rows as columns) */
973 : static GEN
974 36476 : gen_rsolve_lower_unit(GEN L, GEN A, void *E, const struct bb_field *ff,
975 : GEN (*mul)(void *E, GEN a, GEN))
976 : {
977 36476 : long m = lg(L) - 1, m1, n;
978 : GEN L1, L11, L21, L22, A1, A2, X1, X2, X;
979 36476 : pari_sp av = avma;
980 :
981 36476 : if (m == 0) return zeromat(0, lg(A) - 1);
982 36476 : if (m == 1) return rowslice(A, 1, 1);
983 28662 : if (m == 2) return gen_rsolve_lower_unit_2(L, A, E, ff);
984 12974 : m1 = (m + 1)/2;
985 12974 : n = nbrows(L);
986 12974 : L1 = vecslice(L, 1, m1);
987 12974 : L11 = rowslice(L1, 1, m1);
988 12974 : L21 = rowslice(L1, m1 + 1, n);
989 12974 : A1 = rowslice(A, 1, m1);
990 12974 : X1 = gen_rsolve_lower_unit(L11, A1, E, ff, mul);
991 12974 : A2 = rowslice(A, m1 + 1, n);
992 12974 : A2 = gen_matsub(A2, mul(E, L21, X1), E, ff);
993 12974 : if (gc_needed(av, 1)) gerepileall(av, 2, &A2, &X1);
994 12974 : L22 = matslice(L, m1+1,n, m1+1,m);
995 12974 : X2 = gen_rsolve_lower_unit(L22, A2, E, ff, mul);
996 12974 : X = vconcat(X1, X2);
997 12974 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
998 12974 : return X;
999 : }
1000 :
1001 : static GEN
1002 7465 : gen_lsolve_lower_unit_2(GEN L, GEN A, void *E, const struct bb_field *ff)
1003 : {
1004 7465 : GEN X2 = vecslice(A, 2, 2);
1005 7465 : GEN X1 = gen_matsub(vecslice(A, 1, 1),
1006 7465 : gen_matscalmul(X2, gcoeff(L, 2, 1), E, ff), E, ff);
1007 7465 : return shallowconcat(X1, X2);
1008 : }
1009 :
1010 : /* solve L*X = A, L lower triangular with ones on the diagonal
1011 : * (at least as many rows as columns) */
1012 : static GEN
1013 19265 : gen_lsolve_lower_unit(GEN L, GEN A, void *E, const struct bb_field *ff,
1014 : GEN (*mul)(void *E, GEN a, GEN))
1015 : {
1016 19265 : long m = lg(L) - 1, m1;
1017 : GEN L1, L2, L11, L21, L22, A1, A2, X1, X2, X;
1018 19265 : pari_sp av = avma;
1019 :
1020 19265 : if (m <= 1) return A;
1021 15129 : if (m == 2) return gen_lsolve_lower_unit_2(L, A, E, ff);
1022 7664 : m1 = (m + 1)/2;
1023 7664 : L2 = vecslice(L, m1 + 1, m);
1024 7664 : L22 = rowslice(L2, m1 + 1, m);
1025 7664 : A2 = vecslice(A, m1 + 1, m);
1026 7664 : X2 = gen_lsolve_lower_unit(L22, A2, E, ff, mul);
1027 7664 : if (gc_needed(av, 1)) X2 = gerepilecopy(av, X2);
1028 7664 : L1 = vecslice(L, 1, m1);
1029 7664 : L21 = rowslice(L1, m1 + 1, m);
1030 7664 : A1 = vecslice(A, 1, m1);
1031 7664 : A1 = gen_matsub(A1, mul(E, X2, L21), E, ff);
1032 7664 : L11 = rowslice(L1, 1, m1);
1033 7664 : if (gc_needed(av, 1)) gerepileall(av, 3, &A1, &L11, &X2);
1034 7664 : X1 = gen_lsolve_lower_unit(L11, A1, E, ff, mul);
1035 7664 : X = shallowconcat(X1, X2);
1036 7664 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
1037 7664 : return X;
1038 : }
1039 :
1040 : /* destroy A */
1041 : static long
1042 21339 : gen_CUP_basecase(GEN A, GEN *R, GEN *C, GEN *U, GEN *P, void *E, const struct bb_field *ff)
1043 : {
1044 21339 : long i, j, k, m = nbrows(A), n = lg(A) - 1, pr, pc;
1045 : pari_sp av;
1046 : GEN u, v;
1047 :
1048 21339 : if (P) *P = identity_perm(n);
1049 21339 : *R = cgetg(m + 1, t_VECSMALL);
1050 21339 : av = avma;
1051 55654 : for (j = 1, pr = 0; j <= n; j++)
1052 : {
1053 126079 : for (pr++, pc = 0; pr <= m; pr++)
1054 : {
1055 604655 : for (k = j; k <= n; k++)
1056 : {
1057 495244 : v = ff->red(E, gcoeff(A, pr, k));
1058 495244 : gcoeff(A, pr, k) = v;
1059 495244 : if (!pc && !ff->equal0(v)) pc = k;
1060 : }
1061 109411 : if (pc) break;
1062 : }
1063 50983 : if (!pc) break;
1064 34315 : (*R)[j] = pr;
1065 34315 : if (pc != j)
1066 : {
1067 4559 : swap(gel(A, j), gel(A, pc));
1068 4559 : if (P) lswap((*P)[j], (*P)[pc]);
1069 : }
1070 34315 : u = ff->inv(E, gcoeff(A, pr, j));
1071 170309 : for (i = pr + 1; i <= m; i++)
1072 : {
1073 135994 : v = ff->red(E, ff->mul(E, gcoeff(A, i, j), u));
1074 135994 : gcoeff(A, i, j) = v;
1075 135994 : v = ff->neg(E, v);
1076 434171 : for (k = j + 1; k <= n; k++)
1077 298177 : gcoeff(A, i, k) = ff->add(E, gcoeff(A, i, k),
1078 298177 : ff->red(E, ff->mul(E, gcoeff(A, pr, k), v)));
1079 : }
1080 34315 : if (gc_needed(av, 2)) A = gerepilecopy(av, A);
1081 : }
1082 21339 : setlg(*R, j);
1083 21339 : *C = vecslice(A, 1, j - 1);
1084 21339 : if (U) *U = rowpermute(A, *R);
1085 21339 : return j - 1;
1086 : }
1087 :
1088 : static const long gen_CUP_LIMIT = 5;
1089 :
1090 : static long
1091 10979 : gen_CUP(GEN A, GEN *R, GEN *C, GEN *U, GEN *P, void *E, const struct bb_field *ff,
1092 : GEN (*mul)(void *E, GEN a, GEN))
1093 : {
1094 10979 : long m = nbrows(A), m1, n = lg(A) - 1, i, r1, r2, r;
1095 : GEN R1, C1, U1, P1, R2, C2, U2, P2;
1096 : GEN A1, A2, B2, C21, U11, U12, T21, T22;
1097 10979 : pari_sp av = avma;
1098 :
1099 10979 : if (m < gen_CUP_LIMIT || n < gen_CUP_LIMIT)
1100 : /* destroy A; not called at the outermost recursion level */
1101 6284 : return gen_CUP_basecase(A, R, C, U, P, E, ff);
1102 4695 : m1 = (minss(m, n) + 1)/2;
1103 4695 : A1 = rowslice(A, 1, m1);
1104 4695 : A2 = rowslice(A, m1 + 1, m);
1105 4695 : r1 = gen_CUP(A1, &R1, &C1, &U1, &P1, E, ff, mul);
1106 4695 : if (r1 == 0)
1107 : {
1108 556 : r2 = gen_CUP(A2, &R2, &C2, &U2, &P2, E, ff, mul);
1109 556 : *R = cgetg(r2 + 1, t_VECSMALL);
1110 859 : for (i = 1; i <= r2; i++) (*R)[i] = R2[i] + m1;
1111 556 : *C = vconcat(gen_zeromat(m1, r2, E, ff), C2);
1112 556 : *U = U2;
1113 556 : *P = P2;
1114 556 : r = r2;
1115 : }
1116 : else
1117 : {
1118 4139 : U11 = vecslice(U1, 1, r1);
1119 4139 : U12 = vecslice(U1, r1 + 1, n);
1120 4139 : T21 = vecslicepermute(A2, P1, 1, r1);
1121 4139 : T22 = vecslicepermute(A2, P1, r1 + 1, n);
1122 4139 : C21 = gen_lsolve_upper(U11, T21, E, ff, mul);
1123 4139 : if (gc_needed(av, 1))
1124 0 : gerepileall(av, 7, &R1, &C1, &P1, &U11, &U12, &T22, &C21);
1125 4139 : B2 = gen_matsub(T22, mul(E, C21, U12), E, ff);
1126 4139 : r2 = gen_CUP(B2, &R2, &C2, &U2, &P2, E, ff, mul);
1127 4139 : r = r1 + r2;
1128 4139 : *R = cgetg(r + 1, t_VECSMALL);
1129 18917 : for (i = 1; i <= r1; i++) (*R)[i] = R1[i];
1130 19981 : for ( ; i <= r; i++) (*R)[i] = R2[i - r1] + m1;
1131 4139 : *C = shallowconcat(vconcat(C1, C21),
1132 : vconcat(gen_zeromat(m1, r2, E, ff), C2));
1133 4139 : *U = shallowconcat(vconcat(U11, gen_zeromat(r2, r1, E, ff)),
1134 : vconcat(vecpermute(U12, P2), U2));
1135 :
1136 4139 : *P = cgetg(n + 1, t_VECSMALL);
1137 18917 : for (i = 1; i <= r1; i++) (*P)[i] = P1[i];
1138 46740 : for ( ; i <= n; i++) (*P)[i] = P1[P2[i - r1] + r1];
1139 : }
1140 4695 : if (gc_needed(av, 1)) gerepileall(av, 4, R, C, U, P);
1141 4695 : return r;
1142 : }
1143 :
1144 : /* column echelon form */
1145 : static long
1146 26328 : gen_echelon(GEN A, GEN *R, GEN *C, void *E, const struct bb_field *ff,
1147 : GEN (*mul)(void*, GEN, GEN))
1148 : {
1149 26328 : long j, j1, j2, m = nbrows(A), n = lg(A) - 1, n1, r, r1, r2;
1150 : GEN A1, A2, R1, R1c, C1, R2, C2;
1151 : GEN A12, A22, B2, C11, C21, M12;
1152 26328 : pari_sp av = avma;
1153 :
1154 26328 : if (m < gen_CUP_LIMIT || n < gen_CUP_LIMIT)
1155 15055 : return gen_CUP_basecase(shallowcopy(A), R, C, NULL, NULL, E, ff);
1156 :
1157 11273 : n1 = (n + 1)/2;
1158 11273 : A1 = vecslice(A, 1, n1);
1159 11273 : A2 = vecslice(A, n1 + 1, n);
1160 11273 : r1 = gen_echelon(A1, &R1, &C1, E, ff, mul);
1161 11273 : if (!r1) return gen_echelon(A2, R, C, E, ff, mul);
1162 10208 : if (r1 == m) { *R = R1; *C = C1; return r1; }
1163 9997 : R1c = indexcompl(R1, m);
1164 9997 : C11 = rowpermute(C1, R1);
1165 9997 : C21 = rowpermute(C1, R1c);
1166 9997 : A12 = rowpermute(A2, R1);
1167 9997 : A22 = rowpermute(A2, R1c);
1168 9997 : M12 = gen_rsolve_lower_unit(C11, A12, E, ff, mul);
1169 9997 : B2 = gen_matsub(A22, mul(E, C21, M12), E, ff);
1170 9997 : r2 = gen_echelon(B2, &R2, &C2, E, ff, mul);
1171 9997 : if (!r2) { *R = R1; *C = C1; r = r1; }
1172 : else
1173 : {
1174 5271 : R2 = perm_mul(R1c, R2);
1175 5271 : C2 = rowpermute(vconcat(gen_zeromat(r1, r2, E, ff), C2),
1176 : perm_inv(vecsmall_concat(R1, R1c)));
1177 5271 : r = r1 + r2;
1178 5271 : *R = cgetg(r + 1, t_VECSMALL);
1179 5271 : *C = cgetg(r + 1, t_MAT);
1180 36921 : for (j = j1 = j2 = 1; j <= r; j++)
1181 31650 : if (j2 > r2 || (j1 <= r1 && R1[j1] < R2[j2]))
1182 : {
1183 18493 : gel(*C, j) = gel(C1, j1);
1184 18493 : (*R)[j] = R1[j1++];
1185 : }
1186 : else
1187 : {
1188 13157 : gel(*C, j) = gel(C2, j2);
1189 13157 : (*R)[j] = R2[j2++];
1190 : }
1191 : }
1192 9997 : if (gc_needed(av, 1)) gerepileall(av, 2, R, C);
1193 9997 : return r;
1194 : }
1195 :
1196 : static GEN
1197 862 : gen_pivots_CUP(GEN x, long *rr, void *E, const struct bb_field *ff,
1198 : GEN (*mul)(void*, GEN, GEN))
1199 : {
1200 : pari_sp av;
1201 862 : long i, n = lg(x) - 1, r;
1202 862 : GEN R, C, U, P, d = zero_zv(n);
1203 862 : av = avma;
1204 862 : r = gen_CUP(x, &R, &C, &U, &P, E, ff, mul);
1205 6543 : for(i = 1; i <= r; i++)
1206 5681 : d[P[i]] = R[i];
1207 862 : set_avma(av);
1208 862 : *rr = n - r;
1209 862 : return d;
1210 : }
1211 :
1212 : static GEN
1213 140 : gen_det_CUP(GEN a, void *E, const struct bb_field *ff,
1214 : GEN (*mul)(void*, GEN, GEN))
1215 : {
1216 140 : pari_sp av = avma;
1217 : GEN R, C, U, P, d;
1218 140 : long i, n = lg(a) - 1, r;
1219 140 : r = gen_CUP(a, &R, &C, &U, &P, E, ff, mul);
1220 140 : if (r < n)
1221 0 : d = ff->s(E, 0);
1222 : else {
1223 140 : d = ff->s(E, perm_sign(P) == 1 ? 1: - 1);
1224 2730 : for (i = 1; i <= n; i++)
1225 2590 : d = ff->red(E, ff->mul(E, d, gcoeff(U, i, i)));
1226 : }
1227 140 : return gerepileupto(av, d);
1228 : }
1229 :
1230 : static long
1231 35 : gen_matrank(GEN x, void *E, const struct bb_field *ff,
1232 : GEN (*mul)(void*, GEN, GEN))
1233 : {
1234 35 : pari_sp av = avma;
1235 : long r;
1236 35 : if (lg(x) - 1 >= gen_CUP_LIMIT && nbrows(x) >= gen_CUP_LIMIT)
1237 : {
1238 : GEN R, C;
1239 28 : return gc_long(av, gen_echelon(x, &R, &C, E, ff, mul));
1240 : }
1241 7 : (void) gen_Gauss_pivot(x, &r, E, ff);
1242 7 : return gc_long(av, lg(x)-1 - r);
1243 : }
1244 :
1245 : static GEN
1246 63 : gen_invimage_CUP(GEN A, GEN B, void *E, const struct bb_field *ff,
1247 : GEN (*mul)(void*, GEN, GEN))
1248 : {
1249 63 : pari_sp av = avma;
1250 : GEN R, Rc, C, U, P, B1, B2, C1, C2, X, Y, Z;
1251 63 : long r = gen_CUP(A, &R, &C, &U, &P, E, ff, mul);
1252 63 : Rc = indexcompl(R, nbrows(B));
1253 63 : C1 = rowpermute(C, R);
1254 63 : C2 = rowpermute(C, Rc);
1255 63 : B1 = rowpermute(B, R);
1256 63 : B2 = rowpermute(B, Rc);
1257 63 : Z = gen_rsolve_lower_unit(C1, B1, E, ff, mul);
1258 63 : if (!gequal(mul(E, C2, Z), B2))
1259 42 : return NULL;
1260 21 : Y = vconcat(gen_rsolve_upper(vecslice(U, 1, r), Z, E, ff, mul),
1261 21 : gen_zeromat(lg(A) - 1 - r, lg(B) - 1, E, ff));
1262 21 : X = rowpermute(Y, perm_inv(P));
1263 21 : return gerepilecopy(av, X);
1264 : }
1265 :
1266 : static GEN
1267 3881 : gen_ker_echelon(GEN x, void *E, const struct bb_field *ff,
1268 : GEN (*mul)(void*, GEN, GEN))
1269 : {
1270 3881 : pari_sp av = avma;
1271 : GEN R, Rc, C, C1, C2, S, K;
1272 3881 : long n = lg(x) - 1, r;
1273 3881 : r = gen_echelon(shallowtrans(x), &R, &C, E, ff, mul);
1274 3881 : Rc = indexcompl(R, n);
1275 3881 : C1 = rowpermute(C, R);
1276 3881 : C2 = rowpermute(C, Rc);
1277 3881 : S = gen_lsolve_lower_unit(C1, C2, E, ff, mul);
1278 3881 : K = vecpermute(shallowconcat(gen_matneg(S, E, ff), gen_matid(n - r, E, ff)),
1279 : perm_inv(vecsmall_concat(R, Rc)));
1280 3881 : K = shallowtrans(K);
1281 3881 : return gerepilecopy(av, K);
1282 : }
1283 :
1284 : static GEN
1285 84 : gen_deplin_echelon(GEN x, void *E, const struct bb_field *ff,
1286 : GEN (*mul)(void*, GEN, GEN))
1287 : {
1288 84 : pari_sp av = avma;
1289 : GEN R, Rc, C, C1, C2, s, v;
1290 84 : long i, n = lg(x) - 1, r;
1291 84 : r = gen_echelon(shallowtrans(x), &R, &C, E, ff, mul);
1292 84 : if (r == n) return gc_NULL(av);
1293 56 : Rc = indexcompl(R, n);
1294 56 : i = Rc[1];
1295 56 : C1 = rowpermute(C, R);
1296 56 : C2 = rowslice(C, i, i);
1297 56 : s = row(gen_lsolve_lower_unit(C1, C2, E, ff, mul), 1);
1298 56 : settyp(s, t_COL);
1299 56 : v = vecpermute(shallowconcat(gen_colneg(s, E, ff), gen_colei(n - r, 1, E, ff)),
1300 : perm_inv(vecsmall_concat(R, Rc)));
1301 56 : return gerepilecopy(av, v);
1302 : }
1303 :
1304 : static GEN
1305 524 : gen_gauss_CUP(GEN a, GEN b, void *E, const struct bb_field *ff,
1306 : GEN (*mul)(void*, GEN, GEN))
1307 : {
1308 : GEN R, C, U, P, Y;
1309 524 : long n = lg(a) - 1, r;
1310 524 : if (nbrows(a) < n || (r = gen_CUP(a, &R, &C, &U, &P, E, ff, mul)) < n)
1311 56 : return NULL;
1312 468 : Y = gen_rsolve_lower_unit(rowpermute(C, R), rowpermute(b, R), E, ff, mul);
1313 468 : return rowpermute(gen_rsolve_upper(U, Y, E, ff, mul), perm_inv(P));
1314 : }
1315 :
1316 : static GEN
1317 3869 : gen_gauss(GEN a, GEN b, void *E, const struct bb_field *ff,
1318 : GEN (*mul)(void*, GEN, GEN))
1319 : {
1320 3869 : if (lg(a) - 1 >= gen_CUP_LIMIT)
1321 524 : return gen_gauss_CUP(a, b, E, ff, mul);
1322 3345 : return gen_Gauss(a, b, E, ff);
1323 : }
1324 :
1325 : static GEN
1326 5459 : gen_ker_i(GEN x, long deplin, void *E, const struct bb_field *ff,
1327 : GEN (*mul)(void*, GEN, GEN)) {
1328 5459 : if (lg(x) - 1 >= gen_CUP_LIMIT && nbrows(x) >= gen_CUP_LIMIT)
1329 3965 : return deplin? gen_deplin_echelon(x, E, ff, mul): gen_ker_echelon(x, E, ff, mul);
1330 1494 : return gen_ker(x, deplin, E, ff);
1331 : }
1332 :
1333 : static GEN
1334 140 : gen_invimage(GEN A, GEN B, void *E, const struct bb_field *ff,
1335 : GEN (*mul)(void*, GEN, GEN))
1336 : {
1337 140 : long nA = lg(A)-1, nB = lg(B)-1;
1338 :
1339 140 : if (!nB) return cgetg(1, t_MAT);
1340 140 : if (nA + nB >= gen_CUP_LIMIT && nbrows(B) >= gen_CUP_LIMIT)
1341 63 : return gen_invimage_CUP(A, B, E, ff, mul);
1342 77 : return gen_matinvimage(A, B, E, ff);
1343 : }
1344 :
1345 : /* find z such that A z = y. Return NULL if no solution */
1346 : static GEN
1347 70 : gen_matcolinvimage_i(GEN A, GEN y, void *E, const struct bb_field *ff,
1348 : GEN (*mul)(void*, GEN, GEN))
1349 : {
1350 70 : pari_sp av = avma;
1351 70 : long i, l = lg(A);
1352 : GEN M, x, t;
1353 :
1354 70 : M = gen_ker_i(shallowconcat(A, y), 0, E, ff, mul);
1355 70 : i = lg(M) - 1;
1356 70 : if (!i) return gc_NULL(av);
1357 :
1358 70 : x = gel(M, i);
1359 70 : t = gel(x, l);
1360 70 : if (ff->equal0(t)) return gc_NULL(av);
1361 :
1362 49 : t = ff->neg(E, ff->inv(E, t));
1363 49 : setlg(x, l);
1364 175 : for (i = 1; i < l; i++)
1365 126 : gel(x, i) = ff->red(E, ff->mul(E, t, gel(x, i)));
1366 49 : return gerepilecopy(av, x);
1367 : }
1368 :
1369 : static GEN
1370 420 : gen_det_i(GEN a, void *E, const struct bb_field *ff,
1371 : GEN (*mul)(void*, GEN, GEN))
1372 : {
1373 420 : if (lg(a) - 1 >= gen_CUP_LIMIT)
1374 140 : return gen_det_CUP(a, E, ff, mul);
1375 : else
1376 280 : return gen_det(a, E, ff);
1377 : }
1378 :
1379 : static GEN
1380 2411 : gen_pivots(GEN x, long *rr, void *E, const struct bb_field *ff,
1381 : GEN (*mul)(void*, GEN, GEN))
1382 : {
1383 2411 : if (lg(x) - 1 >= gen_CUP_LIMIT && nbrows(x) >= gen_CUP_LIMIT)
1384 862 : return gen_pivots_CUP(x, rr, E, ff, mul);
1385 1549 : return gen_Gauss_pivot(x, rr, E, ff);
1386 : }
1387 :
1388 : /* r = dim Ker x, n = nbrows(x) */
1389 : static GEN
1390 21 : gen_get_suppl(GEN x, GEN d, long n, long r, void *E, const struct bb_field *ff)
1391 : {
1392 : GEN y, c;
1393 21 : long j, k, rx = lg(x)-1; /* != 0 due to init_suppl() */
1394 :
1395 21 : if (rx == n && r == 0) return gcopy(x);
1396 21 : c = zero_zv(n);
1397 21 : y = cgetg(n+1, t_MAT);
1398 : /* c = lines containing pivots (could get it from gauss_pivot, but cheap)
1399 : * In theory r = 0 and d[j] > 0 for all j, but why take chances? */
1400 119 : for (k = j = 1; j<=rx; j++)
1401 98 : if (d[j]) { c[ d[j] ] = 1; gel(y,k++) = gcopy(gel(x,j)); }
1402 203 : for (j=1; j<=n; j++)
1403 182 : if (!c[j]) gel(y,k++) = gen_colei(n, j, E, ff);
1404 21 : return y;
1405 : }
1406 :
1407 : static GEN
1408 21 : gen_suppl(GEN x, void *E, const struct bb_field *ff,
1409 : GEN (*mul)(void*, GEN, GEN))
1410 : {
1411 : GEN d;
1412 21 : long n = nbrows(x), r;
1413 :
1414 21 : if (lg(x) == 1) pari_err_IMPL("suppl [empty matrix]");
1415 21 : d = gen_pivots(x, &r, E, ff, mul);
1416 21 : return gen_get_suppl(x, d, n, r, E, ff);
1417 : }
1418 :
1419 : /*******************************************************************/
1420 : /* */
1421 : /* MATRIX MULTIPLICATION MODULO P */
1422 : /* */
1423 : /*******************************************************************/
1424 :
1425 : GEN
1426 21 : F2xqM_F2xqC_mul(GEN A, GEN B, GEN T) {
1427 : void *E;
1428 21 : const struct bb_field *ff = get_F2xq_field(&E, T);
1429 21 : return gen_matcolmul(A, B, E, ff);
1430 : }
1431 :
1432 : GEN
1433 35 : FlxqM_FlxqC_mul(GEN A, GEN B, GEN T, ulong p) {
1434 : void *E;
1435 35 : const struct bb_field *ff = get_Flxq_field(&E, T, p);
1436 35 : return gen_matcolmul(A, B, E, ff);
1437 : }
1438 :
1439 : GEN
1440 49 : FqM_FqC_mul(GEN A, GEN B, GEN T, GEN p) {
1441 : void *E;
1442 49 : const struct bb_field *ff = get_Fq_field(&E, T, p);
1443 49 : return gen_matcolmul(A, B, E, ff);
1444 : }
1445 :
1446 : GEN
1447 1407 : F2xqM_mul(GEN A, GEN B, GEN T) {
1448 : void *E;
1449 1407 : const struct bb_field *ff = get_F2xq_field(&E, T);
1450 1407 : return gen_matmul(A, B, E, ff);
1451 : }
1452 :
1453 : GEN
1454 146106 : FlxqM_mul(GEN A, GEN B, GEN T, ulong p) {
1455 : void *E;
1456 : const struct bb_field *ff;
1457 146106 : long n = lg(A) - 1;
1458 :
1459 146106 : if (n == 0)
1460 0 : return cgetg(1, t_MAT);
1461 146106 : if (n > 1)
1462 83385 : return FlxqM_mul_Kronecker(A, B, T, p);
1463 62721 : ff = get_Flxq_field(&E, T, p);
1464 62721 : return gen_matmul(A, B, E, ff);
1465 : }
1466 :
1467 : GEN
1468 66423 : FqM_mul(GEN A, GEN B, GEN T, GEN p) {
1469 : void *E;
1470 66423 : long n = lg(A) - 1;
1471 : const struct bb_field *ff;
1472 66423 : if (n == 0)
1473 0 : return cgetg(1, t_MAT);
1474 66423 : if (n > 1)
1475 62804 : return FqM_mul_Kronecker(A, B, T, p);
1476 3619 : ff = get_Fq_field(&E, T, p);
1477 3619 : return gen_matmul(A, B, E, ff);
1478 : }
1479 :
1480 : /*******************************************************************/
1481 : /* */
1482 : /* LINEAR ALGEBRA MODULO P */
1483 : /* */
1484 : /*******************************************************************/
1485 :
1486 : static GEN
1487 0 : _F2xqM_mul(void *E, GEN A, GEN B)
1488 0 : { return F2xqM_mul(A, B, (GEN) E); }
1489 :
1490 : struct _Flxq {
1491 : GEN aut;
1492 : GEN T;
1493 : ulong p;
1494 : };
1495 :
1496 : static GEN
1497 16422 : _FlxqM_mul(void *E, GEN A, GEN B)
1498 : {
1499 16422 : struct _Flxq *D = (struct _Flxq*)E;
1500 16422 : return FlxqM_mul(A, B, D->T, D->p);
1501 : }
1502 :
1503 : static GEN
1504 19213 : _FpM_mul(void *E, GEN A, GEN B)
1505 19213 : { return FpM_mul(A, B, (GEN) E); }
1506 :
1507 : struct _Fq_field
1508 : {
1509 : GEN T, p;
1510 : };
1511 :
1512 : static GEN
1513 6349 : _FqM_mul(void *E, GEN A, GEN B)
1514 : {
1515 6349 : struct _Fq_field *D = (struct _Fq_field*) E;
1516 6349 : return FqM_mul(A, B, D->T, D->p);
1517 : }
1518 :
1519 : static GEN
1520 1301758 : FpM_init(GEN a, GEN p, ulong *pp)
1521 : {
1522 1301758 : if (lgefint(p) == 3)
1523 : {
1524 1297019 : *pp = uel(p,2);
1525 1297019 : return (*pp==2)? ZM_to_F2m(a): ZM_to_Flm(a, *pp);
1526 : }
1527 4739 : *pp = 0; return a;
1528 : }
1529 : static GEN
1530 1288778 : FpM_init3(GEN a, GEN p, ulong *pp)
1531 : {
1532 1288778 : if (lgefint(p) == 3)
1533 : {
1534 1286547 : *pp = uel(p,2);
1535 1286547 : switch(*pp)
1536 : {
1537 711854 : case 2: return ZM_to_F2m(a);
1538 161354 : case 3: return ZM_to_F3m(a);
1539 413339 : default:return ZM_to_Flm(a, *pp);
1540 : }
1541 : }
1542 2231 : *pp = 0; return a;
1543 : }
1544 : GEN
1545 2177 : RgM_Fp_init(GEN a, GEN p, ulong *pp)
1546 : {
1547 2177 : if (lgefint(p) == 3)
1548 : {
1549 1897 : *pp = uel(p,2);
1550 1897 : return (*pp==2)? RgM_to_F2m(a): RgM_to_Flm(a, *pp);
1551 : }
1552 280 : *pp = 0; return RgM_to_FpM(a,p);
1553 : }
1554 : static GEN
1555 189 : RgM_Fp_init3(GEN a, GEN p, ulong *pp)
1556 : {
1557 189 : if (lgefint(p) == 3)
1558 : {
1559 119 : *pp = uel(p,2);
1560 119 : switch(*pp)
1561 : {
1562 21 : case 2: return RgM_to_F2m(a);
1563 7 : case 3: return RgM_to_F3m(a);
1564 91 : default:return RgM_to_Flm(a, *pp);
1565 : }
1566 : }
1567 70 : *pp = 0; return RgM_to_FpM(a,p);
1568 : }
1569 :
1570 : static GEN
1571 315 : FpM_det_gen(GEN a, GEN p)
1572 : {
1573 : void *E;
1574 315 : const struct bb_field *S = get_Fp_field(&E,p);
1575 315 : return gen_det_i(a, E, S, _FpM_mul);
1576 : }
1577 : GEN
1578 3948 : FpM_det(GEN a, GEN p)
1579 : {
1580 3948 : pari_sp av = avma;
1581 : ulong pp, d;
1582 3948 : a = FpM_init(a, p, &pp);
1583 3948 : switch(pp)
1584 : {
1585 315 : case 0: return FpM_det_gen(a, p);
1586 1617 : case 2: d = F2m_det_sp(a); break;
1587 2016 : default:d = Flm_det_sp(a,pp); break;
1588 : }
1589 3633 : set_avma(av); return utoi(d);
1590 : }
1591 :
1592 : GEN
1593 7 : F2xqM_det(GEN a, GEN T)
1594 : {
1595 : void *E;
1596 7 : const struct bb_field *S = get_F2xq_field(&E, T);
1597 7 : return gen_det_i(a, E, S, _F2xqM_mul);
1598 : }
1599 :
1600 : GEN
1601 28 : FlxqM_det(GEN a, GEN T, ulong p) {
1602 : void *E;
1603 28 : const struct bb_field *S = get_Flxq_field(&E, T, p);
1604 28 : return gen_det_i(a, E, S, _FlxqM_mul);
1605 : }
1606 :
1607 : GEN
1608 70 : FqM_det(GEN x, GEN T, GEN p)
1609 : {
1610 : void *E;
1611 70 : const struct bb_field *S = get_Fq_field(&E,T,p);
1612 70 : return gen_det_i(x, E, S, _FqM_mul);
1613 : }
1614 :
1615 : static GEN
1616 857 : FpM_gauss_pivot_gen(GEN x, GEN p, long *rr)
1617 : {
1618 : void *E;
1619 857 : const struct bb_field *S = get_Fp_field(&E,p);
1620 857 : return gen_pivots(x, rr, E, S, _FpM_mul);
1621 : }
1622 :
1623 : static GEN
1624 642269 : FpM_gauss_pivot(GEN x, GEN p, long *rr)
1625 : {
1626 : ulong pp;
1627 642269 : if (lg(x)==1) { *rr = 0; return NULL; }
1628 640148 : x = FpM_init(x, p, &pp);
1629 640158 : switch(pp)
1630 : {
1631 857 : case 0: return FpM_gauss_pivot_gen(x, p, rr);
1632 354695 : case 2: return F2m_gauss_pivot(x, rr);
1633 284606 : default:return Flm_pivots(x, pp, rr, 1);
1634 : }
1635 : }
1636 :
1637 : static GEN
1638 21 : F2xqM_gauss_pivot(GEN x, GEN T, long *rr)
1639 : {
1640 : void *E;
1641 21 : const struct bb_field *S = get_F2xq_field(&E,T);
1642 21 : return gen_pivots(x, rr, E, S, _F2xqM_mul);
1643 : }
1644 :
1645 : static GEN
1646 1407 : FlxqM_gauss_pivot(GEN x, GEN T, ulong p, long *rr) {
1647 : void *E;
1648 1407 : const struct bb_field *S = get_Flxq_field(&E, T, p);
1649 1407 : return gen_pivots(x, rr, E, S, _FlxqM_mul);
1650 : }
1651 :
1652 : static GEN
1653 105 : FqM_gauss_pivot_gen(GEN x, GEN T, GEN p, long *rr)
1654 : {
1655 : void *E;
1656 105 : const struct bb_field *S = get_Fq_field(&E,T,p);
1657 105 : return gen_pivots(x, rr, E, S, _FqM_mul);
1658 : }
1659 : static GEN
1660 1484 : FqM_gauss_pivot(GEN x, GEN T, GEN p, long *rr)
1661 : {
1662 1484 : if (lg(x)==1) { *rr = 0; return NULL; }
1663 1484 : if (!T) return FpM_gauss_pivot(x, p, rr);
1664 1484 : if (lgefint(p) == 3)
1665 : {
1666 1379 : pari_sp av = avma;
1667 1379 : ulong pp = uel(p,2);
1668 1379 : GEN Tp = ZXT_to_FlxT(T, pp);
1669 1379 : GEN d = FlxqM_gauss_pivot(ZXM_to_FlxM(x, pp, get_Flx_var(Tp)), Tp, pp, rr);
1670 1379 : return d ? gerepileuptoleaf(av, d): d;
1671 : }
1672 105 : return FqM_gauss_pivot_gen(x, T, p, rr);
1673 : }
1674 :
1675 : GEN
1676 341342 : FpM_image(GEN x, GEN p)
1677 : {
1678 : long r;
1679 341342 : GEN d = FpM_gauss_pivot(x,p,&r); /* d left on stack for efficiency */
1680 341339 : return image_from_pivot(x,d,r);
1681 : }
1682 :
1683 : GEN
1684 39390 : Flm_image(GEN x, ulong p)
1685 : {
1686 : long r;
1687 39390 : GEN d = Flm_pivots(x, p, &r, 0); /* d left on stack for efficiency */
1688 39390 : return image_from_pivot(x,d,r);
1689 : }
1690 :
1691 : GEN
1692 7 : F2m_image(GEN x)
1693 : {
1694 : long r;
1695 7 : GEN d = F2m_gauss_pivot(F2m_copy(x),&r); /* d left on stack for efficiency */
1696 7 : return image_from_pivot(x,d,r);
1697 : }
1698 :
1699 : GEN
1700 7 : F2xqM_image(GEN x, GEN T)
1701 : {
1702 : long r;
1703 7 : GEN d = F2xqM_gauss_pivot(x,T,&r); /* d left on stack for efficiency */
1704 7 : return image_from_pivot(x,d,r);
1705 : }
1706 :
1707 : GEN
1708 21 : FlxqM_image(GEN x, GEN T, ulong p)
1709 : {
1710 : long r;
1711 21 : GEN d = FlxqM_gauss_pivot(x, T, p, &r); /* d left on stack for efficiency */
1712 21 : return image_from_pivot(x,d,r);
1713 : }
1714 :
1715 : GEN
1716 49 : FqM_image(GEN x, GEN T, GEN p)
1717 : {
1718 : long r;
1719 49 : GEN d = FqM_gauss_pivot(x,T,p,&r); /* d left on stack for efficiency */
1720 49 : return image_from_pivot(x,d,r);
1721 : }
1722 :
1723 : long
1724 56 : FpM_rank(GEN x, GEN p)
1725 : {
1726 56 : pari_sp av = avma;
1727 : long r;
1728 56 : (void)FpM_gauss_pivot(x,p,&r);
1729 56 : return gc_long(av, lg(x)-1 - r);
1730 : }
1731 :
1732 : long
1733 7 : F2xqM_rank(GEN x, GEN T)
1734 : {
1735 7 : pari_sp av = avma;
1736 : long r;
1737 7 : (void)F2xqM_gauss_pivot(x,T,&r);
1738 7 : return gc_long(av, lg(x)-1 - r);
1739 : }
1740 :
1741 : long
1742 35 : FlxqM_rank(GEN x, GEN T, ulong p)
1743 : {
1744 : void *E;
1745 35 : const struct bb_field *S = get_Flxq_field(&E, T, p);
1746 35 : return gen_matrank(x, E, S, _FlxqM_mul);
1747 : }
1748 :
1749 : long
1750 70 : FqM_rank(GEN x, GEN T, GEN p)
1751 : {
1752 70 : pari_sp av = avma;
1753 : long r;
1754 70 : (void)FqM_gauss_pivot(x,T,p,&r);
1755 70 : return gc_long(av, lg(x)-1 - r);
1756 : }
1757 :
1758 : static GEN
1759 35 : FpM_invimage_gen(GEN A, GEN B, GEN p)
1760 : {
1761 : void *E;
1762 35 : const struct bb_field *ff = get_Fp_field(&E, p);
1763 35 : return gen_invimage(A, B, E, ff, _FpM_mul);
1764 : }
1765 :
1766 : GEN
1767 0 : FpM_invimage(GEN A, GEN B, GEN p)
1768 : {
1769 0 : pari_sp av = avma;
1770 : ulong pp;
1771 : GEN y;
1772 :
1773 0 : A = FpM_init(A, p, &pp);
1774 0 : switch(pp)
1775 : {
1776 0 : case 0: return FpM_invimage_gen(A, B, p);
1777 0 : case 2:
1778 0 : y = F2m_invimage(A, ZM_to_F2m(B));
1779 0 : if (!y) return gc_NULL(av);
1780 0 : y = F2m_to_ZM(y);
1781 0 : return gerepileupto(av, y);
1782 0 : default:
1783 0 : y = Flm_invimage_i(A, ZM_to_Flm(B, pp), pp);
1784 0 : if (!y) return gc_NULL(av);
1785 0 : y = Flm_to_ZM(y);
1786 0 : return gerepileupto(av, y);
1787 : }
1788 : }
1789 :
1790 : GEN
1791 21 : F2xqM_invimage(GEN A, GEN B, GEN T) {
1792 : void *E;
1793 21 : const struct bb_field *ff = get_F2xq_field(&E, T);
1794 21 : return gen_invimage(A, B, E, ff, _F2xqM_mul);
1795 : }
1796 :
1797 : GEN
1798 42 : FlxqM_invimage(GEN A, GEN B, GEN T, ulong p) {
1799 : void *E;
1800 42 : const struct bb_field *ff = get_Flxq_field(&E, T, p);
1801 42 : return gen_invimage(A, B, E, ff, _FlxqM_mul);
1802 : }
1803 :
1804 : GEN
1805 42 : FqM_invimage(GEN A, GEN B, GEN T, GEN p) {
1806 : void *E;
1807 42 : const struct bb_field *ff = get_Fq_field(&E, T, p);
1808 42 : return gen_invimage(A, B, E, ff, _FqM_mul);
1809 : }
1810 :
1811 : static GEN
1812 7 : FpM_FpC_invimage_gen(GEN A, GEN y, GEN p)
1813 : {
1814 : void *E;
1815 7 : const struct bb_field *ff = get_Fp_field(&E, p);
1816 7 : return gen_matcolinvimage_i(A, y, E, ff, _FpM_mul);
1817 : }
1818 :
1819 : GEN
1820 317554 : FpM_FpC_invimage(GEN A, GEN x, GEN p)
1821 : {
1822 317554 : pari_sp av = avma;
1823 : ulong pp;
1824 : GEN y;
1825 :
1826 317554 : A = FpM_init(A, p, &pp);
1827 317563 : switch(pp)
1828 : {
1829 7 : case 0: return FpM_FpC_invimage_gen(A, x, p);
1830 197048 : case 2:
1831 197048 : y = F2m_F2c_invimage(A, ZV_to_F2v(x));
1832 197049 : if (!y) return y;
1833 197049 : y = F2c_to_ZC(y);
1834 197047 : return gerepileupto(av, y);
1835 120508 : default:
1836 120508 : y = Flm_Flc_invimage(A, ZV_to_Flv(x, pp), pp);
1837 120508 : if (!y) return y;
1838 120508 : y = Flc_to_ZC(y);
1839 120508 : return gerepileupto(av, y);
1840 : }
1841 : }
1842 :
1843 : GEN
1844 21 : F2xqM_F2xqC_invimage(GEN A, GEN B, GEN T) {
1845 : void *E;
1846 21 : const struct bb_field *ff = get_F2xq_field(&E, T);
1847 21 : return gen_matcolinvimage_i(A, B, E, ff, _F2xqM_mul);
1848 : }
1849 :
1850 : GEN
1851 21 : FlxqM_FlxqC_invimage(GEN A, GEN B, GEN T, ulong p) {
1852 : void *E;
1853 21 : const struct bb_field *ff = get_Flxq_field(&E, T, p);
1854 21 : return gen_matcolinvimage_i(A, B, E, ff, _FlxqM_mul);
1855 : }
1856 :
1857 : GEN
1858 21 : FqM_FqC_invimage(GEN A, GEN B, GEN T, GEN p) {
1859 : void *E;
1860 21 : const struct bb_field *ff = get_Fq_field(&E, T, p);
1861 21 : return gen_matcolinvimage_i(A, B, E, ff, _FqM_mul);
1862 : }
1863 :
1864 : static GEN
1865 2302 : FpM_ker_gen(GEN x, GEN p, long deplin)
1866 : {
1867 : void *E;
1868 2302 : const struct bb_field *S = get_Fp_field(&E,p);
1869 2302 : return gen_ker_i(x, deplin, E, S, _FpM_mul);
1870 : }
1871 : static GEN
1872 1288776 : FpM_ker_i(GEN x, GEN p, long deplin)
1873 : {
1874 1288776 : pari_sp av = avma;
1875 : ulong pp;
1876 : GEN y;
1877 :
1878 1288776 : if (lg(x)==1) return cgetg(1,t_MAT);
1879 1288776 : x = FpM_init3(x, p, &pp);
1880 1288801 : switch(pp)
1881 : {
1882 2232 : case 0: return FpM_ker_gen(x,p,deplin);
1883 711870 : case 2:
1884 711870 : y = F2m_ker_sp(x, deplin);
1885 711882 : if (!y) return gc_NULL(av);
1886 711891 : y = deplin? F2c_to_ZC(y): F2m_to_ZM(y);
1887 711885 : return gerepileupto(av, y);
1888 161354 : case 3:
1889 161354 : y = F3m_ker_sp(x, deplin);
1890 161354 : if (!y) return gc_NULL(av);
1891 161354 : y = deplin? F3c_to_ZC(y): F3m_to_ZM(y);
1892 161354 : return gerepileupto(av, y);
1893 413345 : default:
1894 413345 : y = Flm_ker_sp(x, pp, deplin);
1895 413345 : if (!y) return gc_NULL(av);
1896 413345 : y = deplin? Flc_to_ZC(y): Flm_to_ZM(y);
1897 413345 : return gerepileupto(av, y);
1898 : }
1899 : }
1900 :
1901 : GEN
1902 797274 : FpM_ker(GEN x, GEN p) { return FpM_ker_i(x,p,0); }
1903 :
1904 : static GEN
1905 35 : F2xqM_ker_i(GEN x, GEN T, long deplin)
1906 : {
1907 : const struct bb_field *ff;
1908 : void *E;
1909 :
1910 35 : if (lg(x)==1) return cgetg(1,t_MAT);
1911 35 : ff = get_F2xq_field(&E,T);
1912 35 : return gen_ker_i(x,deplin, E, ff, _F2xqM_mul);
1913 : }
1914 :
1915 : GEN
1916 21 : F2xqM_ker(GEN x, GEN T)
1917 : {
1918 21 : return F2xqM_ker_i(x, T, 0);
1919 : }
1920 :
1921 : static GEN
1922 2926 : FlxqM_ker_i(GEN x, GEN T, ulong p, long deplin) {
1923 : void *E;
1924 2926 : const struct bb_field *S = get_Flxq_field(&E, T, p);
1925 2926 : return gen_ker_i(x, deplin, E, S, _FlxqM_mul);
1926 : }
1927 :
1928 : GEN
1929 2891 : FlxqM_ker(GEN x, GEN T, ulong p)
1930 : {
1931 2891 : return FlxqM_ker_i(x, T, p, 0);
1932 : }
1933 :
1934 : static GEN
1935 126 : FqM_ker_gen(GEN x, GEN T, GEN p, long deplin)
1936 : {
1937 : void *E;
1938 126 : const struct bb_field *S = get_Fq_field(&E,T,p);
1939 126 : return gen_ker_i(x,deplin,E,S,_FqM_mul);
1940 : }
1941 : static GEN
1942 9037 : FqM_ker_i(GEN x, GEN T, GEN p, long deplin)
1943 : {
1944 9037 : if (!T) return FpM_ker_i(x,p,deplin);
1945 2989 : if (lg(x)==1) return cgetg(1,t_MAT);
1946 :
1947 2989 : if (lgefint(p)==3)
1948 : {
1949 2863 : pari_sp ltop=avma;
1950 2863 : ulong l= p[2];
1951 2863 : GEN Tl = ZXT_to_FlxT(T,l);
1952 2863 : GEN Ml = ZXM_to_FlxM(x, l, get_Flx_var(Tl));
1953 2863 : GEN p1 = FlxM_to_ZXM(FlxqM_ker(Ml,Tl,l));
1954 2863 : return gerepileupto(ltop,p1);
1955 : }
1956 126 : return FqM_ker_gen(x, T, p, deplin);
1957 : }
1958 :
1959 : GEN
1960 8960 : FqM_ker(GEN x, GEN T, GEN p) { return FqM_ker_i(x,T,p,0); }
1961 :
1962 : GEN
1963 485473 : FpM_deplin(GEN x, GEN p) { return FpM_ker_i(x,p,1); }
1964 :
1965 : GEN
1966 14 : F2xqM_deplin(GEN x, GEN T)
1967 : {
1968 14 : return F2xqM_ker_i(x, T, 1);
1969 : }
1970 :
1971 : GEN
1972 35 : FlxqM_deplin(GEN x, GEN T, ulong p)
1973 : {
1974 35 : return FlxqM_ker_i(x, T, p, 1);
1975 : }
1976 :
1977 : GEN
1978 77 : FqM_deplin(GEN x, GEN T, GEN p) { return FqM_ker_i(x,T,p,1); }
1979 :
1980 : static GEN
1981 3561 : FpM_gauss_gen(GEN a, GEN b, GEN p)
1982 : {
1983 : void *E;
1984 3561 : const struct bb_field *S = get_Fp_field(&E,p);
1985 3561 : return gen_gauss(a,b, E, S, _FpM_mul);
1986 : }
1987 : /* a an FpM, lg(a)>1; b an FpM or NULL (replace by identity) */
1988 : static GEN
1989 340156 : FpM_gauss_i(GEN a, GEN b, GEN p, ulong *pp)
1990 : {
1991 340156 : long n = nbrows(a);
1992 340157 : a = FpM_init(a,p,pp);
1993 340158 : switch(*pp)
1994 : {
1995 3561 : case 0:
1996 3561 : if (!b) b = matid(n);
1997 3561 : return FpM_gauss_gen(a,b,p);
1998 227307 : case 2:
1999 227307 : if (b) b = ZM_to_F2m(b); else b = matid_F2m(n);
2000 227310 : return F2m_gauss_sp(a,b);
2001 109290 : default:
2002 109290 : if (b) b = ZM_to_Flm(b, *pp); else b = matid_Flm(n);
2003 109290 : return Flm_gauss_sp(a,b, NULL, *pp);
2004 : }
2005 : }
2006 : GEN
2007 35 : FpM_gauss(GEN a, GEN b, GEN p)
2008 : {
2009 35 : pari_sp av = avma;
2010 : ulong pp;
2011 : GEN u;
2012 35 : if (lg(a) == 1 || lg(b)==1) return cgetg(1, t_MAT);
2013 35 : u = FpM_gauss_i(a, b, p, &pp);
2014 35 : if (!u) return gc_NULL(av);
2015 28 : switch(pp)
2016 : {
2017 28 : case 0: return gerepilecopy(av, u);
2018 0 : case 2: u = F2m_to_ZM(u); break;
2019 0 : default: u = Flm_to_ZM(u); break;
2020 : }
2021 0 : return gerepileupto(av, u);
2022 : }
2023 :
2024 : static GEN
2025 84 : F2xqM_gauss_gen(GEN a, GEN b, GEN T)
2026 : {
2027 : void *E;
2028 84 : const struct bb_field *S = get_F2xq_field(&E, T);
2029 84 : return gen_gauss(a, b, E, S, _F2xqM_mul);
2030 : }
2031 :
2032 : GEN
2033 21 : F2xqM_gauss(GEN a, GEN b, GEN T)
2034 : {
2035 21 : pari_sp av = avma;
2036 21 : long n = lg(a)-1;
2037 : GEN u;
2038 21 : if (!n || lg(b)==1) { set_avma(av); return cgetg(1, t_MAT); }
2039 21 : u = F2xqM_gauss_gen(a, b, T);
2040 21 : if (!u) return gc_NULL(av);
2041 14 : return gerepilecopy(av, u);
2042 : }
2043 :
2044 : static GEN
2045 91 : FlxqM_gauss_i(GEN a, GEN b, GEN T, ulong p) {
2046 : void *E;
2047 91 : const struct bb_field *S = get_Flxq_field(&E, T, p);
2048 91 : return gen_gauss(a, b, E, S, _FlxqM_mul);
2049 : }
2050 :
2051 : GEN
2052 21 : FlxqM_gauss(GEN a, GEN b, GEN T, ulong p)
2053 : {
2054 21 : pari_sp av = avma;
2055 21 : long n = lg(a)-1;
2056 : GEN u;
2057 21 : if (!n || lg(b)==1) { set_avma(av); return cgetg(1, t_MAT); }
2058 21 : u = FlxqM_gauss_i(a, b, T, p);
2059 21 : if (!u) return gc_NULL(av);
2060 14 : return gerepilecopy(av, u);
2061 : }
2062 :
2063 : static GEN
2064 133 : FqM_gauss_gen(GEN a, GEN b, GEN T, GEN p)
2065 : {
2066 : void *E;
2067 133 : const struct bb_field *S = get_Fq_field(&E,T,p);
2068 133 : return gen_gauss(a,b,E,S,_FqM_mul);
2069 : }
2070 : GEN
2071 21 : FqM_gauss(GEN a, GEN b, GEN T, GEN p)
2072 : {
2073 21 : pari_sp av = avma;
2074 : GEN u;
2075 : long n;
2076 21 : if (!T) return FpM_gauss(a,b,p);
2077 21 : n = lg(a)-1; if (!n || lg(b)==1) return cgetg(1, t_MAT);
2078 21 : u = FqM_gauss_gen(a,b,T,p);
2079 21 : if (!u) return gc_NULL(av);
2080 14 : return gerepilecopy(av, u);
2081 : }
2082 :
2083 : GEN
2084 14 : FpM_FpC_gauss(GEN a, GEN b, GEN p)
2085 : {
2086 14 : pari_sp av = avma;
2087 : ulong pp;
2088 : GEN u;
2089 14 : if (lg(a) == 1) return cgetg(1, t_COL);
2090 14 : u = FpM_gauss_i(a, mkmat(b), p, &pp);
2091 14 : if (!u) return gc_NULL(av);
2092 14 : switch(pp)
2093 : {
2094 14 : case 0: return gerepilecopy(av, gel(u,1));
2095 0 : case 2: u = F2c_to_ZC(gel(u,1)); break;
2096 0 : default: u = Flc_to_ZC(gel(u,1)); break;
2097 : }
2098 0 : return gerepileupto(av, u);
2099 : }
2100 :
2101 : GEN
2102 28 : F2xqM_F2xqC_gauss(GEN a, GEN b, GEN T)
2103 : {
2104 28 : pari_sp av = avma;
2105 : GEN u;
2106 28 : if (lg(a) == 1) return cgetg(1, t_COL);
2107 28 : u = F2xqM_gauss_gen(a, mkmat(b), T);
2108 28 : if (!u) return gc_NULL(av);
2109 14 : return gerepilecopy(av, gel(u,1));
2110 : }
2111 :
2112 : GEN
2113 14 : FlxqM_FlxqC_gauss(GEN a, GEN b, GEN T, ulong p)
2114 : {
2115 14 : pari_sp av = avma;
2116 : GEN u;
2117 14 : if (lg(a) == 1) return cgetg(1, t_COL);
2118 14 : u = FlxqM_gauss_i(a, mkmat(b), T, p);
2119 14 : if (!u) return gc_NULL(av);
2120 7 : return gerepilecopy(av, gel(u,1));
2121 : }
2122 :
2123 : GEN
2124 14 : FqM_FqC_gauss(GEN a, GEN b, GEN T, GEN p)
2125 : {
2126 14 : pari_sp av = avma;
2127 : GEN u;
2128 14 : if (!T) return FpM_FpC_gauss(a,b,p);
2129 14 : if (lg(a) == 1) return cgetg(1, t_COL);
2130 14 : u = FqM_gauss_gen(a,mkmat(b),T,p);
2131 14 : if (!u) return gc_NULL(av);
2132 7 : return gerepilecopy(av, gel(u,1));
2133 : }
2134 :
2135 : GEN
2136 340107 : FpM_inv(GEN a, GEN p)
2137 : {
2138 340107 : pari_sp av = avma;
2139 : ulong pp;
2140 : GEN u;
2141 340107 : if (lg(a) == 1) return cgetg(1, t_MAT);
2142 340107 : u = FpM_gauss_i(a, NULL, p, &pp);
2143 340109 : if (!u) return gc_NULL(av);
2144 340093 : switch(pp)
2145 : {
2146 3505 : case 0: return gerepilecopy(av, u);
2147 227297 : case 2: u = F2m_to_ZM(u); break;
2148 109291 : default: u = Flm_to_ZM(u); break;
2149 : }
2150 336586 : return gerepileupto(av, u);
2151 : }
2152 :
2153 : GEN
2154 35 : F2xqM_inv(GEN a, GEN T)
2155 : {
2156 35 : pari_sp av = avma;
2157 : GEN u;
2158 35 : if (lg(a) == 1) { set_avma(av); return cgetg(1, t_MAT); }
2159 35 : u = F2xqM_gauss_gen(a, matid_F2xqM(nbrows(a),T), T);
2160 35 : if (!u) return gc_NULL(av);
2161 28 : return gerepilecopy(av, u);
2162 : }
2163 :
2164 : GEN
2165 56 : FlxqM_inv(GEN a, GEN T, ulong p)
2166 : {
2167 56 : pari_sp av = avma;
2168 : GEN u;
2169 56 : if (lg(a) == 1) { set_avma(av); return cgetg(1, t_MAT); }
2170 56 : u = FlxqM_gauss_i(a, matid_FlxqM(nbrows(a),T,p), T,p);
2171 56 : if (!u) return gc_NULL(av);
2172 42 : return gerepilecopy(av, u);
2173 : }
2174 :
2175 : GEN
2176 98 : FqM_inv(GEN a, GEN T, GEN p)
2177 : {
2178 98 : pari_sp av = avma;
2179 : GEN u;
2180 98 : if (!T) return FpM_inv(a,p);
2181 98 : if (lg(a) == 1) return cgetg(1, t_MAT);
2182 98 : u = FqM_gauss_gen(a,matid(nbrows(a)),T,p);
2183 98 : if (!u) return gc_NULL(av);
2184 70 : return gerepilecopy(av, u);
2185 : }
2186 :
2187 : GEN
2188 353909 : FpM_intersect_i(GEN x, GEN y, GEN p)
2189 : {
2190 353909 : long j, lx = lg(x);
2191 : GEN z;
2192 :
2193 353909 : if (lx == 1 || lg(y) == 1) return cgetg(1,t_MAT);
2194 353909 : if (lgefint(p) == 3)
2195 : {
2196 353909 : ulong pp = p[2];
2197 353909 : return Flm_to_ZM(Flm_intersect_i(ZM_to_Flm(x,pp), ZM_to_Flm(y,pp), pp));
2198 : }
2199 0 : z = FpM_ker(shallowconcat(x,y), p);
2200 0 : for (j=lg(z)-1; j; j--) setlg(gel(z,j),lx);
2201 0 : return FpM_mul(x,z,p);
2202 : }
2203 : GEN
2204 0 : FpM_intersect(GEN x, GEN y, GEN p)
2205 : {
2206 0 : pari_sp av = avma;
2207 : GEN z;
2208 0 : if (lgefint(p) == 3)
2209 : {
2210 0 : ulong pp = p[2];
2211 0 : z = Flm_image(Flm_intersect_i(ZM_to_Flm(x,pp), ZM_to_Flm(y,pp), pp), pp);
2212 : }
2213 : else
2214 0 : z = FpM_image(FpM_intersect_i(x,y,p), p);
2215 0 : return gerepileupto(av, z);
2216 : }
2217 :
2218 : static void
2219 272632 : init_suppl(GEN x)
2220 : {
2221 272632 : if (lg(x) == 1) pari_err_IMPL("suppl [empty matrix]");
2222 : /* HACK: avoid overwriting d from gauss_pivot() after set_avma(av) */
2223 272632 : (void)new_chunk(lgcols(x) * 2);
2224 272632 : }
2225 :
2226 : GEN
2227 271085 : FpM_suppl(GEN x, GEN p)
2228 : {
2229 : GEN d;
2230 : long r;
2231 271085 : init_suppl(x); d = FpM_gauss_pivot(x,p, &r);
2232 271088 : return get_suppl(x,d,nbrows(x),r,&col_ei);
2233 : }
2234 :
2235 : GEN
2236 14 : F2m_suppl(GEN x)
2237 : {
2238 : GEN d;
2239 : long r;
2240 14 : init_suppl(x); d = F2m_gauss_pivot(F2m_copy(x), &r);
2241 14 : return get_suppl(x,d,mael(x,1,1),r,&F2v_ei);
2242 : }
2243 :
2244 : GEN
2245 105 : Flm_suppl(GEN x, ulong p)
2246 : {
2247 : GEN d;
2248 : long r;
2249 105 : init_suppl(x); d = Flm_pivots(x, p, &r, 0);
2250 105 : return get_suppl(x,d,nbrows(x),r,&vecsmall_ei);
2251 : }
2252 :
2253 : GEN
2254 7 : F2xqM_suppl(GEN x, GEN T)
2255 : {
2256 : void *E;
2257 7 : const struct bb_field *S = get_F2xq_field(&E, T);
2258 7 : return gen_suppl(x, E, S, _F2xqM_mul);
2259 : }
2260 :
2261 : GEN
2262 14 : FlxqM_suppl(GEN x, GEN T, ulong p)
2263 : {
2264 : void *E;
2265 14 : const struct bb_field *S = get_Flxq_field(&E, T, p);
2266 14 : return gen_suppl(x, E, S, _FlxqM_mul);
2267 : }
2268 :
2269 : GEN
2270 4305 : FqM_suppl(GEN x, GEN T, GEN p)
2271 : {
2272 4305 : pari_sp av = avma;
2273 : GEN d;
2274 : long r;
2275 :
2276 4305 : if (!T) return FpM_suppl(x,p);
2277 1358 : init_suppl(x);
2278 1358 : d = FqM_gauss_pivot(x,T,p,&r);
2279 1358 : set_avma(av); return get_suppl(x,d,nbrows(x),r,&col_ei);
2280 : }
2281 :
2282 : static void
2283 173343 : init_indexrank(GEN x) {
2284 173343 : (void)new_chunk(3 + 2*lg(x)); /* HACK */
2285 173342 : }
2286 :
2287 : GEN
2288 29792 : FpM_indexrank(GEN x, GEN p) {
2289 29792 : pari_sp av = avma;
2290 : long r;
2291 : GEN d;
2292 29792 : init_indexrank(x);
2293 29792 : d = FpM_gauss_pivot(x,p,&r);
2294 29792 : set_avma(av); return indexrank0(lg(x)-1, r, d);
2295 : }
2296 :
2297 : GEN
2298 52877 : Flm_indexrank(GEN x, ulong p) {
2299 52877 : pari_sp av = avma;
2300 : long r;
2301 : GEN d;
2302 52877 : init_indexrank(x);
2303 52876 : d = Flm_pivots(x, p, &r, 0);
2304 52877 : set_avma(av); return indexrank0(lg(x)-1, r, d);
2305 : }
2306 :
2307 : GEN
2308 60 : F2m_indexrank(GEN x) {
2309 60 : pari_sp av = avma;
2310 : long r;
2311 : GEN d;
2312 60 : init_indexrank(x);
2313 60 : d = F2m_gauss_pivot(F2m_copy(x),&r);
2314 60 : set_avma(av); return indexrank0(lg(x)-1, r, d);
2315 : }
2316 :
2317 : GEN
2318 7 : F2xqM_indexrank(GEN x, GEN T) {
2319 7 : pari_sp av = avma;
2320 : long r;
2321 : GEN d;
2322 7 : init_indexrank(x);
2323 7 : d = F2xqM_gauss_pivot(x, T, &r);
2324 7 : set_avma(av); return indexrank0(lg(x) - 1, r, d);
2325 : }
2326 :
2327 : GEN
2328 7 : FlxqM_indexrank(GEN x, GEN T, ulong p) {
2329 7 : pari_sp av = avma;
2330 : long r;
2331 : GEN d;
2332 7 : init_indexrank(x);
2333 7 : d = FlxqM_gauss_pivot(x, T, p, &r);
2334 7 : set_avma(av); return indexrank0(lg(x) - 1, r, d);
2335 : }
2336 :
2337 : GEN
2338 7 : FqM_indexrank(GEN x, GEN T, GEN p) {
2339 7 : pari_sp av = avma;
2340 : long r;
2341 : GEN d;
2342 7 : init_indexrank(x);
2343 7 : d = FqM_gauss_pivot(x, T, p, &r);
2344 7 : set_avma(av); return indexrank0(lg(x) - 1, r, d);
2345 : }
2346 :
2347 : /*******************************************************************/
2348 : /* */
2349 : /* Solve A*X=B (Gauss pivot) */
2350 : /* */
2351 : /*******************************************************************/
2352 : /* x a column, x0 same column in the original input matrix (for reference),
2353 : * c list of pivots so far */
2354 : static long
2355 1892342 : gauss_get_pivot_max(GEN X, GEN X0, long ix, GEN c)
2356 : {
2357 1892342 : GEN p, r, x = gel(X,ix), x0 = gel(X0,ix);
2358 1892342 : long i, k = 0, ex = - (long)HIGHEXPOBIT, lx = lg(x);
2359 1892342 : if (c)
2360 : {
2361 320274 : for (i=1; i<lx; i++)
2362 184004 : if (!c[i])
2363 : {
2364 88238 : long e = gexpo(gel(x,i));
2365 88238 : if (e > ex) { ex = e; k = i; }
2366 : }
2367 : }
2368 : else
2369 : {
2370 6362085 : for (i=ix; i<lx; i++)
2371 : {
2372 4605998 : long e = gexpo(gel(x,i));
2373 4606013 : if (e > ex) { ex = e; k = i; }
2374 : }
2375 : }
2376 1892357 : if (!k) return lx;
2377 1823933 : p = gel(x,k);
2378 1823933 : r = gel(x0,k); if (isrationalzero(r)) r = x0;
2379 1823937 : return cx_approx0(p, r)? lx: k;
2380 : }
2381 : static long
2382 253239 : gauss_get_pivot_padic(GEN X, GEN p, long ix, GEN c)
2383 : {
2384 253239 : GEN x = gel(X, ix);
2385 253239 : long i, k = 0, ex = (long)HIGHVALPBIT, lx = lg(x);
2386 253239 : if (c)
2387 : {
2388 504 : for (i=1; i<lx; i++)
2389 378 : if (!c[i] && !gequal0(gel(x,i)))
2390 : {
2391 245 : long e = gvaluation(gel(x,i), p);
2392 245 : if (e < ex) { ex = e; k = i; }
2393 : }
2394 : }
2395 : else
2396 : {
2397 2312744 : for (i=ix; i<lx; i++)
2398 2059631 : if (!gequal0(gel(x,i)))
2399 : {
2400 1557682 : long e = gvaluation(gel(x,i), p);
2401 1557682 : if (e < ex) { ex = e; k = i; }
2402 : }
2403 : }
2404 253239 : return k? k: lx;
2405 : }
2406 : static long
2407 4235 : gauss_get_pivot_NZ(GEN X, GEN x0/*unused*/, long ix, GEN c)
2408 : {
2409 4235 : GEN x = gel(X, ix);
2410 4235 : long i, lx = lg(x);
2411 : (void)x0;
2412 4235 : if (c)
2413 : {
2414 11872 : for (i=1; i<lx; i++)
2415 11004 : if (!c[i] && !gequal0(gel(x,i))) return i;
2416 : }
2417 : else
2418 : {
2419 2380 : for (i=ix; i<lx; i++)
2420 2366 : if (!gequal0(gel(x,i))) return i;
2421 : }
2422 882 : return lx;
2423 : }
2424 :
2425 : /* Return pivot seeking function appropriate for the domain of the RgM x
2426 : * (first non zero pivot, maximal pivot...)
2427 : * x0 is a reference point used when guessing whether x[i,j] ~ 0
2428 : * (iff x[i,j] << x0[i,j]); typical case: mateigen, Gauss pivot on x - vp.Id,
2429 : * but use original x when deciding whether a prospective pivot is nonzero */
2430 : static pivot_fun
2431 594831 : get_pivot_fun(GEN x, GEN x0, GEN *data)
2432 : {
2433 594831 : long i, j, hx, lx = lg(x);
2434 594831 : int res = t_INT;
2435 594831 : GEN p = NULL;
2436 :
2437 594831 : *data = NULL;
2438 594831 : if (lx == 1) return &gauss_get_pivot_NZ;
2439 594796 : hx = lgcols(x);
2440 2808466 : for (j=1; j<lx; j++)
2441 : {
2442 2213711 : GEN xj = gel(x,j);
2443 13817790 : for (i=1; i<hx; i++)
2444 : {
2445 11604121 : GEN c = gel(xj,i);
2446 11604121 : switch(typ(c))
2447 : {
2448 5672908 : case t_REAL:
2449 5672908 : res = t_REAL;
2450 5672908 : break;
2451 364 : case t_COMPLEX:
2452 364 : if (typ(gel(c,1)) == t_REAL || typ(gel(c,2)) == t_REAL) res = t_REAL;
2453 364 : break;
2454 4385158 : case t_INT: case t_INTMOD: case t_FRAC: case t_FFELT: case t_QUAD:
2455 : case t_POLMOD: /* exact types */
2456 4385158 : break;
2457 1545649 : case t_PADIC:
2458 1545649 : p = gel(c,2);
2459 1545649 : res = t_PADIC;
2460 1545649 : break;
2461 42 : default: return &gauss_get_pivot_NZ;
2462 : }
2463 : }
2464 : }
2465 594755 : switch(res)
2466 : {
2467 563994 : case t_REAL: *data = x0; return &gauss_get_pivot_max;
2468 29477 : case t_PADIC: *data = p; return &gauss_get_pivot_padic;
2469 1284 : default: return &gauss_get_pivot_NZ;
2470 : }
2471 : }
2472 :
2473 : static GEN
2474 660112 : get_col(GEN a, GEN b, GEN p, long li)
2475 : {
2476 660112 : GEN u = cgetg(li+1,t_COL);
2477 : long i, j;
2478 :
2479 660112 : gel(u,li) = gdiv(gel(b,li), p);
2480 2918001 : for (i=li-1; i>0; i--)
2481 : {
2482 2257895 : pari_sp av = avma;
2483 2257895 : GEN m = gel(b,i);
2484 10659594 : for (j=i+1; j<=li; j++) m = gsub(m, gmul(gcoeff(a,i,j), gel(u,j)));
2485 2257880 : gel(u,i) = gerepileupto(av, gdiv(m, gcoeff(a,i,i)));
2486 : }
2487 660106 : return u;
2488 : }
2489 :
2490 : /* bk -= m * bi */
2491 : static void
2492 10684424 : _submul(GEN b, long k, long i, GEN m)
2493 : {
2494 10684424 : gel(b,k) = gsub(gel(b,k), gmul(m, gel(b,i)));
2495 10684391 : }
2496 : static int
2497 1422384 : init_gauss(GEN a, GEN *b, long *aco, long *li, int *iscol)
2498 : {
2499 1422384 : *iscol = *b ? (typ(*b) == t_COL): 0;
2500 1422384 : *aco = lg(a) - 1;
2501 1422384 : if (!*aco) /* a empty */
2502 : {
2503 70 : if (*b && lg(*b) != 1) pari_err_DIM("gauss");
2504 70 : *li = 0; return 0;
2505 : }
2506 1422314 : *li = nbrows(a);
2507 1422314 : if (*li < *aco) pari_err_INV("gauss [no left inverse]", a);
2508 1422314 : if (*b)
2509 : {
2510 1343503 : switch(typ(*b))
2511 : {
2512 121300 : case t_MAT:
2513 121300 : if (lg(*b) == 1) return 0;
2514 121300 : *b = RgM_shallowcopy(*b);
2515 121300 : break;
2516 1222205 : case t_COL:
2517 1222205 : *b = mkmat( leafcopy(*b) );
2518 1222204 : break;
2519 0 : default: pari_err_TYPE("gauss",*b);
2520 : }
2521 1343504 : if (nbrows(*b) != *li) pari_err_DIM("gauss");
2522 : }
2523 : else
2524 78811 : *b = matid(*li);
2525 1422311 : return 1;
2526 : }
2527 :
2528 : static GEN
2529 112 : RgM_inv_FpM(GEN a, GEN p)
2530 : {
2531 : ulong pp;
2532 112 : a = RgM_Fp_init(a, p, &pp);
2533 112 : switch(pp)
2534 : {
2535 35 : case 0:
2536 35 : a = FpM_inv(a,p);
2537 35 : if (a) a = FpM_to_mod(a, p);
2538 35 : break;
2539 35 : case 2:
2540 35 : a = F2m_inv(a);
2541 35 : if (a) a = F2m_to_mod(a);
2542 35 : break;
2543 42 : default:
2544 42 : a = Flm_inv_sp(a, NULL, pp);
2545 42 : if (a) a = Flm_to_mod(a, pp);
2546 : }
2547 112 : return a;
2548 : }
2549 :
2550 : static GEN
2551 42 : RgM_inv_FqM(GEN x, GEN pol, GEN p)
2552 : {
2553 42 : pari_sp av = avma;
2554 42 : GEN b, T = RgX_to_FpX(pol, p);
2555 42 : if (signe(T) == 0) pari_err_OP("^",x,gen_m1);
2556 42 : b = FqM_inv(RgM_to_FqM(x, T, p), T, p);
2557 42 : if (!b) return gc_NULL(av);
2558 28 : return gerepileupto(av, FqM_to_mod(b, T, p));
2559 : }
2560 :
2561 : #define code(t1,t2) ((t1 << 6) | t2)
2562 : static GEN
2563 201059 : RgM_inv_fast(GEN x)
2564 : {
2565 : GEN p, pol;
2566 : long pa;
2567 201059 : long t = RgM_type(x, &p,&pol,&pa);
2568 201060 : switch(t)
2569 : {
2570 45583 : case t_INT: /* Fall back */
2571 45583 : case t_FRAC: return QM_inv(x);
2572 147 : case t_FFELT: return FFM_inv(x, pol);
2573 112 : case t_INTMOD: return RgM_inv_FpM(x, p);
2574 42 : case code(t_POLMOD, t_INTMOD):
2575 42 : return RgM_inv_FqM(x, pol, p);
2576 155176 : default: return gen_0;
2577 : }
2578 : }
2579 : #undef code
2580 :
2581 : static GEN
2582 49 : RgM_RgC_solve_FpC(GEN a, GEN b, GEN p)
2583 : {
2584 49 : pari_sp av = avma;
2585 : ulong pp;
2586 49 : a = RgM_Fp_init(a, p, &pp);
2587 49 : switch(pp)
2588 : {
2589 14 : case 0:
2590 14 : b = RgC_to_FpC(b, p);
2591 14 : a = FpM_FpC_gauss(a,b,p);
2592 14 : return a ? gerepileupto(av, FpC_to_mod(a, p)): NULL;
2593 14 : case 2:
2594 14 : b = RgV_to_F2v(b);
2595 14 : a = F2m_F2c_gauss(a,b);
2596 14 : return a ? gerepileupto(av, F2c_to_mod(a)): NULL;
2597 21 : default:
2598 21 : b = RgV_to_Flv(b, pp);
2599 21 : a = Flm_Flc_gauss(a, b, pp);
2600 21 : return a ? gerepileupto(av, Flc_to_mod(a, pp)): NULL;
2601 : }
2602 : }
2603 :
2604 : static GEN
2605 98 : RgM_solve_FpM(GEN a, GEN b, GEN p)
2606 : {
2607 98 : pari_sp av = avma;
2608 : ulong pp;
2609 98 : a = RgM_Fp_init(a, p, &pp);
2610 98 : switch(pp)
2611 : {
2612 35 : case 0:
2613 35 : b = RgM_to_FpM(b, p);
2614 35 : a = FpM_gauss(a,b,p);
2615 35 : return a ? gerepileupto(av, FpM_to_mod(a, p)): NULL;
2616 21 : case 2:
2617 21 : b = RgM_to_F2m(b);
2618 21 : a = F2m_gauss(a,b);
2619 21 : return a ? gerepileupto(av, F2m_to_mod(a)): NULL;
2620 42 : default:
2621 42 : b = RgM_to_Flm(b, pp);
2622 42 : a = Flm_gauss(a,b,pp);
2623 42 : return a ? gerepileupto(av, Flm_to_mod(a, pp)): NULL;
2624 : }
2625 : }
2626 :
2627 : /* Gaussan Elimination. If a is square, return a^(-1)*b;
2628 : * if a has more rows than columns and b is NULL, return c such that c a = Id.
2629 : * a is a (not necessarily square) matrix
2630 : * b is a matrix or column vector, NULL meaning: take the identity matrix,
2631 : * effectively returning the inverse of a
2632 : * If a and b are empty, the result is the empty matrix.
2633 : *
2634 : * li: number of rows of a and b
2635 : * aco: number of columns of a
2636 : * bco: number of columns of b (if matrix)
2637 : */
2638 : static GEN
2639 760407 : RgM_solve_basecase(GEN a, GEN b)
2640 : {
2641 760407 : pari_sp av = avma;
2642 : long i, j, k, li, bco, aco;
2643 : int iscol;
2644 : pivot_fun pivot;
2645 : GEN p, u, data;
2646 :
2647 760407 : set_avma(av);
2648 :
2649 760407 : if (lg(a)-1 == 2 && nbrows(a) == 2) {
2650 : /* 2x2 matrix, start by inverting a */
2651 288670 : GEN u = gcoeff(a,1,1), v = gcoeff(a,1,2);
2652 288670 : GEN w = gcoeff(a,2,1), x = gcoeff(a,2,2);
2653 288670 : GEN D = gsub(gmul(u,x), gmul(v,w)), ainv;
2654 288661 : if (gequal0(D)) return NULL;
2655 288660 : ainv = mkmat2(mkcol2(x, gneg(w)), mkcol2(gneg(v), u));
2656 288666 : ainv = gmul(ainv, ginv(D));
2657 288659 : if (b) ainv = gmul(ainv, b);
2658 288663 : return gerepileupto(av, ainv);
2659 : }
2660 :
2661 471737 : if (!init_gauss(a, &b, &aco, &li, &iscol)) return cgetg(1, iscol?t_COL:t_MAT);
2662 471737 : pivot = get_pivot_fun(a, a, &data);
2663 471737 : a = RgM_shallowcopy(a);
2664 471736 : bco = lg(b)-1;
2665 471736 : if(DEBUGLEVEL>4) err_printf("Entering gauss\n");
2666 :
2667 471736 : p = NULL; /* gcc -Wall */
2668 1669149 : for (i=1; i<=aco; i++)
2669 : {
2670 : /* k is the line where we find the pivot */
2671 1669139 : k = pivot(a, data, i, NULL);
2672 1669156 : if (k > li) return NULL;
2673 1669129 : if (k != i)
2674 : { /* exchange the lines s.t. k = i */
2675 1349124 : for (j=i; j<=aco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
2676 1010796 : for (j=1; j<=bco; j++) swap(gcoeff(b,i,j), gcoeff(b,k,j));
2677 : }
2678 1669129 : p = gcoeff(a,i,i);
2679 1669129 : if (i == aco) break;
2680 :
2681 3859828 : for (k=i+1; k<=li; k++)
2682 : {
2683 2662423 : GEN m = gcoeff(a,k,i);
2684 2662423 : if (!gequal0(m))
2685 : {
2686 2026829 : m = gdiv(m,p);
2687 8384789 : for (j=i+1; j<=aco; j++) _submul(gel(a,j),k,i,m);
2688 6353533 : for (j=1; j<=bco; j++) _submul(gel(b,j),k,i,m);
2689 : }
2690 : }
2691 1197405 : if (gc_needed(av,1))
2692 : {
2693 12 : if(DEBUGMEM>1) pari_warn(warnmem,"gauss. i=%ld",i);
2694 12 : gerepileall(av,2, &a,&b);
2695 : }
2696 : }
2697 :
2698 471720 : if(DEBUGLEVEL>4) err_printf("Solving the triangular system\n");
2699 471720 : u = cgetg(bco+1,t_MAT);
2700 1131815 : for (j=1; j<=bco; j++) gel(u,j) = get_col(a,gel(b,j),p,aco);
2701 471703 : return gerepilecopy(av, iscol? gel(u,1): u);
2702 : }
2703 :
2704 : static GEN
2705 570643 : RgM_RgC_solve_fast(GEN x, GEN y)
2706 : {
2707 : GEN p, pol;
2708 : long pa;
2709 570643 : long t = RgM_RgC_type(x, y, &p,&pol,&pa);
2710 570646 : switch(t)
2711 : {
2712 13552 : case t_INT: return ZM_gauss(x, y);
2713 42 : case t_FRAC: return QM_gauss(x, y);
2714 49 : case t_INTMOD: return RgM_RgC_solve_FpC(x, y, p);
2715 56 : case t_FFELT: return FFM_FFC_gauss(x, y, pol);
2716 556947 : default: return gen_0;
2717 : }
2718 : }
2719 :
2720 : static GEN
2721 48509 : RgM_solve_fast(GEN x, GEN y)
2722 : {
2723 : GEN p, pol;
2724 : long pa;
2725 48509 : long t = RgM_type2(x, y, &p,&pol,&pa);
2726 48510 : switch(t)
2727 : {
2728 49 : case t_INT: return ZM_gauss(x, y);
2729 14 : case t_FRAC: return QM_gauss(x, y);
2730 98 : case t_INTMOD: return RgM_solve_FpM(x, y, p);
2731 63 : case t_FFELT: return FFM_gauss(x, y, pol);
2732 48286 : default: return gen_0;
2733 : }
2734 : }
2735 :
2736 : GEN
2737 619152 : RgM_solve(GEN a, GEN b)
2738 : {
2739 619152 : pari_sp av = avma;
2740 : GEN u;
2741 619152 : if (!b) return RgM_inv(a);
2742 619152 : u = typ(b)==t_MAT ? RgM_solve_fast(a, b): RgM_RgC_solve_fast(a, b);
2743 619156 : if (!u) return gc_NULL(av);
2744 619058 : if (u != gen_0) return u;
2745 605233 : return RgM_solve_basecase(a, b);
2746 : }
2747 :
2748 : GEN
2749 201059 : RgM_inv(GEN a)
2750 : {
2751 201059 : GEN b = RgM_inv_fast(a);
2752 201046 : return b==gen_0? RgM_solve_basecase(a, NULL): b;
2753 : }
2754 :
2755 : /* assume dim A >= 1, A invertible + upper triangular */
2756 : static GEN
2757 832424 : RgM_inv_upper_ind(GEN A, long index)
2758 : {
2759 832424 : long n = lg(A)-1, i = index, j;
2760 832424 : GEN u = zerocol(n);
2761 832430 : gel(u,i) = ginv(gcoeff(A,i,i));
2762 2389933 : for (i--; i>0; i--)
2763 : {
2764 1557492 : pari_sp av = avma;
2765 1557492 : GEN m = gneg(gmul(gcoeff(A,i,i+1),gel(u,i+1))); /* j = i+1 */
2766 6632405 : for (j=i+2; j<=n; j++) m = gsub(m, gmul(gcoeff(A,i,j),gel(u,j)));
2767 1557461 : gel(u,i) = gerepileupto(av, gdiv(m, gcoeff(A,i,i)));
2768 : }
2769 832441 : return u;
2770 : }
2771 : GEN
2772 277952 : RgM_inv_upper(GEN A)
2773 : {
2774 : long i, l;
2775 277952 : GEN B = cgetg_copy(A, &l);
2776 1110373 : for (i = 1; i < l; i++) gel(B,i) = RgM_inv_upper_ind(A, i);
2777 277963 : return B;
2778 : }
2779 :
2780 : static GEN
2781 2671127 : split_realimag_col(GEN z, long r1, long r2)
2782 : {
2783 2671127 : long i, ru = r1+r2;
2784 2671127 : GEN x = cgetg(ru+r2+1,t_COL), y = x + r2;
2785 7054195 : for (i=1; i<=r1; i++) {
2786 4383068 : GEN a = gel(z,i);
2787 4383068 : if (typ(a) == t_COMPLEX) a = gel(a,1); /* paranoia: a should be real */
2788 4383068 : gel(x,i) = a;
2789 : }
2790 5354966 : for ( ; i<=ru; i++) {
2791 2683839 : GEN b, a = gel(z,i);
2792 2683839 : if (typ(a) == t_COMPLEX) { b = gel(a,2); a = gel(a,1); } else b = gen_0;
2793 2683839 : gel(x,i) = a;
2794 2683839 : gel(y,i) = b;
2795 : }
2796 2671127 : return x;
2797 : }
2798 : GEN
2799 1345947 : split_realimag(GEN x, long r1, long r2)
2800 : {
2801 : long i,l; GEN y;
2802 1345947 : if (typ(x) == t_COL) return split_realimag_col(x,r1,r2);
2803 666094 : y = cgetg_copy(x, &l);
2804 2657371 : for (i=1; i<l; i++) gel(y,i) = split_realimag_col(gel(x,i), r1, r2);
2805 666095 : return y;
2806 : }
2807 :
2808 : /* assume M = (r1+r2) x (r1+2r2) matrix and y compatible vector or matrix
2809 : * r1 first lines of M,y are real. Solve the system obtained by splitting
2810 : * real and imaginary parts. */
2811 : GEN
2812 604096 : RgM_solve_realimag(GEN M, GEN y)
2813 : {
2814 604096 : long l = lg(M), r2 = l - lgcols(M), r1 = l-1 - 2*r2;
2815 604097 : return RgM_solve(split_realimag(M, r1,r2),
2816 : split_realimag(y, r1,r2));
2817 : }
2818 :
2819 : GEN
2820 434 : gauss(GEN a, GEN b)
2821 : {
2822 : GEN z;
2823 434 : long t = typ(b);
2824 434 : if (typ(a)!=t_MAT) pari_err_TYPE("gauss",a);
2825 434 : if (t!=t_COL && t!=t_MAT) pari_err_TYPE("gauss",b);
2826 434 : z = RgM_solve(a,b);
2827 434 : if (!z) pari_err_INV("gauss",a);
2828 329 : return z;
2829 : }
2830 :
2831 : static GEN
2832 948431 : ZlM_gauss_ratlift(GEN a, GEN b, ulong p, long e, GEN C)
2833 : {
2834 948431 : pari_sp av = avma, av2;
2835 : GEN bb, xi, xb, pi, q, B, r;
2836 : long i, f, k;
2837 : ulong mask;
2838 948431 : if (!C) {
2839 0 : C = Flm_inv(ZM_to_Flm(a, p), p);
2840 0 : if (!C) pari_err_INV("ZlM_gauss", a);
2841 : }
2842 948431 : k = f = ZM_max_lg(a)-1;
2843 948431 : mask = quadratic_prec_mask((e+f-1)/f);
2844 948430 : pi = q = powuu(p, f);
2845 948403 : bb = b;
2846 948403 : C = ZpM_invlift(FpM_red(a, q), Flm_to_ZM(C), utoipos(p), f);
2847 948420 : av2 = avma;
2848 948420 : xb = xi = FpM_mul(C, b, q);
2849 987610 : for (i = f; i <= e; i+=f)
2850 : {
2851 158706 : if (i==k)
2852 : {
2853 153664 : k = (mask&1UL) ? 2*k-f: 2*k;
2854 153664 : mask >>= 1;
2855 153664 : B = sqrti(shifti(pi,-1));
2856 153664 : r = FpM_ratlift(xb, pi, B, B, NULL);
2857 153664 : if (r)
2858 : {
2859 126927 : GEN dr, nr = Q_remove_denom(r,&dr);
2860 126927 : if (ZM_equal(ZM_mul(a,nr), dr? ZM_Z_mul(b,dr): b))
2861 : {
2862 119520 : if (DEBUGLEVEL>=4)
2863 0 : err_printf("ZlM_gauss: early solution: %ld/%ld\n",i,e);
2864 119520 : return gerepilecopy(av, r);
2865 : }
2866 : }
2867 : }
2868 39186 : bb = ZM_Z_divexact(ZM_sub(bb, ZM_mul(a, xi)), q);
2869 39186 : if (gc_needed(av,2))
2870 : {
2871 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZlM_gauss. i=%ld/%ld",i,e);
2872 0 : gerepileall(av2,3, &pi,&bb,&xb);
2873 : }
2874 39186 : xi = FpM_mul(C, bb, q);
2875 39186 : xb = ZM_add(xb, ZM_Z_mul(xi, pi));
2876 39186 : pi = mulii(pi, q);
2877 : }
2878 828904 : B = sqrti(shifti(pi,-1));
2879 828918 : return gerepileupto(av, FpM_ratlift(xb, pi, B, B, NULL));
2880 : }
2881 :
2882 : /* Dixon p-adic lifting algorithm.
2883 : * Numer. Math. 40, 137-141 (1982), DOI: 10.1007/BF01459082 */
2884 : GEN
2885 950647 : ZM_gauss(GEN a, GEN b)
2886 : {
2887 950647 : pari_sp av = avma, av2;
2888 : int iscol;
2889 : long n, ncol, i, m, elim;
2890 : ulong p;
2891 : GEN C, delta, nb, nmin, res;
2892 : forprime_t S;
2893 :
2894 950647 : if (!init_gauss(a, &b, &n, &ncol, &iscol)) return cgetg(1, iscol?t_COL:t_MAT);
2895 950574 : if (n < ncol)
2896 : {
2897 28 : GEN y = ZM_indexrank(a), y1 = gel(y,1), y2 = gel(y,2);
2898 28 : if (lg(y2)-1 != n) return NULL;
2899 28 : a = rowpermute(a, y1);
2900 28 : b = rowpermute(b, y1);
2901 : }
2902 : /* a is square and invertible */
2903 950574 : nb = gen_0; ncol = lg(b);
2904 2064417 : for (i = 1; i < ncol; i++)
2905 : {
2906 1113853 : GEN ni = gnorml2(gel(b, i));
2907 1113841 : if (cmpii(nb, ni) < 0) nb = ni;
2908 : }
2909 950564 : if (!signe(nb)) {set_avma(av); return iscol? zerocol(n): zeromat(n,lg(b)-1);}
2910 948436 : delta = gen_1; nmin = nb;
2911 4083928 : for (i = 1; i <= n; i++)
2912 : {
2913 3135575 : GEN ni = gnorml2(gel(a, i));
2914 3135687 : if (cmpii(ni, nmin) < 0)
2915 : {
2916 80983 : delta = mulii(delta, nmin); nmin = ni;
2917 : }
2918 : else
2919 3054705 : delta = mulii(delta, ni);
2920 : }
2921 948353 : if (!signe(nmin)) return NULL;
2922 948339 : elim = expi(delta)+1;
2923 948407 : av2 = avma;
2924 948407 : init_modular_big(&S);
2925 : for(;;)
2926 : {
2927 948424 : p = u_forprime_next(&S);
2928 948423 : C = Flm_inv_sp(ZM_to_Flm(a, p), NULL, p);
2929 948436 : if (C) break;
2930 7 : elim -= expu(p);
2931 7 : if (elim < 0) return NULL;
2932 0 : set_avma(av2);
2933 : }
2934 : /* N.B. Our delta/lambda are SQUARES of those in the paper
2935 : * log(delta lambda) / log p, where lambda is 3+sqrt(5) / 2,
2936 : * whose log is < 1, hence + 1 (to cater for rounding errors) */
2937 948429 : m = (long)ceil((dbllog2(delta)*M_LN2 + 1) / log((double)p));
2938 948432 : res = ZlM_gauss_ratlift(a, b, p, m, C);
2939 948438 : if (iscol) return gerepilecopy(av, gel(res, 1));
2940 79664 : return gerepileupto(av, res);
2941 : }
2942 :
2943 : /* #C = n, C[z[i]] = K[i], complete by 0s */
2944 : static GEN
2945 14 : RgC_inflate(GEN K, GEN z, long n)
2946 : {
2947 14 : GEN c = zerocol(n);
2948 14 : long j, l = lg(K);
2949 28 : for (j = 1; j < l; j++) gel(c, z[j]) = gel(K, j);
2950 14 : return c;
2951 : }
2952 : /* in place: C[i] *= cB / v[i] */
2953 : static void
2954 6069 : QC_normalize(GEN C, GEN v, GEN cB)
2955 : {
2956 6069 : long l = lg(C), i;
2957 45521 : for (i = 1; i < l; i++)
2958 : {
2959 39452 : GEN c = cB, k = gel(C,i), d = gel(v,i);
2960 39452 : if (d)
2961 : {
2962 23814 : if (isintzero(d)) { gel(C,i) = gen_0; continue; }
2963 23814 : c = div_content(c, d);
2964 : }
2965 39452 : gel(C,i) = c? gmul(k,c): k;
2966 : }
2967 6069 : }
2968 :
2969 : /* same as above, M rational; if flag = 1, call indexrank and return 1 sol */
2970 : GEN
2971 6062 : QM_gauss_i(GEN M, GEN B, long flag)
2972 : {
2973 6062 : pari_sp av = avma;
2974 : long i, l, n;
2975 6062 : int col = typ(B) == t_COL;
2976 6062 : GEN K, cB, N = cgetg_copy(M, &l), v = cgetg(l, t_VEC), z2 = NULL;
2977 :
2978 45542 : for (i = 1; i < l; i++)
2979 39480 : gel(N,i) = Q_primitive_part(gel(M,i), &gel(v,i));
2980 6062 : if (flag)
2981 : {
2982 301 : GEN z = ZM_indexrank(N), z1 = gel(z,1);
2983 301 : z2 = gel(z,2);
2984 301 : N = shallowmatextract(N, z1, z2);
2985 301 : B = col? vecpermute(B,z1): rowpermute(B,z1);
2986 301 : if (lg(z2) == l) z2 = NULL; else v = vecpermute(v, z2);
2987 : }
2988 6062 : B = Q_primitive_part(B, &cB);
2989 6062 : K = ZM_gauss(N, B); if (!K) return gc_NULL(av);
2990 6062 : n = l - 1;
2991 6062 : if (col)
2992 : {
2993 6034 : QC_normalize(K, v, cB);
2994 6034 : if (z2) K = RgC_inflate(K, z2, n);
2995 : }
2996 : else
2997 : {
2998 28 : long lK = lg(K);
2999 63 : for (i = 1; i < lK; i++)
3000 : {
3001 35 : QC_normalize(gel(K,i), v, cB);
3002 35 : if (z2) gel(K,i) = RgC_inflate(gel(K,i), z2, n);
3003 : }
3004 : }
3005 6062 : return gerepilecopy(av, K);
3006 : }
3007 : GEN
3008 5761 : QM_gauss(GEN M, GEN B) { return QM_gauss_i(M, B, 0); }
3009 :
3010 : static GEN
3011 589781 : ZM_inv_slice(GEN A, GEN P, GEN *mod)
3012 : {
3013 589781 : pari_sp av = avma;
3014 589781 : long i, n = lg(P)-1;
3015 : GEN H, T;
3016 589781 : if (n == 1)
3017 : {
3018 574133 : ulong p = uel(P,1);
3019 574133 : GEN Hp, a = ZM_to_Flm(A, p);
3020 574135 : Hp = Flm_adjoint(a, p);
3021 574131 : Hp = gerepileupto(av, Flm_to_ZM(Hp));
3022 574135 : *mod = utoipos(p); return Hp;
3023 : }
3024 15648 : T = ZV_producttree(P);
3025 15648 : A = ZM_nv_mod_tree(A, P, T);
3026 15648 : H = cgetg(n+1, t_VEC);
3027 77282 : for(i=1; i <= n; i++)
3028 61634 : gel(H,i) = Flm_adjoint(gel(A, i), uel(P,i));
3029 15648 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P,T));
3030 15647 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
3031 : }
3032 :
3033 : static GEN
3034 545250 : RgM_true_Hadamard(GEN a)
3035 : {
3036 545250 : pari_sp av = avma;
3037 545250 : long n = lg(a)-1, i;
3038 : GEN B;
3039 545250 : if (n == 0) return gen_1;
3040 545250 : a = RgM_gtofp(a, LOWDEFAULTPREC);
3041 545249 : B = gnorml2(gel(a,1));
3042 2286832 : for (i = 2; i <= n; i++) B = gmul(B, gnorml2(gel(a,i)));
3043 545253 : return gerepileuptoint(av, ceil_safe(sqrtr(B)));
3044 : }
3045 :
3046 : GEN
3047 589781 : ZM_inv_worker(GEN P, GEN A)
3048 : {
3049 589781 : GEN V = cgetg(3, t_VEC);
3050 589781 : gel(V,1) = ZM_inv_slice(A, P, &gel(V,2));
3051 589781 : return V;
3052 : }
3053 :
3054 : static GEN
3055 41110 : ZM_inv0(GEN A, GEN *pden)
3056 : {
3057 41110 : if (pden) *pden = gen_1;
3058 41110 : (void)A; return cgetg(1, t_MAT);
3059 : }
3060 : static GEN
3061 66683 : ZM_inv1(GEN A, GEN *pden)
3062 : {
3063 66683 : GEN a = gcoeff(A,1,1);
3064 66683 : long s = signe(a);
3065 66683 : if (!s) return NULL;
3066 66683 : if (pden) *pden = absi(a);
3067 66683 : retmkmat(mkcol(s == 1? gen_1: gen_m1));
3068 : }
3069 : static GEN
3070 122234 : ZM_inv2(GEN A, GEN *pden)
3071 : {
3072 : GEN a, b, c, d, D, cA;
3073 : long s;
3074 122234 : A = Q_primitive_part(A, &cA);
3075 122234 : a = gcoeff(A,1,1); b = gcoeff(A,1,2);
3076 122234 : c = gcoeff(A,2,1); d = gcoeff(A,2,2);
3077 122234 : D = subii(mulii(a,d), mulii(b,c)); /* left on stack */
3078 122227 : s = signe(D);
3079 122227 : if (!s) return NULL;
3080 122227 : if (s < 0) D = negi(D);
3081 122229 : if (pden) *pden = mul_denom(D, cA);
3082 122229 : if (s > 0)
3083 81558 : retmkmat2(mkcol2(icopy(d), negi(c)), mkcol2(negi(b), icopy(a)));
3084 : else
3085 40671 : retmkmat2(mkcol2(negi(d), icopy(c)), mkcol2(icopy(b), negi(a)));
3086 : }
3087 :
3088 : /* to be used when denom(M^(-1)) << det(M) and a sharp multiple is
3089 : * not available. Return H primitive such that M*H = den*Id */
3090 : GEN
3091 0 : ZM_inv_ratlift(GEN M, GEN *pden)
3092 : {
3093 0 : pari_sp av2, av = avma;
3094 : GEN Hp, q, H;
3095 : ulong p;
3096 0 : long m = lg(M)-1;
3097 : forprime_t S;
3098 : pari_timer ti;
3099 :
3100 0 : if (m == 0) return ZM_inv0(M,pden);
3101 0 : if (m == 1 && nbrows(M)==1) return ZM_inv1(M,pden);
3102 0 : if (m == 2 && nbrows(M)==2) return ZM_inv2(M,pden);
3103 :
3104 0 : if (DEBUGLEVEL>5) timer_start(&ti);
3105 0 : init_modular_big(&S);
3106 0 : av2 = avma;
3107 0 : H = NULL;
3108 0 : while ((p = u_forprime_next(&S)))
3109 : {
3110 : GEN Mp, B, Hr;
3111 0 : Mp = ZM_to_Flm(M,p);
3112 0 : Hp = Flm_inv_sp(Mp, NULL, p);
3113 0 : if (!Hp) continue;
3114 0 : if (!H)
3115 : {
3116 0 : H = ZM_init_CRT(Hp, p);
3117 0 : q = utoipos(p);
3118 : }
3119 : else
3120 0 : ZM_incremental_CRT(&H, Hp, &q, p);
3121 0 : B = sqrti(shifti(q,-1));
3122 0 : Hr = FpM_ratlift(H,q,B,B,NULL);
3123 0 : if (DEBUGLEVEL>5)
3124 0 : timer_printf(&ti,"ZM_inv mod %lu (ratlift=%ld)", p,!!Hr);
3125 0 : if (Hr) {/* DONE ? */
3126 0 : GEN Hl = Q_remove_denom(Hr, pden);
3127 0 : if (ZM_isscalar(ZM_mul(Hl, M), *pden)) { H = Hl; break; }
3128 : }
3129 :
3130 0 : if (gc_needed(av,2))
3131 : {
3132 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZM_inv_ratlift");
3133 0 : gerepileall(av2, 2, &H, &q);
3134 : }
3135 : }
3136 0 : if (!*pden) *pden = gen_1;
3137 0 : return gc_all(av, 2, &H, pden);
3138 : }
3139 :
3140 : GEN
3141 72492 : FpM_ratlift_worker(GEN A, GEN mod, GEN B)
3142 : {
3143 : long l, i;
3144 72492 : GEN H = cgetg_copy(A, &l);
3145 147560 : for (i = 1; i < l; i++)
3146 : {
3147 75085 : GEN c = FpC_ratlift(gel(A,i), mod, B, B, NULL);
3148 75070 : gel(H,i) = c? c: gen_0;
3149 : }
3150 72475 : return H;
3151 : }
3152 : static int
3153 575531 : can_ratlift(GEN x, GEN mod, GEN B)
3154 : {
3155 575531 : pari_sp av = avma;
3156 : GEN a, b;
3157 575531 : return gc_bool(av, Fp_ratlift(x, mod, B, B, &a,&b));
3158 : }
3159 : static GEN
3160 635847 : FpM_ratlift_parallel(GEN A, GEN mod, GEN B)
3161 : {
3162 635847 : pari_sp av = avma;
3163 : GEN worker;
3164 635847 : long i, l = lg(A), m = mt_nbthreads();
3165 635853 : int test = !!B;
3166 :
3167 635853 : if (l == 1 || lgcols(A) == 1) return gcopy(A);
3168 635853 : if (!B) B = sqrti(shifti(mod,-1));
3169 635854 : if (m == 1 || l == 2 || lgcols(A) < 10)
3170 : {
3171 629045 : A = FpM_ratlift(A, mod, B, B, NULL);
3172 629047 : return A? A: gc_NULL(av);
3173 : }
3174 : /* test one coefficient first */
3175 6809 : if (test && !can_ratlift(gcoeff(A,1,1), mod, B)) return gc_NULL(av);
3176 6722 : worker = snm_closure(is_entry("_FpM_ratlift_worker"), mkvec2(mod,B));
3177 6722 : A = gen_parapply_slice(worker, A, m);
3178 74355 : for (i = 1; i < l; i++) if (typ(gel(A,i)) != t_COL) return gc_NULL(av);
3179 5842 : return A;
3180 : }
3181 :
3182 : static GEN
3183 568922 : ZM_adj_ratlift(GEN A, GEN H, GEN mod, GEN T)
3184 : {
3185 568922 : pari_sp av = avma;
3186 : GEN B, D, g;
3187 568922 : D = ZMrow_ZC_mul(H, gel(A,1), 1);
3188 568921 : if (T) D = mulii(T, D);
3189 568921 : g = gcdii(D, mod);
3190 568915 : if (!equali1(g))
3191 : {
3192 14 : mod = diviiexact(mod, g);
3193 14 : H = FpM_red(H, mod);
3194 : }
3195 568913 : D = Fp_inv(Fp_red(D, mod), mod);
3196 : /* test 1 coeff first */
3197 568914 : B = sqrti(shifti(mod,-1));
3198 568917 : if (!can_ratlift(Fp_mul(D, gcoeff(A,1,1), mod), mod, B)) return gc_NULL(av);
3199 554960 : H = FpM_Fp_mul(H, D, mod);
3200 554960 : H = FpM_ratlift_parallel(H, mod, B);
3201 554971 : return H? H: gc_NULL(av);
3202 : }
3203 :
3204 : /* if (T) return T A^(-1) in Mn(Q), else B in Mn(Z) such that A B = den*Id */
3205 : static GEN
3206 775284 : ZM_inv_i(GEN A, GEN *pden, GEN T)
3207 : {
3208 775284 : pari_sp av = avma;
3209 775284 : long m = lg(A)-1, n, k1 = 1, k2;
3210 775284 : GEN H = NULL, D, H1 = NULL, mod1 = NULL, worker;
3211 : ulong bnd, mask;
3212 : forprime_t S;
3213 : pari_timer ti;
3214 :
3215 775284 : if (m == 0) return ZM_inv0(A,pden);
3216 734174 : if (pden) *pden = gen_1;
3217 734174 : if (nbrows(A) < m) return NULL;
3218 734169 : if (m == 1 && nbrows(A)==1 && !T) return ZM_inv1(A,pden);
3219 667486 : if (m == 2 && nbrows(A)==2 && !T) return ZM_inv2(A,pden);
3220 :
3221 545252 : if (DEBUGLEVEL>=5) timer_start(&ti);
3222 545252 : init_modular_big(&S);
3223 545252 : bnd = expi(RgM_true_Hadamard(A));
3224 545254 : worker = snm_closure(is_entry("_ZM_inv_worker"), mkvec(A));
3225 545254 : gen_inccrt("ZM_inv_r", worker, NULL, k1, 0, &S, &H1, &mod1, nmV_chinese_center, FpM_center);
3226 545254 : n = (bnd+1)/expu(S.p)+1;
3227 545254 : if (DEBUGLEVEL>=5) timer_printf(&ti,"inv (%ld/%ld primes)", k1, n);
3228 545254 : mask = quadratic_prec_mask(n);
3229 545254 : for (k2 = 0;;)
3230 44356 : {
3231 : GEN Hr;
3232 589610 : if (k2 > 0)
3233 : {
3234 38747 : gen_inccrt("ZM_inv_r", worker, NULL, k2, 0, &S, &H1, &mod1,nmV_chinese_center,FpM_center);
3235 38747 : k1 += k2;
3236 38747 : if (DEBUGLEVEL>=5) timer_printf(&ti,"CRT (%ld/%ld primes)", k1, n);
3237 : }
3238 589610 : if (mask == 1) break;
3239 568922 : k2 = (mask&1UL) ? k1-1: k1;
3240 568922 : mask >>= 1;
3241 :
3242 568922 : Hr = ZM_adj_ratlift(A, H1, mod1, T);
3243 568923 : if (DEBUGLEVEL>=5) timer_printf(&ti,"ratlift (%ld/%ld primes)", k1, n);
3244 568923 : if (Hr) {/* DONE ? */
3245 527454 : GEN Hl = Q_primpart(Hr), R = ZM_mul(Hl, A), d = gcoeff(R,1,1);
3246 527453 : if (gsigne(d) < 0) { d = gneg(d); Hl = ZM_neg(Hl); }
3247 527453 : if (DEBUGLEVEL>=5) timer_printf(&ti,"mult (%ld/%ld primes)", k1, n);
3248 527453 : if (equali1(d))
3249 : {
3250 437626 : if (ZM_isidentity(R)) { H = Hl; break; }
3251 : }
3252 89827 : else if (ZM_isscalar(R, d))
3253 : {
3254 86940 : if (T) T = gdiv(T,d);
3255 83641 : else if (pden) *pden = d;
3256 86940 : H = Hl; break;
3257 : }
3258 : }
3259 : }
3260 545254 : if (!H)
3261 : {
3262 : GEN d;
3263 20688 : H = H1;
3264 20688 : D = ZMrow_ZC_mul(H, gel(A,1), 1);
3265 20688 : if (signe(D)==0) pari_err_INV("ZM_inv", A);
3266 20688 : if (T) T = gdiv(T, D);
3267 : else
3268 : {
3269 19766 : d = gcdii(Q_content_safe(H), D);
3270 19766 : if (signe(D) < 0) d = negi(d);
3271 19766 : if (!equali1(d))
3272 : {
3273 11962 : H = ZM_Z_divexact(H, d);
3274 11962 : D = diviiexact(D, d);
3275 : }
3276 19767 : if (pden) *pden = D;
3277 : }
3278 : }
3279 545255 : if (T && !isint1(T)) H = ZM_Q_mul(H, T);
3280 545255 : return gc_all(av, pden? 2: 1, &H, pden);
3281 : }
3282 : GEN
3283 714748 : ZM_inv(GEN A, GEN *pden) { return ZM_inv_i(A, pden, NULL); }
3284 :
3285 : /* same as above, M rational */
3286 : GEN
3287 60535 : QM_inv(GEN M)
3288 : {
3289 60535 : pari_sp av = avma;
3290 : GEN den, dM, K;
3291 60535 : M = Q_remove_denom(M, &dM);
3292 60535 : K = ZM_inv_i(M, &den, dM);
3293 60535 : if (!K) return gc_NULL(av);
3294 60528 : if (den && !equali1(den)) K = ZM_Q_mul(K, ginv(den));
3295 60515 : return gerepileupto(av, K);
3296 : }
3297 :
3298 : static GEN
3299 105218 : ZM_ker_filter(GEN A, GEN P)
3300 : {
3301 105218 : long i, j, l = lg(A), n = 1, d = lg(gmael(A,1,1));
3302 105218 : GEN B, Q, D = gmael(A,1,2);
3303 215177 : for (i=2; i<l; i++)
3304 : {
3305 109959 : GEN Di = gmael(A,i,2);
3306 109959 : long di = lg(gmael(A,i,1));
3307 109959 : int c = vecsmall_lexcmp(D, Di);
3308 109959 : if (di==d && c==0) n++;
3309 45588 : else if (d > di || (di==d && c>0))
3310 37680 : { n = 1; d = di; D = Di; }
3311 : }
3312 105218 : B = cgetg(n+1, t_VEC);
3313 105219 : Q = cgetg(n+1, typ(P));
3314 320394 : for (i=1, j=1; i<l; i++)
3315 : {
3316 215176 : if (lg(gmael(A,i,1))==d && vecsmall_lexcmp(D, gmael(A,i,2))==0)
3317 : {
3318 169587 : gel(B,j) = gmael(A,i,1);
3319 169587 : Q[j] = P[i];
3320 169587 : j++;
3321 : }
3322 : }
3323 105218 : return mkvec3(B,Q,D);
3324 : }
3325 :
3326 : static GEN
3327 69555 : ZM_ker_chinese(GEN A, GEN P, GEN *mod)
3328 : {
3329 69555 : GEN BQD = ZM_ker_filter(A, P);
3330 69555 : return mkvec2(nmV_chinese_center(gel(BQD,1), gel(BQD,2), mod), gel(BQD,3));
3331 : }
3332 :
3333 : static GEN
3334 129629 : ZM_ker_slice(GEN A, GEN P, GEN *mod)
3335 : {
3336 129629 : pari_sp av = avma;
3337 129629 : long i, n = lg(P)-1;
3338 : GEN BQD, D, H, T, Q;
3339 129629 : if (n == 1)
3340 : {
3341 93962 : ulong p = uel(P,1);
3342 93962 : GEN K = Flm_ker_sp(ZM_to_Flm(A, p), p, 2);
3343 93962 : *mod = utoipos(p); return mkvec2(Flm_to_ZM(gel(K,1)), gel(K,2));
3344 : }
3345 35667 : T = ZV_producttree(P);
3346 35666 : A = ZM_nv_mod_tree(A, P, T);
3347 35667 : H = cgetg(n+1, t_VEC);
3348 111498 : for(i=1 ; i <= n; i++)
3349 75835 : gel(H,i) = Flm_ker_sp(gel(A, i), P[i], 2);
3350 35663 : BQD = ZM_ker_filter(H, P); Q = gel(BQD,2);
3351 35663 : if (lg(Q) != lg(P)) T = ZV_producttree(Q);
3352 35663 : H = nmV_chinese_center_tree_seq(gel(BQD,1), Q, T, ZV_chinesetree(Q,T));
3353 35665 : *mod = gmael(T, lg(T)-1, 1);
3354 35665 : D = gel(BQD, 3);
3355 35665 : gerepileall(av, 3, &H, &D, mod);
3356 35664 : return mkvec2(H,D);
3357 : }
3358 :
3359 : GEN
3360 129629 : ZM_ker_worker(GEN P, GEN A)
3361 : {
3362 129629 : GEN V = cgetg(3, t_VEC);
3363 129629 : gel(V,1) = ZM_ker_slice(A, P, &gel(V,2));
3364 129626 : return V;
3365 : }
3366 :
3367 : /* assume lg(A) > 1 */
3368 : static GEN
3369 62891 : ZM_ker_i(GEN A)
3370 : {
3371 : pari_sp av;
3372 62891 : long k, m = lg(A)-1;
3373 62891 : GEN HD = NULL, mod = gen_1, worker;
3374 : forprime_t S;
3375 :
3376 62891 : if (m >= 2*nbrows(A))
3377 : {
3378 3052 : GEN v = ZM_indexrank(A), y = gel(v,2), z = indexcompl(y, m);
3379 : GEN B, A1, A1i, d;
3380 3052 : A = rowpermute(A, gel(v,1)); /* same kernel */
3381 3052 : A1 = vecpermute(A, y); /* maximal rank submatrix */
3382 3052 : B = vecpermute(A, z);
3383 3052 : A1i = ZM_inv(A1, &d);
3384 3052 : if (!d) d = gen_1;
3385 3052 : B = vconcat(ZM_mul(ZM_neg(A1i), B), scalarmat_shallow(d, lg(B)-1));
3386 3052 : if (!gequal(y, identity_perm(lg(y)-1)))
3387 657 : B = rowpermute(B, perm_inv(shallowconcat(y,z)));
3388 3052 : return vec_Q_primpart(B);
3389 : }
3390 59839 : init_modular_big(&S);
3391 59839 : worker = snm_closure(is_entry("_ZM_ker_worker"), mkvec(A));
3392 59839 : av = avma;
3393 59839 : for (k = 1;; k <<= 1)
3394 65342 : {
3395 : pari_timer ti;
3396 : GEN H, Hr;
3397 125181 : gen_inccrt_i("ZM_ker", worker, NULL, (k+1)>>1, 0,
3398 : &S, &HD, &mod, ZM_ker_chinese, NULL);
3399 125181 : gerepileall(av, 2, &HD, &mod);
3400 140724 : H = gel(HD, 1); if (lg(H) == 1) return H;
3401 80885 : if (DEBUGLEVEL >= 4) timer_start(&ti);
3402 80885 : Hr = FpM_ratlift_parallel(H, mod, NULL);
3403 80885 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"ZM_ker: ratlift (%ld)",!!Hr);
3404 80885 : if (Hr)
3405 : {
3406 : GEN MH;
3407 70141 : Hr = vec_Q_primpart(Hr);
3408 70141 : MH = ZM_mul(A, Hr);
3409 70141 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"ZM_ker: QM_mul");
3410 70141 : if (ZM_equal0(MH)) return Hr;
3411 : }
3412 : }
3413 : }
3414 :
3415 : GEN
3416 47141 : ZM_ker(GEN M)
3417 : {
3418 47141 : pari_sp av = avma;
3419 47141 : long l = lg(M)-1;
3420 47141 : if (l==0) return cgetg(1, t_MAT);
3421 47141 : if (lgcols(M)==1) return matid(l);
3422 47141 : return gerepilecopy(av, ZM_ker_i(M));
3423 : }
3424 :
3425 : GEN
3426 16569 : QM_ker(GEN M)
3427 : {
3428 16569 : pari_sp av = avma;
3429 16569 : long l = lg(M)-1;
3430 16569 : if (l==0) return cgetg(1, t_MAT);
3431 16527 : if (lgcols(M)==1) return matid(l);
3432 15666 : return gerepilecopy(av, ZM_ker_i(row_Q_primpart(M)));
3433 : }
3434 :
3435 : /* x a ZM. Return a multiple of the determinant of the lattice generated by
3436 : * the columns of x. From Algorithm 2.2.6 in GTM138 */
3437 : GEN
3438 47549 : detint(GEN A)
3439 : {
3440 47549 : if (typ(A) != t_MAT) pari_err_TYPE("detint",A);
3441 47549 : RgM_check_ZM(A, "detint");
3442 47549 : return ZM_detmult(A);
3443 : }
3444 : GEN
3445 155070 : ZM_detmult(GEN A)
3446 : {
3447 155070 : pari_sp av1, av = avma;
3448 : GEN B, c, v, piv;
3449 155070 : long rg, i, j, k, m, n = lg(A) - 1;
3450 :
3451 155070 : if (!n) return gen_1;
3452 155070 : m = nbrows(A);
3453 155070 : if (n < m) return gen_0;
3454 154993 : c = zero_zv(m);
3455 154993 : av1 = avma;
3456 154993 : B = zeromatcopy(m,m);
3457 154993 : v = cgetg(m+1, t_COL);
3458 154993 : piv = gen_1; rg = 0;
3459 680643 : for (k=1; k<=n; k++)
3460 : {
3461 680629 : GEN pivprec = piv;
3462 680629 : long t = 0;
3463 4908045 : for (i=1; i<=m; i++)
3464 : {
3465 4227417 : pari_sp av2 = avma;
3466 : GEN vi;
3467 4227417 : if (c[i]) continue;
3468 :
3469 2454271 : vi = mulii(piv, gcoeff(A,i,k));
3470 22463686 : for (j=1; j<=m; j++)
3471 20009375 : if (c[j]) vi = addii(vi, mulii(gcoeff(B,j,i),gcoeff(A,j,k)));
3472 2454311 : if (!t && signe(vi)) t = i;
3473 2454311 : gel(v,i) = gerepileuptoint(av2, vi);
3474 : }
3475 680628 : if (!t) continue;
3476 : /* at this point c[t] = 0 */
3477 :
3478 680537 : if (++rg >= m) { /* full rank; mostly done */
3479 154978 : GEN det = gel(v,t); /* last on stack */
3480 154978 : if (++k > n)
3481 154839 : det = absi(det);
3482 : else
3483 : {
3484 : /* improve further; at this point c[i] is set for all i != t */
3485 139 : gcoeff(B,t,t) = piv; v = centermod(gel(B,t), det);
3486 446 : for ( ; k<=n; k++)
3487 307 : det = gcdii(det, ZV_dotproduct(v, gel(A,k)));
3488 : }
3489 154978 : return gerepileuptoint(av, det);
3490 : }
3491 :
3492 525559 : piv = gel(v,t);
3493 4071946 : for (i=1; i<=m; i++)
3494 : {
3495 : GEN mvi;
3496 3546388 : if (c[i] || i == t) continue;
3497 :
3498 1773195 : gcoeff(B,t,i) = mvi = negi(gel(v,i));
3499 17551825 : for (j=1; j<=m; j++)
3500 15778631 : if (c[j]) /* implies j != t */
3501 : {
3502 4077414 : pari_sp av2 = avma;
3503 4077414 : GEN z = addii(mulii(gcoeff(B,j,i), piv), mulii(gcoeff(B,j,t), mvi));
3504 4077413 : if (rg > 1) z = diviiexact(z, pivprec);
3505 4077412 : gcoeff(B,j,i) = gerepileuptoint(av2, z);
3506 : }
3507 : }
3508 525558 : c[t] = k;
3509 525558 : if (gc_needed(av,1))
3510 : {
3511 0 : if(DEBUGMEM>1) pari_warn(warnmem,"detint. k=%ld",k);
3512 0 : gerepileall(av1, 2, &piv,&B); v = zerovec(m);
3513 : }
3514 : }
3515 14 : return gc_const(av, gen_0);
3516 : }
3517 :
3518 : /* Reduce x modulo (invertible) y */
3519 : GEN
3520 13468 : closemodinvertible(GEN x, GEN y)
3521 : {
3522 13468 : return gmul(y, ground(RgM_solve(y,x)));
3523 : }
3524 : GEN
3525 7 : reducemodinvertible(GEN x, GEN y)
3526 : {
3527 7 : return gsub(x, closemodinvertible(x,y));
3528 : }
3529 : GEN
3530 0 : reducemodlll(GEN x,GEN y)
3531 : {
3532 0 : return reducemodinvertible(x, ZM_lll(y, 0.75, LLL_INPLACE));
3533 : }
3534 :
3535 : /*******************************************************************/
3536 : /* */
3537 : /* KERNEL of an m x n matrix */
3538 : /* return n - rk(x) linearly independent vectors */
3539 : /* */
3540 : /*******************************************************************/
3541 : static GEN
3542 28 : RgM_deplin_i(GEN x0)
3543 : {
3544 28 : pari_sp av = avma, av2;
3545 28 : long i, j, k, nl, nc = lg(x0)-1;
3546 : GEN D, x, y, c, l, d, ck;
3547 :
3548 28 : if (!nc) return NULL;
3549 28 : nl = nbrows(x0);
3550 28 : c = zero_zv(nl);
3551 28 : l = cgetg(nc+1, t_VECSMALL); /* not initialized */
3552 28 : av2 = avma;
3553 28 : x = RgM_shallowcopy(x0);
3554 28 : d = const_vec(nl, gen_1); /* pivot list */
3555 28 : ck = NULL; /* gcc -Wall */
3556 98 : for (k=1; k<=nc; k++)
3557 : {
3558 91 : ck = gel(x,k);
3559 196 : for (j=1; j<k; j++)
3560 : {
3561 105 : GEN cj = gel(x,j), piv = gel(d,j), q = gel(ck,l[j]);
3562 420 : for (i=1; i<=nl; i++)
3563 315 : if (i!=l[j]) gel(ck,i) = gsub(gmul(piv, gel(ck,i)), gmul(q, gel(cj,i)));
3564 : }
3565 :
3566 91 : i = gauss_get_pivot_NZ(x, NULL, k, c);
3567 91 : if (i > nl) break;
3568 70 : if (gc_needed(av,1))
3569 : {
3570 0 : if (DEBUGMEM>1) pari_warn(warnmem,"deplin k = %ld/%ld",k,nc);
3571 0 : gerepileall(av2, 2, &x, &d);
3572 0 : ck = gel(x,k);
3573 : }
3574 70 : gel(d,k) = gel(ck,i);
3575 70 : c[i] = k; l[k] = i; /* pivot d[k] in x[i,k] */
3576 : }
3577 28 : if (k > nc) return gc_NULL(av);
3578 21 : if (k == 1) { set_avma(av); return scalarcol_shallow(gen_1,nc); }
3579 21 : y = cgetg(nc+1,t_COL);
3580 21 : gel(y,1) = gcopy(gel(ck, l[1]));
3581 49 : for (D=gel(d,1),j=2; j<k; j++)
3582 : {
3583 28 : gel(y,j) = gmul(gel(ck, l[j]), D);
3584 28 : D = gmul(D, gel(d,j));
3585 : }
3586 21 : gel(y,j) = gneg(D);
3587 21 : for (j++; j<=nc; j++) gel(y,j) = gen_0;
3588 21 : y = primitive_part(y, &c);
3589 21 : return c? gerepileupto(av, y): gerepilecopy(av, y);
3590 : }
3591 : static GEN
3592 0 : RgV_deplin(GEN v)
3593 : {
3594 0 : pari_sp av = avma;
3595 0 : long n = lg(v)-1;
3596 0 : GEN y, p = NULL;
3597 0 : if (n <= 1)
3598 : {
3599 0 : if (n == 1 && gequal0(gel(v,1))) return mkcol(gen_1);
3600 0 : return cgetg(1, t_COL);
3601 : }
3602 0 : if (gequal0(gel(v,1))) return scalarcol_shallow(gen_1, n);
3603 0 : v = primpart(mkvec2(gel(v,1),gel(v,2)));
3604 0 : if (RgV_is_FpV(v, &p) && p) v = centerlift(v);
3605 0 : y = zerocol(n);
3606 0 : gel(y,1) = gneg(gel(v,2));
3607 0 : gel(y,2) = gcopy(gel(v,1));
3608 0 : return gerepileupto(av, y);
3609 :
3610 : }
3611 :
3612 : static GEN
3613 105 : RgM_deplin_FpM(GEN x, GEN p)
3614 : {
3615 105 : pari_sp av = avma;
3616 : ulong pp;
3617 105 : x = RgM_Fp_init3(x, p, &pp);
3618 105 : switch(pp)
3619 : {
3620 35 : case 0:
3621 35 : x = FpM_ker_gen(x,p,1);
3622 35 : if (!x) return gc_NULL(av);
3623 21 : x = FpC_center(x,p,shifti(p,-1));
3624 21 : break;
3625 14 : case 2:
3626 14 : x = F2m_ker_sp(x,1);
3627 14 : if (!x) return gc_NULL(av);
3628 7 : x = F2c_to_ZC(x); break;
3629 0 : case 3:
3630 0 : x = F3m_ker_sp(x,1);
3631 0 : if (!x) return gc_NULL(av);
3632 0 : x = F3c_to_ZC(x); break;
3633 56 : default:
3634 56 : x = Flm_ker_sp(x,pp,1);
3635 56 : if (!x) return gc_NULL(av);
3636 35 : x = Flv_center(x, pp, pp>>1);
3637 35 : x = zc_to_ZC(x);
3638 35 : break;
3639 : }
3640 63 : return gerepileupto(av, x);
3641 : }
3642 :
3643 : /* FIXME: implement direct modular ZM_deplin ? */
3644 : static GEN
3645 119 : QM_deplin(GEN M)
3646 : {
3647 119 : pari_sp av = avma;
3648 119 : long l = lg(M)-1;
3649 : GEN k;
3650 119 : if (l==0) return NULL;
3651 84 : if (lgcols(M)==1) return col_ei(l, 1);
3652 84 : k = ZM_ker_i(row_Q_primpart(M));
3653 84 : if (lg(k)== 1) return gc_NULL(av);
3654 70 : return gerepilecopy(av, gel(k,1));
3655 : }
3656 :
3657 : static GEN
3658 42 : RgM_deplin_FqM(GEN x, GEN pol, GEN p)
3659 : {
3660 42 : pari_sp av = avma;
3661 42 : GEN b, T = RgX_to_FpX(pol, p);
3662 42 : if (signe(T) == 0) pari_err_OP("deplin",x,pol);
3663 42 : b = FqM_deplin(RgM_to_FqM(x, T, p), T, p);
3664 42 : return gerepileupto(av, b);
3665 : }
3666 :
3667 : #define code(t1,t2) ((t1 << 6) | t2)
3668 : static GEN
3669 378 : RgM_deplin_fast(GEN x)
3670 : {
3671 : GEN p, pol;
3672 : long pa;
3673 378 : long t = RgM_type(x, &p,&pol,&pa);
3674 378 : switch(t)
3675 : {
3676 119 : case t_INT: /* fall through */
3677 119 : case t_FRAC: return QM_deplin(x);
3678 84 : case t_FFELT: return FFM_deplin(x, pol);
3679 105 : case t_INTMOD: return RgM_deplin_FpM(x, p);
3680 42 : case code(t_POLMOD, t_INTMOD):
3681 42 : return RgM_deplin_FqM(x, pol, p);
3682 28 : default: return gen_0;
3683 : }
3684 : }
3685 : #undef code
3686 :
3687 : static GEN
3688 378 : RgM_deplin(GEN x)
3689 : {
3690 378 : GEN z = RgM_deplin_fast(x);
3691 378 : if (z!= gen_0) return z;
3692 28 : return RgM_deplin_i(x);
3693 : }
3694 :
3695 : GEN
3696 378 : deplin(GEN x)
3697 : {
3698 378 : switch(typ(x))
3699 : {
3700 378 : case t_MAT:
3701 : {
3702 378 : GEN z = RgM_deplin(x);
3703 378 : if (z) return z;
3704 140 : return cgetg(1, t_COL);
3705 : }
3706 0 : case t_VEC: return RgV_deplin(x);
3707 0 : default: pari_err_TYPE("deplin",x);
3708 : }
3709 : return NULL;/*LCOV_EXCL_LINE*/
3710 : }
3711 :
3712 : /*******************************************************************/
3713 : /* */
3714 : /* GAUSS REDUCTION OF MATRICES (m lines x n cols) */
3715 : /* (kernel, image, complementary image, rank) */
3716 : /* */
3717 : /*******************************************************************/
3718 : /* return the transform of x under a standard Gauss pivot.
3719 : * x0 is a reference point when guessing whether x[i,j] ~ 0
3720 : * (iff x[i,j] << x0[i,j])
3721 : * Set r = dim ker(x). d[k] contains the index of the first nonzero pivot
3722 : * in column k */
3723 : static GEN
3724 1056 : gauss_pivot_ker(GEN x, GEN x0, GEN *dd, long *rr)
3725 : {
3726 : GEN c, d, p, data;
3727 : pari_sp av;
3728 : long i, j, k, r, t, n, m;
3729 : pivot_fun pivot;
3730 :
3731 1056 : n=lg(x)-1; if (!n) { *dd=NULL; *rr=0; return cgetg(1,t_MAT); }
3732 1056 : m=nbrows(x); r=0;
3733 1056 : pivot = get_pivot_fun(x, x0, &data);
3734 1056 : x = RgM_shallowcopy(x);
3735 1056 : c = zero_zv(m);
3736 1056 : d = cgetg(n+1,t_VECSMALL);
3737 1056 : av=avma;
3738 6260 : for (k=1; k<=n; k++)
3739 : {
3740 5204 : j = pivot(x, data, k, c);
3741 5204 : if (j > m)
3742 : {
3743 1113 : r++; d[k]=0;
3744 5068 : for(j=1; j<k; j++)
3745 3955 : if (d[j]) gcoeff(x,d[j],k) = gclone(gcoeff(x,d[j],k));
3746 : }
3747 : else
3748 : { /* pivot for column k on row j */
3749 4091 : c[j]=k; d[k]=j; p = gdiv(gen_m1,gcoeff(x,j,k));
3750 4091 : gcoeff(x,j,k) = gen_m1;
3751 : /* x[j,] /= - x[j,k] */
3752 21907 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = gmul(p,gcoeff(x,j,i));
3753 39026 : for (t=1; t<=m; t++)
3754 34935 : if (t!=j)
3755 : { /* x[t,] -= 1 / x[j,k] x[j,] */
3756 30844 : p = gcoeff(x,t,k); gcoeff(x,t,k) = gen_0;
3757 30844 : if (gequal0(p)) continue;
3758 81212 : for (i=k+1; i<=n; i++)
3759 65446 : gcoeff(x,t,i) = gadd(gcoeff(x,t,i),gmul(p,gcoeff(x,j,i)));
3760 15766 : if (gc_needed(av,1)) gerepile_gauss_ker(x,k,t,av);
3761 : }
3762 : }
3763 : }
3764 1056 : *dd=d; *rr=r; return x;
3765 : }
3766 :
3767 : /* r = dim ker(x).
3768 : * Returns d:
3769 : * d[k] != 0 contains the index of a nonzero pivot in column k
3770 : * d[k] == 0 if column k is a linear combination of the (k-1) first ones */
3771 : GEN
3772 154120 : RgM_pivots(GEN x0, GEN data, long *rr, pivot_fun pivot)
3773 : {
3774 : GEN x, c, d, p;
3775 154120 : long i, j, k, r, t, m, n = lg(x0)-1;
3776 : pari_sp av;
3777 :
3778 154120 : if (RgM_is_ZM(x0)) return ZM_pivots(x0, rr);
3779 135934 : if (!n) { *rr = 0; return NULL; }
3780 :
3781 135934 : d = cgetg(n+1, t_VECSMALL);
3782 135934 : x = RgM_shallowcopy(x0);
3783 135934 : m = nbrows(x); r = 0;
3784 135934 : c = zero_zv(m);
3785 135960 : av = avma;
3786 3523459 : for (k=1; k<=n; k++)
3787 : {
3788 3387524 : j = pivot(x, data, k, c);
3789 3387530 : if (j > m) { r++; d[k] = 0; }
3790 : else
3791 : {
3792 273059 : c[j] = k; d[k] = j; p = gdiv(gen_m1, gcoeff(x,j,k));
3793 10062321 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = gmul(p,gcoeff(x,j,i));
3794 :
3795 1096388 : for (t=1; t<=m; t++)
3796 823360 : if (!c[t]) /* no pivot on that line yet */
3797 : {
3798 315532 : p = gcoeff(x,t,k); gcoeff(x,t,k) = gen_0;
3799 22365244 : for (i=k+1; i<=n; i++)
3800 22049712 : gcoeff(x,t,i) = gadd(gcoeff(x,t,i), gmul(p, gcoeff(x,j,i)));
3801 315532 : if (gc_needed(av,1)) gerepile_gauss(x,k,t,av,j,c);
3802 : }
3803 10335532 : for (i=k; i<=n; i++) gcoeff(x,j,i) = gen_0; /* dummy */
3804 : }
3805 : }
3806 135935 : *rr = r; return gc_const((pari_sp)d, d);
3807 : }
3808 :
3809 : static long
3810 356424 : ZM_count_0_cols(GEN M)
3811 : {
3812 356424 : long i, l = lg(M), n = 0;
3813 1699649 : for (i = 1; i < l; i++)
3814 1343224 : if (ZV_equal0(gel(M,i))) n++;
3815 356425 : return n;
3816 : }
3817 :
3818 : static void indexrank_all(long m, long n, long r, GEN d, GEN *prow, GEN *pcol);
3819 : /* As RgM_pivots, integer entries. Set *rr = dim Ker M0 */
3820 : GEN
3821 379750 : ZM_pivots(GEN M0, long *rr)
3822 : {
3823 379750 : GEN d, dbest = NULL;
3824 : long m, mm, n, nn, i, imax, rmin, rbest, zc;
3825 379750 : int beenthere = 0;
3826 379750 : pari_sp av, av0 = avma;
3827 : forprime_t S;
3828 :
3829 379750 : rbest = n = lg(M0)-1;
3830 379750 : if (n == 0) { *rr = 0; return NULL; }
3831 356424 : zc = ZM_count_0_cols(M0);
3832 356424 : if (n == zc) { *rr = zc; return zero_zv(n); }
3833 :
3834 356144 : m = nbrows(M0);
3835 356141 : rmin = maxss(zc, n-m);
3836 356140 : init_modular_small(&S);
3837 356141 : if (n <= m) { nn = n; mm = m; } else { nn = m; mm = n; }
3838 356141 : imax = (nn < 16)? 1: (nn < 64)? 2: 3; /* heuristic */
3839 :
3840 : for(;;)
3841 0 : {
3842 : GEN row, col, M, KM, IM, RHS, X, cX;
3843 : long rk;
3844 379786 : for (av = avma, i = 0;; set_avma(av), i++)
3845 23645 : {
3846 379786 : ulong p = u_forprime_next(&S);
3847 : long rp;
3848 379789 : if (!p) pari_err_OVERFLOW("ZM_pivots [ran out of primes]");
3849 379789 : d = Flm_pivots(ZM_to_Flm(M0, p), p, &rp, 1);
3850 379790 : if (rp == rmin) { rbest = rp; goto END; } /* maximal rank, return */
3851 45776 : if (rp < rbest) { /* save best r so far */
3852 22133 : rbest = rp;
3853 22133 : guncloneNULL(dbest);
3854 22133 : dbest = gclone(d);
3855 22133 : if (beenthere) break;
3856 : }
3857 45776 : if (!beenthere && i >= imax) break;
3858 : }
3859 22131 : beenthere = 1;
3860 : /* Dubious case: there is (probably) a non trivial kernel */
3861 22131 : indexrank_all(m,n, rbest, dbest, &row, &col);
3862 22131 : M = rowpermute(vecpermute(M0, col), row);
3863 22131 : rk = n - rbest; /* (probable) dimension of image */
3864 22131 : if (n > m) M = shallowtrans(M);
3865 22131 : IM = vecslice(M,1,rk);
3866 22131 : KM = vecslice(M,rk+1, nn);
3867 22131 : M = rowslice(IM, 1,rk); /* square maximal rank */
3868 22131 : X = ZM_gauss(M, rowslice(KM, 1,rk));
3869 22131 : RHS = rowslice(KM,rk+1,mm);
3870 22131 : M = rowslice(IM,rk+1,mm);
3871 22131 : X = Q_remove_denom(X, &cX);
3872 22131 : if (cX) RHS = ZM_Z_mul(RHS, cX);
3873 22131 : if (ZM_equal(ZM_mul(M, X), RHS)) { d = vecsmall_copy(dbest); goto END; }
3874 0 : set_avma(av);
3875 : }
3876 356145 : END:
3877 356145 : *rr = rbest; guncloneNULL(dbest);
3878 356144 : return gerepileuptoleaf(av0, d);
3879 : }
3880 :
3881 : /* set *pr = dim Ker x */
3882 : static GEN
3883 59093 : gauss_pivot(GEN x, long *pr) {
3884 : GEN data;
3885 59093 : pivot_fun pivot = get_pivot_fun(x, x, &data);
3886 59093 : return RgM_pivots(x, data, pr, pivot);
3887 : }
3888 :
3889 : /* compute ker(x), x0 is a reference point when guessing whether x[i,j] ~ 0
3890 : * (iff x[i,j] << x0[i,j]) */
3891 : static GEN
3892 1056 : ker_aux(GEN x, GEN x0)
3893 : {
3894 1056 : pari_sp av = avma;
3895 : GEN d,y;
3896 : long i,j,k,r,n;
3897 :
3898 1056 : x = gauss_pivot_ker(x,x0,&d,&r);
3899 1056 : if (!r) { set_avma(av); return cgetg(1,t_MAT); }
3900 1015 : n = lg(x)-1; y=cgetg(r+1,t_MAT);
3901 2128 : for (j=k=1; j<=r; j++,k++)
3902 : {
3903 1113 : GEN p = cgetg(n+1,t_COL);
3904 :
3905 4704 : gel(y,j) = p; while (d[k]) k++;
3906 5068 : for (i=1; i<k; i++)
3907 3955 : if (d[i])
3908 : {
3909 3773 : GEN p1=gcoeff(x,d[i],k);
3910 3773 : gel(p,i) = gcopy(p1); gunclone(p1);
3911 : }
3912 : else
3913 182 : gel(p,i) = gen_0;
3914 1883 : gel(p,k) = gen_1; for (i=k+1; i<=n; i++) gel(p,i) = gen_0;
3915 : }
3916 1015 : return gerepileupto(av,y);
3917 : }
3918 :
3919 : static GEN
3920 84 : RgM_ker_FpM(GEN x, GEN p)
3921 : {
3922 84 : pari_sp av = avma;
3923 : ulong pp;
3924 84 : x = RgM_Fp_init3(x, p, &pp);
3925 84 : switch(pp)
3926 : {
3927 35 : case 0: x = FpM_to_mod(FpM_ker_gen(x,p,0),p); break;
3928 7 : case 2: x = F2m_to_mod(F2m_ker_sp(x,0)); break;
3929 7 : case 3: x = F3m_to_mod(F3m_ker_sp(x,0)); break;
3930 35 : default:x = Flm_to_mod(Flm_ker_sp(x,pp,0), pp); break;
3931 : }
3932 84 : return gerepileupto(av, x);
3933 : }
3934 :
3935 : static GEN
3936 91 : RgM_ker_FqM(GEN x, GEN pol, GEN p)
3937 : {
3938 91 : pari_sp av = avma;
3939 91 : GEN b, T = RgX_to_FpX(pol, p);
3940 91 : if (signe(T) == 0) pari_err_OP("ker",x,pol);
3941 84 : b = FqM_ker(RgM_to_FqM(x, T, p), T, p);
3942 84 : return gerepileupto(av, FqM_to_mod(b, T, p));
3943 : }
3944 :
3945 : #define code(t1,t2) ((t1 << 6) | t2)
3946 : static GEN
3947 8484 : RgM_ker_fast(GEN x)
3948 : {
3949 : GEN p, pol;
3950 : long pa;
3951 8484 : long t = RgM_type(x, &p,&pol,&pa);
3952 8484 : switch(t)
3953 : {
3954 7462 : case t_INT: /* fall through */
3955 7462 : case t_FRAC: return QM_ker(x);
3956 77 : case t_FFELT: return FFM_ker(x, pol);
3957 84 : case t_INTMOD: return RgM_ker_FpM(x, p);
3958 91 : case code(t_POLMOD, t_INTMOD):
3959 91 : return RgM_ker_FqM(x, pol, p);
3960 770 : default: return NULL;
3961 : }
3962 : }
3963 : #undef code
3964 :
3965 : GEN
3966 8484 : ker(GEN x)
3967 : {
3968 8484 : GEN b = RgM_ker_fast(x);
3969 8477 : if (b) return b;
3970 770 : return ker_aux(x,x);
3971 : }
3972 :
3973 : GEN
3974 46221 : matker0(GEN x,long flag)
3975 : {
3976 46221 : if (typ(x)!=t_MAT) pari_err_TYPE("matker",x);
3977 46221 : if (!flag) return ker(x);
3978 45934 : RgM_check_ZM(x, "matker");
3979 45934 : return ZM_ker(x);
3980 : }
3981 :
3982 : static GEN
3983 63 : RgM_image_FpM(GEN x, GEN p)
3984 : {
3985 63 : pari_sp av = avma;
3986 : ulong pp;
3987 63 : x = RgM_Fp_init(x, p, &pp);
3988 63 : switch(pp)
3989 : {
3990 28 : case 0: x = FpM_to_mod(FpM_image(x,p),p); break;
3991 7 : case 2: x = F2m_to_mod(F2m_image(x)); break;
3992 28 : default:x = Flm_to_mod(Flm_image(x,pp), pp); break;
3993 : }
3994 63 : return gerepileupto(av, x);
3995 : }
3996 :
3997 : static GEN
3998 35 : RgM_image_FqM(GEN x, GEN pol, GEN p)
3999 : {
4000 35 : pari_sp av = avma;
4001 35 : GEN b, T = RgX_to_FpX(pol, p);
4002 35 : if (signe(T) == 0) pari_err_OP("image",x,pol);
4003 28 : b = FqM_image(RgM_to_FqM(x, T, p), T, p);
4004 28 : return gerepileupto(av, FqM_to_mod(b, T, p));
4005 : }
4006 :
4007 : GEN
4008 5971 : QM_image_shallow(GEN A)
4009 : {
4010 5971 : A = vec_Q_primpart(A);
4011 5971 : return vecpermute(A, ZM_indeximage(A));
4012 : }
4013 : GEN
4014 5201 : QM_image(GEN A)
4015 : {
4016 5201 : pari_sp av = avma;
4017 5201 : return gerepilecopy(av, QM_image_shallow(A));
4018 : }
4019 :
4020 : #define code(t1,t2) ((t1 << 6) | t2)
4021 : static GEN
4022 5362 : RgM_image_fast(GEN x)
4023 : {
4024 : GEN p, pol;
4025 : long pa;
4026 5362 : long t = RgM_type(x, &p,&pol,&pa);
4027 5362 : switch(t)
4028 : {
4029 5201 : case t_INT: /* fall through */
4030 5201 : case t_FRAC: return QM_image(x);
4031 49 : case t_FFELT: return FFM_image(x, pol);
4032 63 : case t_INTMOD: return RgM_image_FpM(x, p);
4033 35 : case code(t_POLMOD, t_INTMOD):
4034 35 : return RgM_image_FqM(x, pol, p);
4035 14 : default: return NULL;
4036 : }
4037 : }
4038 : #undef code
4039 :
4040 : GEN
4041 5362 : image(GEN x)
4042 : {
4043 : GEN d, M;
4044 : long r;
4045 :
4046 5362 : if (typ(x)!=t_MAT) pari_err_TYPE("matimage",x);
4047 5362 : M = RgM_image_fast(x);
4048 5355 : if (M) return M;
4049 14 : d = gauss_pivot(x,&r); /* d left on stack for efficiency */
4050 14 : return image_from_pivot(x,d,r);
4051 : }
4052 :
4053 : static GEN
4054 84 : imagecompl_aux(GEN x, GEN(*PIVOT)(GEN,long*))
4055 : {
4056 84 : pari_sp av = avma;
4057 : GEN d,y;
4058 : long j,i,r;
4059 :
4060 84 : if (typ(x)!=t_MAT) pari_err_TYPE("imagecompl",x);
4061 84 : (void)new_chunk(lg(x) * 4 + 1); /* HACK */
4062 84 : d = PIVOT(x,&r); /* if (!d) then r = 0 */
4063 84 : set_avma(av); y = cgetg(r+1,t_VECSMALL);
4064 126 : for (i=j=1; j<=r; i++)
4065 42 : if (!d[i]) y[j++] = i;
4066 84 : return y;
4067 : }
4068 : GEN
4069 84 : imagecompl(GEN x) { return imagecompl_aux(x, &gauss_pivot); }
4070 : GEN
4071 0 : ZM_imagecompl(GEN x) { return imagecompl_aux(x, &ZM_pivots); }
4072 :
4073 : static GEN
4074 28 : RgM_RgC_invimage_FpC(GEN A, GEN y, GEN p)
4075 : {
4076 28 : pari_sp av = avma;
4077 : ulong pp;
4078 : GEN x;
4079 28 : A = RgM_Fp_init(A,p,&pp);
4080 28 : switch(pp)
4081 : {
4082 7 : case 0:
4083 7 : y = RgC_to_FpC(y,p);
4084 7 : x = FpM_FpC_invimage(A, y, p);
4085 7 : return x ? gerepileupto(av, FpC_to_mod(x,p)): NULL;
4086 7 : case 2:
4087 7 : y = RgV_to_F2v(y);
4088 7 : x = F2m_F2c_invimage(A, y);
4089 7 : return x ? gerepileupto(av, F2c_to_mod(x)): NULL;
4090 14 : default:
4091 14 : y = RgV_to_Flv(y,pp);
4092 14 : x = Flm_Flc_invimage(A, y, pp);
4093 14 : return x ? gerepileupto(av, Flc_to_mod(x,pp)): NULL;
4094 : }
4095 : }
4096 :
4097 : static GEN
4098 2149 : RgM_RgC_invimage_fast(GEN x, GEN y)
4099 : {
4100 : GEN p, pol;
4101 : long pa;
4102 2149 : long t = RgM_RgC_type(x, y, &p,&pol,&pa);
4103 2149 : switch(t)
4104 : {
4105 28 : case t_INTMOD: return RgM_RgC_invimage_FpC(x, y, p);
4106 63 : case t_FFELT: return FFM_FFC_invimage(x, y, pol);
4107 2058 : default: return gen_0;
4108 : }
4109 : }
4110 :
4111 : GEN
4112 2254 : RgM_RgC_invimage(GEN A, GEN y)
4113 : {
4114 2254 : pari_sp av = avma;
4115 2254 : long i, l = lg(A);
4116 : GEN M, x, t;
4117 2254 : if (l==1) return NULL;
4118 2149 : if (lg(y) != lgcols(A)) pari_err_DIM("inverseimage");
4119 2149 : M = RgM_RgC_invimage_fast(A, y);
4120 2149 : if (!M) return gc_NULL(av);
4121 2128 : if (M != gen_0) return M;
4122 2058 : M = ker(shallowconcat(A, y));
4123 2058 : i = lg(M)-1;
4124 2058 : if (!i) return gc_NULL(av);
4125 :
4126 1799 : x = gel(M,i); t = gel(x,l);
4127 1799 : if (gequal0(t)) return gc_NULL(av);
4128 :
4129 1764 : t = gneg_i(t); setlg(x,l);
4130 1764 : return gerepileupto(av, RgC_Rg_div(x, t));
4131 : }
4132 :
4133 : /* Return X such that m X = v (t_COL or t_MAT), resp. an empty t_COL / t_MAT
4134 : * if no solution exist */
4135 : GEN
4136 2415 : inverseimage(GEN m, GEN v)
4137 : {
4138 : GEN y;
4139 2415 : if (typ(m)!=t_MAT) pari_err_TYPE("inverseimage",m);
4140 2415 : switch(typ(v))
4141 : {
4142 2177 : case t_COL:
4143 2177 : y = RgM_RgC_invimage(m,v);
4144 2177 : return y? y: cgetg(1,t_COL);
4145 238 : case t_MAT:
4146 238 : y = RgM_invimage(m, v);
4147 238 : return y? y: cgetg(1,t_MAT);
4148 : }
4149 0 : pari_err_TYPE("inverseimage",v);
4150 : return NULL;/*LCOV_EXCL_LINE*/
4151 : }
4152 :
4153 : static GEN
4154 84 : RgM_invimage_FpM(GEN A, GEN B, GEN p)
4155 : {
4156 84 : pari_sp av = avma;
4157 : ulong pp;
4158 : GEN x;
4159 84 : A = RgM_Fp_init(A,p,&pp);
4160 84 : switch(pp)
4161 : {
4162 35 : case 0:
4163 35 : B = RgM_to_FpM(B,p);
4164 35 : x = FpM_invimage_gen(A, B, p);
4165 35 : return x ? gerepileupto(av, FpM_to_mod(x, p)): x;
4166 7 : case 2:
4167 7 : B = RgM_to_F2m(B);
4168 7 : x = F2m_invimage_i(A, B);
4169 7 : return x ? gerepileupto(av, F2m_to_mod(x)): x;
4170 42 : default:
4171 42 : B = RgM_to_Flm(B,pp);
4172 42 : x = Flm_invimage_i(A, B, pp);
4173 42 : return x ? gerepileupto(av, Flm_to_mod(x, pp)): x;
4174 : }
4175 : }
4176 :
4177 : static GEN
4178 252 : RgM_invimage_fast(GEN x, GEN y)
4179 : {
4180 : GEN p, pol;
4181 : long pa;
4182 252 : long t = RgM_type2(x, y, &p,&pol,&pa);
4183 252 : switch(t)
4184 : {
4185 84 : case t_INTMOD: return RgM_invimage_FpM(x, y, p);
4186 105 : case t_FFELT: return FFM_invimage(x, y, pol);
4187 63 : default: return gen_0;
4188 : }
4189 : }
4190 :
4191 : /* find Z such that A Z = B. Return NULL if no solution */
4192 : GEN
4193 252 : RgM_invimage(GEN A, GEN B)
4194 : {
4195 252 : pari_sp av = avma;
4196 : GEN d, x, X, Y;
4197 252 : long i, j, nY, nA = lg(A)-1, nB = lg(B)-1;
4198 252 : X = RgM_invimage_fast(A, B);
4199 252 : if (!X) return gc_NULL(av);
4200 140 : if (X != gen_0) return X;
4201 63 : x = ker(shallowconcat(RgM_neg(A), B));
4202 : /* AX = BY, Y in strict upper echelon form with pivots = 1.
4203 : * We must find T such that Y T = Id_nB then X T = Z. This exists iff
4204 : * Y has at least nB columns and full rank */
4205 63 : nY = lg(x)-1;
4206 63 : if (nY < nB) return gc_NULL(av);
4207 49 : Y = rowslice(x, nA+1, nA+nB); /* nB rows */
4208 49 : d = cgetg(nB+1, t_VECSMALL);
4209 441 : for (i = nB, j = nY; i >= 1; i--, j--)
4210 : {
4211 546 : for (; j>=1; j--)
4212 532 : if (!gequal0(gcoeff(Y,i,j))) { d[i] = j; break; }
4213 406 : if (!j) return gc_NULL(av);
4214 : }
4215 : /* reduce to the case Y square, upper triangular with 1s on diagonal */
4216 35 : Y = vecpermute(Y, d);
4217 35 : x = vecpermute(x, d);
4218 35 : X = rowslice(x, 1, nA);
4219 35 : return gerepileupto(av, RgM_mul(X, RgM_inv_upper(Y)));
4220 : }
4221 :
4222 : static GEN
4223 70 : RgM_suppl_FpM(GEN x, GEN p)
4224 : {
4225 70 : pari_sp av = avma;
4226 : ulong pp;
4227 70 : x = RgM_Fp_init(x, p, &pp);
4228 70 : switch(pp)
4229 : {
4230 21 : case 0: x = FpM_to_mod(FpM_suppl(x,p), p); break;
4231 14 : case 2: x = F2m_to_mod(F2m_suppl(x)); break;
4232 35 : default:x = Flm_to_mod(Flm_suppl(x,pp), pp); break;
4233 : }
4234 70 : return gerepileupto(av, x);
4235 : }
4236 :
4237 : static GEN
4238 175 : RgM_suppl_fast(GEN x)
4239 : {
4240 : GEN p, pol;
4241 : long pa;
4242 175 : long t = RgM_type(x,&p,&pol,&pa);
4243 175 : switch(t)
4244 : {
4245 70 : case t_INTMOD: return RgM_suppl_FpM(x, p);
4246 35 : case t_FFELT: return FFM_suppl(x, pol);
4247 70 : default: return NULL;
4248 : }
4249 : }
4250 :
4251 : /* x is an n x k matrix, rank(x) = k <= n. Return an invertible n x n matrix
4252 : * whose first k columns are given by x. If rank(x) < k, undefined result. */
4253 : GEN
4254 175 : suppl(GEN x)
4255 : {
4256 175 : pari_sp av = avma;
4257 : GEN d, M;
4258 : long r;
4259 175 : if (typ(x)!=t_MAT) pari_err_TYPE("suppl",x);
4260 175 : M = RgM_suppl_fast(x);
4261 175 : if (M) return M;
4262 70 : init_suppl(x);
4263 70 : d = gauss_pivot(x,&r);
4264 70 : set_avma(av); return get_suppl(x,d,nbrows(x),r,&col_ei);
4265 : }
4266 :
4267 : GEN
4268 7 : image2(GEN x)
4269 : {
4270 7 : pari_sp av = avma;
4271 : long k, n, i;
4272 : GEN A, B;
4273 :
4274 7 : if (typ(x)!=t_MAT) pari_err_TYPE("image2",x);
4275 7 : if (lg(x) == 1) return cgetg(1,t_MAT);
4276 7 : A = ker(x); k = lg(A)-1;
4277 7 : if (!k) { set_avma(av); return gcopy(x); }
4278 7 : A = suppl(A); n = lg(A)-1;
4279 7 : B = cgetg(n-k+1, t_MAT);
4280 21 : for (i = k+1; i <= n; i++) gel(B,i-k) = RgM_RgC_mul(x, gel(A,i));
4281 7 : return gerepileupto(av, B);
4282 : }
4283 :
4284 : GEN
4285 210 : matimage0(GEN x,long flag)
4286 : {
4287 210 : switch(flag)
4288 : {
4289 203 : case 0: return image(x);
4290 7 : case 1: return image2(x);
4291 0 : default: pari_err_FLAG("matimage");
4292 : }
4293 : return NULL; /* LCOV_EXCL_LINE */
4294 : }
4295 :
4296 : static long
4297 126 : RgM_rank_FpM(GEN x, GEN p)
4298 : {
4299 126 : pari_sp av = avma;
4300 : ulong pp;
4301 : long r;
4302 126 : x = RgM_Fp_init(x,p,&pp);
4303 126 : switch(pp)
4304 : {
4305 28 : case 0: r = FpM_rank(x,p); break;
4306 63 : case 2: r = F2m_rank(x); break;
4307 35 : default:r = Flm_rank(x,pp); break;
4308 : }
4309 126 : return gc_long(av, r);
4310 : }
4311 :
4312 : static long
4313 49 : RgM_rank_FqM(GEN x, GEN pol, GEN p)
4314 : {
4315 49 : pari_sp av = avma;
4316 : long r;
4317 49 : GEN T = RgX_to_FpX(pol, p);
4318 49 : if (signe(T) == 0) pari_err_OP("rank",x,pol);
4319 42 : r = FqM_rank(RgM_to_FqM(x, T, p), T, p);
4320 42 : return gc_long(av,r);
4321 : }
4322 :
4323 : #define code(t1,t2) ((t1 << 6) | t2)
4324 : static long
4325 294 : RgM_rank_fast(GEN x)
4326 : {
4327 : GEN p, pol;
4328 : long pa;
4329 294 : long t = RgM_type(x,&p,&pol,&pa);
4330 294 : switch(t)
4331 : {
4332 42 : case t_INT: return ZM_rank(x);
4333 0 : case t_FRAC: return QM_rank(x);
4334 126 : case t_INTMOD: return RgM_rank_FpM(x, p);
4335 70 : case t_FFELT: return FFM_rank(x, pol);
4336 49 : case code(t_POLMOD, t_INTMOD):
4337 49 : return RgM_rank_FqM(x, pol, p);
4338 7 : default: return -1;
4339 : }
4340 : }
4341 : #undef code
4342 :
4343 : long
4344 294 : rank(GEN x)
4345 : {
4346 294 : pari_sp av = avma;
4347 : long r;
4348 :
4349 294 : if (typ(x)!=t_MAT) pari_err_TYPE("rank",x);
4350 294 : r = RgM_rank_fast(x);
4351 287 : if (r >= 0) return r;
4352 7 : (void)gauss_pivot(x, &r);
4353 7 : return gc_long(av, lg(x)-1 - r);
4354 : }
4355 :
4356 : /* d a t_VECSMALL of integers in 1..n. Return the vector of the d[i]
4357 : * followed by the missing indices */
4358 : static GEN
4359 44262 : perm_complete(GEN d, long n)
4360 : {
4361 44262 : GEN y = cgetg(n+1, t_VECSMALL);
4362 44262 : long i, j = 1, k = n, l = lg(d);
4363 44262 : pari_sp av = avma;
4364 44262 : char *T = stack_calloc(n+1);
4365 221966 : for (i = 1; i < l; i++) T[d[i]] = 1;
4366 436651 : for (i = 1; i <= n; i++)
4367 392389 : if (T[i]) y[j++] = i; else y[k--] = i;
4368 44262 : return gc_const(av, y);
4369 : }
4370 :
4371 : /* n = dim x, r = dim Ker(x), d from gauss_pivot */
4372 : static GEN
4373 5971 : indeximage0(long n, long r, GEN d)
4374 : {
4375 : long i, j;
4376 : GEN v;
4377 :
4378 5971 : r = n - r; /* now r = dim Im(x) */
4379 5971 : v = cgetg(r+1,t_VECSMALL);
4380 32571 : if (d) for (i=j=1; j<=n; j++)
4381 26600 : if (d[j]) v[i++] = j;
4382 5971 : return v;
4383 : }
4384 : /* x an m x n t_MAT, n > 0, r = dim Ker(x), d from gauss_pivot */
4385 : static void
4386 22131 : indexrank_all(long m, long n, long r, GEN d, GEN *prow, GEN *pcol)
4387 : {
4388 22131 : GEN IR = indexrank0(n, r, d);
4389 22131 : *prow = perm_complete(gel(IR,1), m);
4390 22131 : *pcol = perm_complete(gel(IR,2), n);
4391 22131 : }
4392 :
4393 : static GEN
4394 28 : RgM_indexrank_FpM(GEN x, GEN p)
4395 : {
4396 28 : pari_sp av = avma;
4397 : ulong pp;
4398 : GEN r;
4399 28 : x = RgM_Fp_init(x,p,&pp);
4400 28 : switch(pp)
4401 : {
4402 7 : case 0: r = FpM_indexrank(x,p); break;
4403 7 : case 2: r = F2m_indexrank(x); break;
4404 14 : default: r = Flm_indexrank(x,pp); break;
4405 : }
4406 28 : return gerepileupto(av, r);
4407 : }
4408 :
4409 : static GEN
4410 0 : RgM_indexrank_FqM(GEN x, GEN pol, GEN p)
4411 : {
4412 0 : pari_sp av = avma;
4413 0 : GEN r, T = RgX_to_FpX(pol, p);
4414 0 : if (signe(T) == 0) pari_err_OP("indexrank",x,pol);
4415 0 : r = FqM_indexrank(RgM_to_FqM(x, T, p), T, p);
4416 0 : return gerepileupto(av, r);
4417 : }
4418 :
4419 : #define code(t1,t2) ((t1 << 6) | t2)
4420 : static GEN
4421 60507 : RgM_indexrank_fast(GEN x)
4422 : {
4423 : GEN p, pol;
4424 : long pa;
4425 60507 : long t = RgM_type(x,&p,&pol,&pa);
4426 60507 : switch(t)
4427 : {
4428 392 : case t_INT: return ZM_indexrank(x);
4429 1148 : case t_FRAC: return QM_indexrank(x);
4430 28 : case t_INTMOD: return RgM_indexrank_FpM(x, p);
4431 21 : case t_FFELT: return FFM_indexrank(x, pol);
4432 0 : case code(t_POLMOD, t_INTMOD):
4433 0 : return RgM_indexrank_FqM(x, pol, p);
4434 58918 : default: return NULL;
4435 : }
4436 : }
4437 : #undef code
4438 :
4439 : GEN
4440 60507 : indexrank(GEN x)
4441 : {
4442 : pari_sp av;
4443 : long r;
4444 : GEN d;
4445 60507 : if (typ(x)!=t_MAT) pari_err_TYPE("indexrank",x);
4446 60507 : d = RgM_indexrank_fast(x);
4447 60507 : if (d) return d;
4448 58918 : av = avma;
4449 58918 : init_indexrank(x);
4450 58918 : d = gauss_pivot(x, &r);
4451 58917 : set_avma(av); return indexrank0(lg(x)-1, r, d);
4452 : }
4453 :
4454 : GEN
4455 5971 : ZM_indeximage(GEN x) {
4456 5971 : pari_sp av = avma;
4457 : long r;
4458 : GEN d;
4459 5971 : init_indexrank(x);
4460 5971 : d = ZM_pivots(x,&r);
4461 5971 : set_avma(av); return indeximage0(lg(x)-1, r, d);
4462 : }
4463 : long
4464 100646 : ZM_rank(GEN x) {
4465 100646 : pari_sp av = avma;
4466 : long r;
4467 100646 : (void)ZM_pivots(x,&r);
4468 100648 : return gc_long(av, lg(x)-1-r);
4469 : }
4470 : GEN
4471 25704 : ZM_indexrank(GEN x) {
4472 25704 : pari_sp av = avma;
4473 : long r;
4474 : GEN d;
4475 25704 : init_indexrank(x);
4476 25704 : d = ZM_pivots(x,&r);
4477 25704 : set_avma(av); return indexrank0(lg(x)-1, r, d);
4478 : }
4479 :
4480 : long
4481 0 : QM_rank(GEN x)
4482 : {
4483 0 : pari_sp av = avma;
4484 0 : long r = ZM_rank(Q_primpart(x));
4485 0 : set_avma(av);
4486 0 : return r;
4487 : }
4488 :
4489 : GEN
4490 1148 : QM_indexrank(GEN x)
4491 : {
4492 1148 : pari_sp av = avma;
4493 1148 : GEN r = ZM_indexrank(Q_primpart(x));
4494 1148 : return gerepileupto(av, r);
4495 : }
4496 :
4497 : /*******************************************************************/
4498 : /* */
4499 : /* ZabM */
4500 : /* */
4501 : /*******************************************************************/
4502 :
4503 : static GEN
4504 1276 : FpXM_ratlift(GEN a, GEN q)
4505 : {
4506 : GEN B, y;
4507 1276 : long i, j, l = lg(a), n;
4508 1276 : B = sqrti(shifti(q,-1));
4509 1276 : y = cgetg(l, t_MAT);
4510 1276 : if (l==1) return y;
4511 1276 : n = lgcols(a);
4512 3059 : for (i=1; i<l; i++)
4513 : {
4514 2404 : GEN yi = cgetg(n, t_COL);
4515 32311 : for (j=1; j<n; j++)
4516 : {
4517 30528 : GEN v = FpX_ratlift(gmael(a,i,j), q, B, B, NULL);
4518 30528 : if (!v) return NULL;
4519 29907 : gel(yi, j) = RgX_renormalize(v);
4520 : }
4521 1783 : gel(y,i) = yi;
4522 : }
4523 655 : return y;
4524 : }
4525 :
4526 : static GEN
4527 4476 : FlmV_recover_pre(GEN a, GEN M, ulong p, ulong pi, long sv)
4528 : {
4529 4476 : GEN a1 = gel(a,1);
4530 4476 : long i, j, k, l = lg(a1), n, lM = lg(M);
4531 4476 : GEN v = cgetg(lM, t_VECSMALL);
4532 4476 : GEN y = cgetg(l, t_MAT);
4533 4476 : if (l==1) return y;
4534 4476 : n = lgcols(a1);
4535 22462 : for (i=1; i<l; i++)
4536 : {
4537 17985 : GEN yi = cgetg(n, t_COL);
4538 347211 : for (j=1; j<n; j++)
4539 : {
4540 4673669 : for (k=1; k<lM; k++) uel(v,k) = umael(gel(a,k),i,j);
4541 329225 : gel(yi, j) = Flm_Flc_mul_pre_Flx(M, v, p, pi, sv);
4542 : }
4543 17986 : gel(y,i) = yi;
4544 : }
4545 4477 : return y;
4546 : }
4547 :
4548 : static GEN
4549 0 : FlkM_inv(GEN M, GEN P, ulong p)
4550 : {
4551 0 : ulong pi = get_Fl_red(p);
4552 0 : GEN R = Flx_roots(P, p);
4553 0 : long l = lg(R), i;
4554 0 : GEN W = Flv_invVandermonde(R, 1UL, p);
4555 0 : GEN V = cgetg(l, t_VEC);
4556 0 : for(i=1; i<l; i++)
4557 : {
4558 0 : GEN pows = Fl_powers_pre(uel(R,i), degpol(P), p, pi);
4559 0 : GEN H = Flm_inv_sp(FlxM_eval_powers_pre(M, pows, p, pi), NULL, p);
4560 0 : if (!H) return NULL;
4561 0 : gel(V, i) = H;
4562 : }
4563 0 : return FlmV_recover_pre(V, W, p, pi, P[1]);
4564 : }
4565 :
4566 : static GEN
4567 3200 : FlkM_adjoint(GEN M, GEN P, ulong p)
4568 : {
4569 3200 : ulong pi = get_Fl_red(p);
4570 3200 : GEN R = Flx_roots(P, p);
4571 3201 : long l = lg(R), i;
4572 3201 : GEN W = Flv_invVandermonde(R, 1UL, p);
4573 3201 : GEN V = cgetg(l, t_VEC);
4574 15520 : for(i=1; i<l; i++)
4575 : {
4576 12320 : GEN pows = Fl_powers_pre(uel(R,i), degpol(P), p, pi);
4577 12320 : gel(V, i) = Flm_adjoint(FlxM_eval_powers_pre(M, pows, p, pi), p);
4578 : }
4579 3200 : return FlmV_recover_pre(V, W, p, pi, P[1]);
4580 : }
4581 :
4582 : static GEN
4583 1978 : ZabM_inv_slice(GEN A, GEN Q, GEN P, GEN *mod)
4584 : {
4585 1978 : pari_sp av = avma;
4586 1978 : long i, n = lg(P)-1, w = varn(Q);
4587 : GEN H, T;
4588 1978 : if (n == 1)
4589 : {
4590 1548 : ulong p = uel(P,1);
4591 1548 : GEN Qp = ZX_to_Flx(Q, p);
4592 1548 : GEN Ap = ZXM_to_FlxM(A, p, get_Flx_var(Qp));
4593 1548 : GEN Hp = FlkM_adjoint(Ap, Qp, p);
4594 1548 : Hp = gerepileupto(av, FlxM_to_ZXM(Hp));
4595 1548 : *mod = utoipos(p); return Hp;
4596 : }
4597 430 : T = ZV_producttree(P);
4598 430 : A = ZXM_nv_mod_tree(A, P, T, w);
4599 430 : Q = ZX_nv_mod_tree(Q, P, T);
4600 430 : H = cgetg(n+1, t_VEC);
4601 2083 : for(i=1; i <= n; i++)
4602 : {
4603 1653 : ulong p = P[i];
4604 1653 : GEN a = gel(A,i), q = gel(Q, i);
4605 1653 : gel(H,i) = FlkM_adjoint(a, q, p);
4606 : }
4607 430 : H = nxMV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P,T));
4608 430 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
4609 : }
4610 :
4611 : GEN
4612 1978 : ZabM_inv_worker(GEN P, GEN A, GEN Q)
4613 : {
4614 1978 : GEN V = cgetg(3, t_VEC);
4615 1978 : gel(V,1) = ZabM_inv_slice(A, Q, P, &gel(V,2));
4616 1978 : return V;
4617 : }
4618 :
4619 : static GEN
4620 5467 : vecnorml1(GEN a)
4621 : {
4622 : long i, l;
4623 5467 : GEN g = cgetg_copy(a, &l);
4624 60214 : for (i=1; i<l; i++)
4625 54747 : gel(g, i) = gnorml1_fake(gel(a,i));
4626 5467 : return g;
4627 : }
4628 :
4629 : static GEN
4630 1820 : ZabM_true_Hadamard(GEN a)
4631 : {
4632 1820 : pari_sp av = avma;
4633 1820 : long n = lg(a)-1, i;
4634 : GEN B;
4635 1820 : if (n == 0) return gen_1;
4636 1820 : if (n == 1) return gnorml1_fake(gcoeff(a,1,1));
4637 1176 : B = gen_1;
4638 6643 : for (i = 1; i <= n; i++)
4639 5467 : B = gmul(B, gnorml2(RgC_gtofp(vecnorml1(gel(a,i)),DEFAULTPREC)));
4640 1176 : return gerepileuptoint(av, ceil_safe(sqrtr_abs(B)));
4641 : }
4642 :
4643 : GEN
4644 1820 : ZabM_inv(GEN A, GEN Q, long n, GEN *pt_den)
4645 : {
4646 1820 : pari_sp av = avma;
4647 : forprime_t S;
4648 : GEN bnd, H, D, d, mod, worker;
4649 1820 : if (lg(A) == 1)
4650 : {
4651 0 : if (pt_den) *pt_den = gen_1;
4652 0 : return cgetg(1, t_MAT);
4653 : }
4654 1820 : bnd = ZabM_true_Hadamard(A);
4655 1820 : worker = snm_closure(is_entry("_ZabM_inv_worker"), mkvec2(A, Q));
4656 1820 : u_forprime_arith_init(&S, HIGHBIT+1, ULONG_MAX, 1, n);
4657 1820 : H = gen_crt("ZabM_inv", worker, &S, NULL, expi(bnd), 0, &mod,
4658 : nxMV_chinese_center, FpXM_center);
4659 1820 : D = RgMrow_RgC_mul(H, gel(A,1), 1);
4660 1820 : D = ZX_rem(D, Q);
4661 1820 : d = Z_content(mkvec2(H, D));
4662 1820 : if (d)
4663 : {
4664 511 : D = ZX_Z_divexact(D, d);
4665 511 : H = Q_div_to_int(H, d);
4666 : }
4667 1820 : if (!pt_den) return gerepileupto(av, H);
4668 1820 : *pt_den = D; return gc_all(av, 2, &H, pt_den);
4669 : }
4670 :
4671 : GEN
4672 0 : ZabM_inv_ratlift(GEN M, GEN P, long n, GEN *pden)
4673 : {
4674 0 : pari_sp av2, av = avma;
4675 : GEN q, H;
4676 0 : ulong m = LONG_MAX>>1;
4677 0 : ulong p= 1 + m - (m % n);
4678 0 : long lM = lg(M);
4679 0 : if (lM == 1) { *pden = gen_1; return cgetg(1,t_MAT); }
4680 :
4681 0 : av2 = avma;
4682 0 : H = NULL;
4683 : for(;;)
4684 0 : {
4685 : GEN Hp, Pp, Mp, Hr;
4686 0 : do p += n; while(!uisprime(p));
4687 0 : Pp = ZX_to_Flx(P, p);
4688 0 : Mp = ZXM_to_FlxM(M, p, get_Flx_var(Pp));
4689 0 : Hp = FlkM_inv(Mp, Pp, p);
4690 0 : if (!Hp) continue;
4691 0 : if (!H)
4692 : {
4693 0 : H = ZXM_init_CRT(Hp, degpol(P)-1, p);
4694 0 : q = utoipos(p);
4695 : }
4696 : else
4697 0 : ZXM_incremental_CRT(&H, Hp, &q, p);
4698 0 : Hr = FpXM_ratlift(H, q);
4699 0 : if (DEBUGLEVEL>5) err_printf("ZabM_inv mod %ld (ratlift=%ld)\n", p,!!Hr);
4700 0 : if (Hr) {/* DONE ? */
4701 0 : GEN Hl = Q_remove_denom(Hr, pden);
4702 0 : GEN MH = ZXQM_mul(Hl, M, P);
4703 0 : if (*pden)
4704 0 : { if (RgM_isscalar(MH, *pden)) { H = Hl; break; }}
4705 : else
4706 0 : { if (RgM_isidentity(MH)) { H = Hl; *pden = gen_1; break; } }
4707 : }
4708 :
4709 0 : if (gc_needed(av,2))
4710 : {
4711 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZabM_inv");
4712 0 : gerepileall(av2, 2, &H, &q);
4713 : }
4714 : }
4715 0 : return gc_all(av, 2, &H, pden);
4716 : }
4717 :
4718 : static GEN
4719 1276 : FlkM_ker(GEN M, GEN P, ulong p)
4720 : {
4721 1276 : ulong pi = get_Fl_red(p);
4722 1276 : GEN R = Flx_roots(P, p);
4723 1276 : long l = lg(R), i, dP = degpol(P), r;
4724 : GEN M1, K, D;
4725 1276 : GEN W = Flv_invVandermonde(R, 1UL, p);
4726 1276 : GEN V = cgetg(l, t_VEC);
4727 1276 : M1 = FlxM_eval_powers_pre(M, Fl_powers_pre(uel(R,1), dP, p, pi), p, pi);
4728 1276 : K = Flm_ker_sp(M1, p, 2);
4729 1276 : r = lg(gel(K,1)); D = gel(K,2);
4730 1276 : gel(V, 1) = gel(K,1);
4731 2652 : for(i=2; i<l; i++)
4732 : {
4733 1376 : GEN Mi = FlxM_eval_powers_pre(M, Fl_powers_pre(uel(R,i), dP, p, pi), p, pi);
4734 1376 : GEN K = Flm_ker_sp(Mi, p, 2);
4735 1376 : if (lg(gel(K,1)) != r || !zv_equal(D, gel(K,2))) return NULL;
4736 1376 : gel(V, i) = gel(K,1);
4737 : }
4738 1276 : return mkvec2(FlmV_recover_pre(V, W, p, pi, P[1]), D);
4739 : }
4740 :
4741 : static int
4742 655 : ZabM_ker_check(GEN M, GEN H, ulong p, GEN P, long n)
4743 : {
4744 : GEN pow;
4745 655 : long j, l = lg(H);
4746 : ulong pi, r;
4747 3899 : do p += n; while(!uisprime(p));
4748 655 : pi = get_Fl_red(p);
4749 655 : P = ZX_to_Flx(P, p);
4750 655 : r = Flx_oneroot(P, p);
4751 655 : pow = Fl_powers_pre(r, degpol(P),p,pi);
4752 655 : M = ZXM_to_FlxM(M, p, P[1]); M = FlxM_eval_powers_pre(M, pow, p, pi);
4753 655 : H = ZXM_to_FlxM(H, p, P[1]); H = FlxM_eval_powers_pre(H, pow, p, pi);
4754 2178 : for (j = 1; j < l; j++)
4755 1555 : if (!zv_equal0(Flm_Flc_mul_pre(M, gel(H,j), p, pi))) return 0;
4756 623 : return 1;
4757 : }
4758 :
4759 : GEN
4760 623 : ZabM_ker(GEN M, GEN P, long n)
4761 : {
4762 623 : pari_sp av = avma;
4763 : pari_timer ti;
4764 623 : GEN q, H = NULL, D = NULL;
4765 623 : ulong m = LONG_MAX>>1;
4766 623 : ulong p = 1 + m - (m % n);
4767 :
4768 623 : if (DEBUGLEVEL>5) timer_start(&ti);
4769 : for(;;)
4770 653 : {
4771 : GEN Kp, Hp, Dp, Pp, Mp, Hr;
4772 22341 : do p += n; while(!uisprime(p));
4773 1276 : Pp = ZX_to_Flx(P, p);
4774 1276 : Mp = ZXM_to_FlxM(M, p, get_Flx_var(Pp));
4775 1276 : Kp = FlkM_ker(Mp, Pp, p);
4776 1276 : if (!Kp) continue;
4777 1276 : Hp = gel(Kp,1); Dp = gel(Kp,2);
4778 1276 : if (H && (lg(Hp)>lg(H) || (lg(Hp)==lg(H) && vecsmall_lexcmp(Dp,D)>0))) continue;
4779 1276 : if (!H || (lg(Hp)<lg(H) || vecsmall_lexcmp(Dp,D)<0))
4780 : {
4781 623 : H = ZXM_init_CRT(Hp, degpol(P)-1, p); D = Dp;
4782 623 : q = utoipos(p);
4783 : }
4784 : else
4785 653 : ZXM_incremental_CRT(&H, Hp, &q, p);
4786 1276 : Hr = FpXM_ratlift(H, q);
4787 1276 : if (DEBUGLEVEL>5) timer_printf(&ti,"ZabM_ker mod %ld (ratlift=%ld)", p,!!Hr);
4788 1276 : if (Hr) {/* DONE ? */
4789 655 : GEN Hl = vec_Q_primpart(Hr);
4790 655 : if (ZabM_ker_check(M, Hl, p, P, n)) { H = Hl; break; }
4791 : }
4792 :
4793 653 : if (gc_needed(av,2))
4794 : {
4795 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZabM_ker");
4796 0 : gerepileall(av, 3, &H, &D, &q);
4797 : }
4798 : }
4799 623 : return gerepilecopy(av, H);
4800 : }
4801 :
4802 : GEN
4803 2387 : ZabM_indexrank(GEN M, GEN P, long n)
4804 : {
4805 2387 : pari_sp av = avma;
4806 2387 : ulong m = LONG_MAX>>1;
4807 2387 : ulong p = 1+m-(m%n), D = degpol(P);
4808 2387 : long lM = lg(M), lmax = 0, c = 0;
4809 : GEN v;
4810 : for(;;)
4811 735 : {
4812 : GEN R, Pp, Mp, K;
4813 : ulong pi;
4814 : long l;
4815 61415 : do p += n; while (!uisprime(p));
4816 3122 : pi = get_Fl_red(p);
4817 3122 : Pp = ZX_to_Flx(P, p);
4818 3122 : R = Flx_roots(Pp, p);
4819 3122 : Mp = ZXM_to_FlxM(M, p, get_Flx_var(Pp));
4820 3122 : K = FlxM_eval_powers_pre(Mp, Fl_powers_pre(uel(R,1), D,p,pi), p,pi);
4821 3122 : v = Flm_indexrank(K, p);
4822 3122 : l = lg(gel(v,2));
4823 3122 : if (l == lM) break;
4824 980 : if (lmax >= 0 && l > lmax) { lmax = l; c = 0; } else c++;
4825 980 : if (c > 2)
4826 : { /* probably not maximal rank, expensive check */
4827 245 : lM -= lg(ZabM_ker(M, P, n))-1; /* actual rank (+1) */
4828 245 : if (lmax == lM) break;
4829 0 : lmax = -1; /* disable check */
4830 : }
4831 : }
4832 2387 : return gerepileupto(av, v);
4833 : }
4834 :
4835 : #if 0
4836 : GEN
4837 : ZabM_gauss(GEN M, GEN P, long n, GEN *den)
4838 : {
4839 : pari_sp av = avma;
4840 : GEN v, S, W;
4841 : v = ZabM_indexrank(M, P, n);
4842 : S = shallowmatextract(M,gel(v,1),gel(v,2));
4843 : W = ZabM_inv(S, P, n, den);
4844 : return gc_all(av,2,&W,den);
4845 : }
4846 : #endif
4847 :
4848 : GEN
4849 140 : ZabM_pseudoinv(GEN M, GEN P, long n, GEN *pv, GEN *den)
4850 : {
4851 140 : GEN v = ZabM_indexrank(M, P, n);
4852 140 : if (pv) *pv = v;
4853 140 : M = shallowmatextract(M,gel(v,1),gel(v,2));
4854 140 : return ZabM_inv(M, P, n, den);
4855 : }
4856 : GEN
4857 4977 : ZM_pseudoinv(GEN M, GEN *pv, GEN *den)
4858 : {
4859 4977 : GEN v = ZM_indexrank(M);
4860 4977 : if (pv) *pv = v;
4861 4977 : M = shallowmatextract(M,gel(v,1),gel(v,2));
4862 4977 : return ZM_inv(M, den);
4863 : }
4864 :
4865 : /*******************************************************************/
4866 : /* */
4867 : /* Structured Elimination */
4868 : /* */
4869 : /*******************************************************************/
4870 :
4871 : static void
4872 111654 : rem_col(GEN c, long i, GEN iscol, GEN Wrow, long *rcol, long *rrow)
4873 : {
4874 111654 : long lc = lg(c), k;
4875 111654 : iscol[i] = 0; (*rcol)--;
4876 1078678 : for (k = 1; k < lc; ++k)
4877 : {
4878 967024 : Wrow[c[k]]--;
4879 967024 : if (Wrow[c[k]]==0) (*rrow)--;
4880 : }
4881 111654 : }
4882 :
4883 : static void
4884 6278 : rem_singleton(GEN M, GEN iscol, GEN Wrow, long idx, long *rcol, long *rrow)
4885 : {
4886 : long i, j;
4887 6278 : long nbcol = lg(iscol)-1, last;
4888 : do
4889 : {
4890 8353 : last = 0;
4891 19439662 : for (i = 1; i <= nbcol; ++i)
4892 19431309 : if (iscol[i])
4893 : {
4894 10129482 : GEN c = idx ? gmael(M, i, idx): gel(M,i);
4895 10129482 : long lc = lg(c);
4896 97291642 : for (j = 1; j < lc; ++j)
4897 87184471 : if (Wrow[c[j]] == 1)
4898 : {
4899 22311 : rem_col(c, i, iscol, Wrow, rcol, rrow);
4900 22311 : last=1; break;
4901 : }
4902 : }
4903 8353 : } while (last);
4904 6278 : }
4905 :
4906 : static GEN
4907 6106 : fill_wcol(GEN M, GEN iscol, GEN Wrow, long *w, GEN wcol)
4908 : {
4909 6106 : long nbcol = lg(iscol)-1;
4910 : long i, j, m, last;
4911 : GEN per;
4912 15300 : for (m = 2, last=0; !last ; m++)
4913 : {
4914 22448205 : for (i = 1; i <= nbcol; ++i)
4915 : {
4916 22439011 : wcol[i] = 0;
4917 22439011 : if (iscol[i])
4918 : {
4919 11521072 : GEN c = gmael(M, i, 1);
4920 11521072 : long lc = lg(c);
4921 107254564 : for (j = 1; j < lc; ++j)
4922 95733492 : if (Wrow[c[j]] == m) { wcol[i]++; last = 1; }
4923 : }
4924 : }
4925 : }
4926 6106 : per = vecsmall_indexsort(wcol);
4927 6106 : *w = wcol[per[nbcol]];
4928 6106 : return per;
4929 : }
4930 :
4931 : /* M is a RgMs with nbrow rows, A a list of row indices.
4932 : Eliminate rows of M with a single entry that do not belong to A,
4933 : and the corresponding columns. Also eliminate columns until #colums=#rows.
4934 : Return pcol and prow:
4935 : pcol is a map from the new columns indices to the old one.
4936 : prow is a map from the old rows indices to the new one (0 if removed).
4937 : */
4938 :
4939 : void
4940 119 : RgMs_structelim_col(GEN M, long nbcol, long nbrow, GEN A, GEN *p_col, GEN *p_row)
4941 : {
4942 119 : long i, j, k, lA = lg(A);
4943 119 : GEN prow = cgetg(nbrow+1, t_VECSMALL);
4944 119 : GEN pcol = zero_zv(nbcol);
4945 119 : pari_sp av = avma;
4946 119 : long rcol = nbcol, rrow = 0, imin = nbcol - usqrt(nbcol);
4947 119 : GEN iscol = const_vecsmall(nbcol, 1);
4948 119 : GEN Wrow = zero_zv(nbrow);
4949 119 : GEN wcol = cgetg(nbcol+1, t_VECSMALL);
4950 119 : pari_sp av2 = avma;
4951 126763 : for (i = 1; i <= nbcol; ++i)
4952 : {
4953 126644 : GEN F = gmael(M, i, 1);
4954 126644 : long l = lg(F)-1;
4955 1115936 : for (j = 1; j <= l; ++j) Wrow[F[j]]++;
4956 : }
4957 119 : for (j = 1; j < lA; ++j)
4958 : {
4959 0 : if (Wrow[A[j]] == 0) { *p_col=NULL; return; }
4960 0 : Wrow[A[j]] = -1;
4961 : }
4962 235298 : for (i = 1; i <= nbrow; ++i)
4963 235179 : if (Wrow[i]) rrow++;
4964 119 : rem_singleton(M, iscol, Wrow, 1, &rcol, &rrow);
4965 119 : if (rcol < rrow) pari_err_BUG("RgMs_structelim, rcol<rrow");
4966 6225 : while (rcol > rrow)
4967 : {
4968 : long w;
4969 6106 : GEN per = fill_wcol(M, iscol, Wrow, &w, wcol);
4970 95449 : for (i = nbcol; i>=imin && wcol[per[i]]>=w && rcol>rrow; i--)
4971 89343 : rem_col(gmael(M, per[i], 1), per[i], iscol, Wrow, &rcol, &rrow);
4972 6106 : rem_singleton(M, iscol, Wrow, 1, &rcol, &rrow); set_avma(av2);
4973 : }
4974 126763 : for (j = 1, i = 1; i <= nbcol; ++i)
4975 126644 : if (iscol[i]) pcol[j++] = i;
4976 119 : setlg(pcol,j);
4977 235298 : for (k = 1, i = 1; i <= nbrow; ++i) prow[i] = Wrow[i]? k++: 0;
4978 119 : *p_col = pcol; *p_row = prow; set_avma(av);
4979 : }
4980 :
4981 : void
4982 0 : RgMs_structelim(GEN M, long nbrow, GEN A, GEN *p_col, GEN *p_row)
4983 0 : { RgMs_structelim_col(M, lg(M)-1, nbrow, A, p_col, p_row); }
4984 :
4985 : GEN
4986 53 : F2Ms_colelim(GEN M, long nbrow)
4987 : {
4988 53 : long i,j, nbcol = lg(M)-1, rcol = nbcol, rrow = 0;
4989 53 : GEN pcol = zero_zv(nbcol);
4990 53 : pari_sp av = avma;
4991 53 : GEN iscol = const_vecsmall(nbcol, 1), Wrow = zero_zv(nbrow);
4992 85968 : for (i = 1; i <= nbcol; ++i)
4993 : {
4994 85915 : GEN F = gel(M, i);
4995 85915 : long l = lg(F)-1;
4996 1587632 : for (j = 1; j <= l; ++j) Wrow[F[j]]++;
4997 : }
4998 53 : rem_singleton(M, iscol, Wrow, 0, &rcol, &rrow);
4999 85968 : for (j = 1, i = 1; i <= nbcol; ++i)
5000 85915 : if (iscol[i]) pcol[j++] = i;
5001 53 : fixlg(pcol,j); return gc_const(av, pcol);
5002 : }
5003 :
5004 : /*******************************************************************/
5005 : /* */
5006 : /* EIGENVECTORS */
5007 : /* (independent eigenvectors, sorted by increasing eigenvalue) */
5008 : /* */
5009 : /*******************************************************************/
5010 : /* assume x is square of dimension > 0 */
5011 : static int
5012 34 : RgM_is_symmetric_cx(GEN x, long bit)
5013 : {
5014 34 : pari_sp av = avma;
5015 34 : long i, j, l = lg(x);
5016 201 : for (i = 1; i < l; i++)
5017 670 : for (j = 1; j < i; j++)
5018 : {
5019 503 : GEN a = gcoeff(x,i,j), b = gcoeff(x,j,i), c = gsub(a,b);
5020 503 : if (!gequal0(c) && gexpo(c) - gexpo(a) > -bit) return gc_long(av,0);
5021 : }
5022 21 : return gc_long(av,1);
5023 : }
5024 : static GEN
5025 34 : eigen_err(int exact, GEN x, long flag, long prec)
5026 : {
5027 34 : pari_sp av = avma;
5028 34 : if (RgM_is_symmetric_cx(x, prec2nbits(prec) - 10))
5029 : { /* approximately symmetric: recover */
5030 21 : x = jacobi(x, prec); if (flag) return x;
5031 14 : return gerepilecopy(av, gel(x,2));
5032 : }
5033 13 : if (exact)
5034 : {
5035 6 : GEN y = mateigen(x, flag, precdbl(prec));
5036 6 : return gerepilecopy(av, gprec_wtrunc(y, prec));
5037 : }
5038 7 : pari_err_PREC("mateigen");
5039 : return NULL; /* LCOV_EXCL_LINE */
5040 : }
5041 : GEN
5042 104 : mateigen(GEN x, long flag, long prec)
5043 : {
5044 : GEN y, R, T;
5045 104 : long k, l, ex, n = lg(x);
5046 : int exact;
5047 104 : pari_sp av = avma;
5048 :
5049 104 : if (typ(x)!=t_MAT) pari_err_TYPE("eigen",x);
5050 104 : if (n != 1 && n != lgcols(x)) pari_err_DIM("eigen");
5051 104 : if (flag < 0 || flag > 1) pari_err_FLAG("mateigen");
5052 104 : if (n == 1)
5053 : {
5054 14 : if (flag) retmkvec2(cgetg(1,t_VEC), cgetg(1,t_MAT));
5055 7 : return cgetg(1,t_VEC);
5056 : }
5057 90 : if (n == 2)
5058 : {
5059 14 : if (flag) retmkvec2(mkveccopy(gcoeff(x,1,1)), matid(1));
5060 7 : return matid(1);
5061 : }
5062 :
5063 76 : ex = 16 - prec2nbits(prec);
5064 76 : T = charpoly(x,0);
5065 76 : exact = RgX_is_QX(T);
5066 76 : if (exact)
5067 : {
5068 41 : T = ZX_radical( Q_primpart(T) );
5069 41 : R = nfrootsQ(T);
5070 41 : if (lg(R)-1 < degpol(T))
5071 : { /* add missing complex roots */
5072 27 : GEN r = cleanroots(RgX_div(T, roots_to_pol(R, 0)), prec);
5073 27 : settyp(r, t_VEC);
5074 27 : R = shallowconcat(R, r);
5075 : }
5076 : }
5077 : else
5078 : {
5079 35 : GEN r1, v = vectrunc_init(lg(T));
5080 : long e;
5081 35 : R = cleanroots(T,prec);
5082 35 : r1 = NULL;
5083 231 : for (k = 1; k < lg(R); k++)
5084 : {
5085 196 : GEN r2 = gel(R,k), r = grndtoi(r2, &e);
5086 196 : if (e < ex) r2 = r;
5087 196 : if (r1)
5088 : {
5089 161 : r = gsub(r1,r2);
5090 161 : if (gequal0(r) || gexpo(r) < ex) continue;
5091 : }
5092 154 : vectrunc_append(v, r2);
5093 154 : r1 = r2;
5094 : }
5095 35 : R = v;
5096 : }
5097 : /* R = distinct complex roots of charpoly(x) */
5098 76 : l = lg(R); y = cgetg(l, t_VEC);
5099 328 : for (k = 1; k < l; k++)
5100 : {
5101 286 : GEN F = ker_aux(RgM_Rg_sub_shallow(x, gel(R,k)), x);
5102 286 : long d = lg(F)-1;
5103 286 : if (!d) { set_avma(av); return eigen_err(exact, x, flag, prec); }
5104 252 : gel(y,k) = F;
5105 252 : if (flag) gel(R,k) = const_vec(d, gel(R,k));
5106 : }
5107 42 : y = shallowconcat1(y);
5108 42 : if (lg(y) > n) { set_avma(av); return eigen_err(exact, x, flag, prec); }
5109 : /* lg(y) < n if x is not diagonalizable */
5110 42 : if (flag) y = mkvec2(shallowconcat1(R), y);
5111 42 : return gerepilecopy(av,y);
5112 : }
5113 : GEN
5114 0 : eigen(GEN x, long prec) { return mateigen(x, 0, prec); }
5115 :
5116 : /*******************************************************************/
5117 : /* */
5118 : /* DETERMINANT */
5119 : /* */
5120 : /*******************************************************************/
5121 :
5122 : GEN
5123 26390 : det0(GEN a,long flag)
5124 : {
5125 26390 : switch(flag)
5126 : {
5127 26376 : case 0: return det(a);
5128 14 : case 1: return det2(a);
5129 0 : default: pari_err_FLAG("matdet");
5130 : }
5131 : return NULL; /* LCOV_EXCL_LINE */
5132 : }
5133 :
5134 : /* M a 2x2 matrix, returns det(M) */
5135 : static GEN
5136 83375 : RgM_det2(GEN M)
5137 : {
5138 83375 : pari_sp av = avma;
5139 83375 : GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
5140 83375 : GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
5141 83375 : return gerepileupto(av, gsub(gmul(a,d), gmul(b,c)));
5142 : }
5143 : /* M a 2x2 ZM, returns det(M) */
5144 : static GEN
5145 6364 : ZM_det2(GEN M)
5146 : {
5147 6364 : pari_sp av = avma;
5148 6364 : GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
5149 6364 : GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
5150 6364 : return gerepileuptoint(av, subii(mulii(a,d), mulii(b, c)));
5151 : }
5152 : /* M a 3x3 ZM, return det(M) */
5153 : static GEN
5154 92862 : ZM_det3(GEN M)
5155 : {
5156 92862 : pari_sp av = avma;
5157 92862 : GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2), c = gcoeff(M,1,3);
5158 92862 : GEN d = gcoeff(M,2,1), e = gcoeff(M,2,2), f = gcoeff(M,2,3);
5159 92862 : GEN g = gcoeff(M,3,1), h = gcoeff(M,3,2), i = gcoeff(M,3,3);
5160 92862 : GEN t, D = signe(i)? mulii(subii(mulii(a,e), mulii(b,d)), i): gen_0;
5161 92862 : if (signe(g))
5162 : {
5163 64876 : t = mulii(subii(mulii(b,f), mulii(c,e)), g);
5164 64876 : D = addii(D, t);
5165 : }
5166 92862 : if (signe(h))
5167 : {
5168 74816 : t = mulii(subii(mulii(c,d), mulii(a,f)), h);
5169 74816 : D = addii(D, t);
5170 : }
5171 92862 : return gerepileuptoint(av, D);
5172 : }
5173 :
5174 : static GEN
5175 62709 : det_simple_gauss(GEN a, GEN data, pivot_fun pivot)
5176 : {
5177 62709 : pari_sp av = avma;
5178 62709 : long i,j,k, s = 1, nbco = lg(a)-1;
5179 62709 : GEN p, x = gen_1;
5180 :
5181 62709 : a = RgM_shallowcopy(a);
5182 403734 : for (i=1; i<nbco; i++)
5183 : {
5184 341033 : k = pivot(a, data, i, NULL);
5185 341034 : if (k > nbco) return gerepilecopy(av, gcoeff(a,i,i));
5186 341027 : if (k != i)
5187 : { /* exchange the lines s.t. k = i */
5188 1524596 : for (j=i; j<=nbco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
5189 149915 : s = -s;
5190 : }
5191 341027 : p = gcoeff(a,i,i);
5192 :
5193 341027 : x = gmul(x,p);
5194 2340414 : for (k=i+1; k<=nbco; k++)
5195 : {
5196 1999389 : GEN m = gcoeff(a,i,k);
5197 1999389 : if (gequal0(m)) continue;
5198 :
5199 1397183 : m = gdiv(m,p);
5200 13539150 : for (j=i+1; j<=nbco; j++)
5201 12141969 : gcoeff(a,j,k) = gsub(gcoeff(a,j,k), gmul(m,gcoeff(a,j,i)));
5202 : }
5203 341025 : if (gc_needed(av,2))
5204 : {
5205 0 : if(DEBUGMEM>1) pari_warn(warnmem,"det. col = %ld",i);
5206 0 : gerepileall(av,2, &a,&x);
5207 : }
5208 : }
5209 62701 : if (s < 0) x = gneg_i(x);
5210 62702 : return gerepileupto(av, gmul(x, gcoeff(a,nbco,nbco)));
5211 : }
5212 :
5213 : GEN
5214 114779 : det2(GEN a)
5215 : {
5216 : GEN data;
5217 : pivot_fun pivot;
5218 114779 : long n = lg(a)-1;
5219 114779 : if (typ(a)!=t_MAT) pari_err_TYPE("det2",a);
5220 114779 : if (!n) return gen_1;
5221 114779 : if (n != nbrows(a)) pari_err_DIM("det2");
5222 114779 : if (n == 1) return gcopy(gcoeff(a,1,1));
5223 77204 : if (n == 2) return RgM_det2(a);
5224 28864 : pivot = get_pivot_fun(a, a, &data);
5225 28864 : return det_simple_gauss(a, data, pivot);
5226 : }
5227 :
5228 : /* Assumes a a square t_MAT of dimension n > 0. Returns det(a) using
5229 : * Gauss-Bareiss. */
5230 : static GEN
5231 448 : det_bareiss(GEN a)
5232 : {
5233 448 : pari_sp av = avma;
5234 448 : long nbco = lg(a)-1,i,j,k,s = 1;
5235 : GEN p, pprec;
5236 :
5237 448 : a = RgM_shallowcopy(a);
5238 1274 : for (pprec=gen_1,i=1; i<nbco; i++,pprec=p)
5239 : {
5240 826 : int diveuc = (gequal1(pprec)==0);
5241 : GEN ci;
5242 :
5243 826 : p = gcoeff(a,i,i);
5244 826 : if (gequal0(p))
5245 : {
5246 0 : k=i+1; while (k<=nbco && gequal0(gcoeff(a,i,k))) k++;
5247 0 : if (k>nbco) return gerepilecopy(av, p);
5248 0 : swap(gel(a,k), gel(a,i)); s = -s;
5249 0 : p = gcoeff(a,i,i);
5250 : }
5251 826 : ci = gel(a,i);
5252 2184 : for (k=i+1; k<=nbco; k++)
5253 : {
5254 1358 : GEN ck = gel(a,k), m = gel(ck,i);
5255 1358 : if (gequal0(m))
5256 : {
5257 0 : if (gequal1(p))
5258 : {
5259 0 : if (diveuc)
5260 0 : gel(a,k) = gdiv(gel(a,k), pprec);
5261 : }
5262 : else
5263 0 : for (j=i+1; j<=nbco; j++)
5264 : {
5265 0 : GEN p1 = gmul(p, gel(ck,j));
5266 0 : if (diveuc) p1 = gdiv(p1,pprec);
5267 0 : gel(ck,j) = p1;
5268 : }
5269 : }
5270 : else
5271 4088 : for (j=i+1; j<=nbco; j++)
5272 : {
5273 2730 : pari_sp av2 = avma;
5274 2730 : GEN p1 = gsub(gmul(p,gel(ck,j)), gmul(m,gel(ci,j)));
5275 2730 : if (diveuc) p1 = gdiv(p1,pprec);
5276 2730 : gel(ck,j) = gerepileupto(av2, p1);
5277 : }
5278 1358 : if (gc_needed(av,2))
5279 : {
5280 0 : if(DEBUGMEM>1) pari_warn(warnmem,"det. col = %ld",i);
5281 0 : gerepileall(av,2, &a,&pprec);
5282 0 : ci = gel(a,i);
5283 0 : p = gcoeff(a,i,i);
5284 : }
5285 : }
5286 : }
5287 448 : p = gcoeff(a,nbco,nbco);
5288 448 : p = (s < 0)? gneg(p): gcopy(p);
5289 448 : return gerepileupto(av, p);
5290 : }
5291 :
5292 : /* count nonzero entries in col j, at most 'max' of them.
5293 : * Return their indices */
5294 : static GEN
5295 1400 : col_count_non_zero(GEN a, long j, long max)
5296 : {
5297 1400 : GEN v = cgetg(max+1, t_VECSMALL);
5298 1400 : GEN c = gel(a,j);
5299 1400 : long i, l = lg(a), k = 1;
5300 5278 : for (i = 1; i < l; i++)
5301 5040 : if (!gequal0(gel(c,i)))
5302 : {
5303 4788 : if (k > max) return NULL; /* fail */
5304 3626 : v[k++] = i;
5305 : }
5306 238 : setlg(v, k); return v;
5307 : }
5308 : /* count nonzero entries in row i, at most 'max' of them.
5309 : * Return their indices */
5310 : static GEN
5311 1386 : row_count_non_zero(GEN a, long i, long max)
5312 : {
5313 1386 : GEN v = cgetg(max+1, t_VECSMALL);
5314 1386 : long j, l = lg(a), k = 1;
5315 5222 : for (j = 1; j < l; j++)
5316 4998 : if (!gequal0(gcoeff(a,i,j)))
5317 : {
5318 4774 : if (k > max) return NULL; /* fail */
5319 3612 : v[k++] = j;
5320 : }
5321 224 : setlg(v, k); return v;
5322 : }
5323 :
5324 : static GEN det_develop(GEN a, long max, double bound);
5325 : /* (-1)^(i+j) a[i,j] * det RgM_minor(a,i,j) */
5326 : static GEN
5327 406 : coeff_det(GEN a, long i, long j, long max, double bound)
5328 : {
5329 406 : GEN c = gcoeff(a, i, j);
5330 406 : c = gmul(c, det_develop(RgM_minor(a, i,j), max, bound));
5331 406 : if (odd(i+j)) c = gneg(c);
5332 406 : return c;
5333 : }
5334 : /* a square t_MAT, 'bound' a rough upper bound for the number of
5335 : * multiplications we are willing to pay while developing rows/columns before
5336 : * switching to Gaussian elimination */
5337 : static GEN
5338 644 : det_develop(GEN M, long max, double bound)
5339 : {
5340 644 : pari_sp av = avma;
5341 644 : long i,j, n = lg(M)-1, lbest = max+2, best_col = 0, best_row = 0;
5342 644 : GEN best = NULL;
5343 :
5344 644 : if (bound < 1.) return det_bareiss(M); /* too costly now */
5345 :
5346 420 : switch(n)
5347 : {
5348 0 : case 0: return gen_1;
5349 0 : case 1: return gcopy(gcoeff(M,1,1));
5350 14 : case 2: return RgM_det2(M);
5351 : }
5352 406 : if (max > ((n+2)>>1)) max = (n+2)>>1;
5353 1792 : for (j = 1; j <= n; j++)
5354 : {
5355 1400 : pari_sp av2 = avma;
5356 1400 : GEN v = col_count_non_zero(M, j, max);
5357 : long lv;
5358 1400 : if (!v || (lv = lg(v)) >= lbest) { set_avma(av2); continue; }
5359 182 : if (lv == 1) { set_avma(av); return gen_0; }
5360 182 : if (lv == 2) {
5361 14 : set_avma(av);
5362 14 : return gerepileupto(av, coeff_det(M,v[1],j,max,bound));
5363 : }
5364 168 : best = v; lbest = lv; best_col = j;
5365 : }
5366 1778 : for (i = 1; i <= n; i++)
5367 : {
5368 1386 : pari_sp av2 = avma;
5369 1386 : GEN v = row_count_non_zero(M, i, max);
5370 : long lv;
5371 1386 : if (!v || (lv = lg(v)) >= lbest) { set_avma(av2); continue; }
5372 0 : if (lv == 1) { set_avma(av); return gen_0; }
5373 0 : if (lv == 2) {
5374 0 : set_avma(av);
5375 0 : return gerepileupto(av, coeff_det(M,i,v[1],max,bound));
5376 : }
5377 0 : best = v; lbest = lv; best_row = i;
5378 : }
5379 392 : if (best_row)
5380 : {
5381 0 : double d = lbest-1;
5382 0 : GEN s = NULL;
5383 : long k;
5384 0 : bound /= d*d*d;
5385 0 : for (k = 1; k < lbest; k++)
5386 : {
5387 0 : GEN c = coeff_det(M, best_row, best[k], max, bound);
5388 0 : s = s? gadd(s, c): c;
5389 : }
5390 0 : return gerepileupto(av, s);
5391 : }
5392 392 : if (best_col)
5393 : {
5394 168 : double d = lbest-1;
5395 168 : GEN s = NULL;
5396 : long k;
5397 168 : bound /= d*d*d;
5398 560 : for (k = 1; k < lbest; k++)
5399 : {
5400 392 : GEN c = coeff_det(M, best[k], best_col, max, bound);
5401 392 : s = s? gadd(s, c): c;
5402 : }
5403 168 : return gerepileupto(av, s);
5404 : }
5405 224 : return det_bareiss(M);
5406 : }
5407 :
5408 : /* area of parallelogram bounded by (v1,v2) */
5409 : static GEN
5410 61810 : parallelogramarea(GEN v1, GEN v2)
5411 61810 : { return gsub(gmul(gnorml2(v1), gnorml2(v2)), gsqr(RgV_dotproduct(v1, v2))); }
5412 :
5413 : /* Square of Hadamard bound for det(a), a square matrix.
5414 : * Slight improvement: instead of using the column norms, use the area of
5415 : * the parallelogram formed by pairs of consecutive vectors */
5416 : GEN
5417 19397 : RgM_Hadamard(GEN a)
5418 : {
5419 19397 : pari_sp av = avma;
5420 19397 : long n = lg(a)-1, i;
5421 : GEN B;
5422 19397 : if (n == 0) return gen_1;
5423 19397 : if (n == 1) return gsqr(gcoeff(a,1,1));
5424 19397 : a = RgM_gtofp(a, LOWDEFAULTPREC);
5425 19397 : B = gen_1;
5426 81207 : for (i = 1; i <= n/2; i++)
5427 61810 : B = gmul(B, parallelogramarea(gel(a,2*i-1), gel(a,2*i)));
5428 19397 : if (odd(n)) B = gmul(B, gnorml2(gel(a, n)));
5429 19397 : return gerepileuptoint(av, ceil_safe(B));
5430 : }
5431 :
5432 : /* If B=NULL, assume B=A' */
5433 : static GEN
5434 20570 : ZM_det_slice(GEN A, GEN P, GEN *mod)
5435 : {
5436 20570 : pari_sp av = avma;
5437 20570 : long i, n = lg(P)-1;
5438 : GEN H, T;
5439 20570 : if (n == 1)
5440 : {
5441 0 : ulong Hp, p = uel(P,1);
5442 0 : GEN a = ZM_to_Flm(A, p);
5443 0 : Hp = Flm_det_sp(a, p);
5444 0 : set_avma(av); *mod = utoipos(p); return utoi(Hp);
5445 : }
5446 20570 : T = ZV_producttree(P);
5447 20570 : A = ZM_nv_mod_tree(A, P, T);
5448 20570 : H = cgetg(n+1, t_VECSMALL);
5449 85601 : for(i=1; i <= n; i++)
5450 : {
5451 65031 : ulong p = P[i];
5452 65031 : GEN a = gel(A,i);
5453 65031 : H[i] = Flm_det_sp(a, p);
5454 : }
5455 20570 : H = ZV_chinese_tree(H, P, T, ZV_chinesetree(P,T));
5456 20570 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
5457 : }
5458 :
5459 : GEN
5460 20570 : ZM_det_worker(GEN P, GEN A)
5461 : {
5462 20570 : GEN V = cgetg(3, t_VEC);
5463 20570 : gel(V,1) = ZM_det_slice(A, P, &gel(V,2));
5464 20570 : return V;
5465 : }
5466 :
5467 : GEN
5468 120148 : ZM_det(GEN M)
5469 : {
5470 120148 : const long DIXON_THRESHOLD = 40;
5471 : pari_sp av, av2;
5472 120148 : long i, n = lg(M)-1;
5473 : ulong p, Dp;
5474 : forprime_t S;
5475 : pari_timer ti;
5476 : GEN H, D, mod, h, q, v, worker;
5477 : #ifdef LONG_IS_64BIT
5478 102984 : const ulong PMAX = 18446744073709551557UL;
5479 : #else
5480 17164 : const ulong PMAX = 4294967291UL;
5481 : #endif
5482 :
5483 120148 : switch(n)
5484 : {
5485 7 : case 0: return gen_1;
5486 1518 : case 1: return icopy(gcoeff(M,1,1));
5487 6364 : case 2: return ZM_det2(M);
5488 92862 : case 3: return ZM_det3(M);
5489 : }
5490 19397 : if (DEBUGLEVEL>=4) timer_start(&ti);
5491 19397 : av = avma; h = RgM_Hadamard(M); /* |D| <= sqrt(h) */
5492 19397 : if (!signe(h)) { set_avma(av); return gen_0; }
5493 19397 : h = sqrti(h);
5494 19397 : if (lgefint(h) == 3 && (ulong)h[2] <= (PMAX >> 1))
5495 : { /* h < p/2 => direct result */
5496 6747 : p = PMAX;
5497 6747 : Dp = Flm_det_sp(ZM_to_Flm(M, p), p);
5498 6747 : set_avma(av);
5499 6747 : if (!Dp) return gen_0;
5500 6747 : return (Dp <= (p>>1))? utoipos(Dp): utoineg(p - Dp);
5501 : }
5502 12650 : q = gen_1; Dp = 1;
5503 12650 : init_modular_big(&S);
5504 12650 : p = 0; /* -Wall */
5505 12650 : while (cmpii(q, h) <= 0 && (p = u_forprime_next(&S)))
5506 : {
5507 12650 : av2 = avma; Dp = Flm_det_sp(ZM_to_Flm(M, p), p);
5508 12650 : set_avma(av2);
5509 12650 : if (Dp) break;
5510 0 : q = muliu(q, p);
5511 : }
5512 12650 : if (!p) pari_err_OVERFLOW("ZM_det [ran out of primes]");
5513 12650 : if (!Dp) { set_avma(av); return gen_0; }
5514 12650 : if (mt_nbthreads() > 1 || n <= DIXON_THRESHOLD)
5515 12650 : D = q; /* never competitive when bound is sharp even with 2 threads */
5516 : else
5517 : {
5518 0 : av2 = avma;
5519 0 : v = cgetg(n+1, t_COL);
5520 0 : gel(v, 1) = gen_1; /* ensure content(v) = 1 */
5521 0 : for (i = 2; i <= n; i++) gel(v, i) = stoi(random_Fl(15) - 7);
5522 0 : D = Q_denom(ZM_gauss(M, v));
5523 0 : if (expi(D) < expi(h) >> 1)
5524 : { /* First try unlucky, try once more */
5525 0 : for (i = 2; i <= n; i++) gel(v, i) = stoi(random_Fl(15) - 7);
5526 0 : D = lcmii(D, Q_denom(ZM_gauss(M, v)));
5527 : }
5528 0 : D = gerepileuptoint(av2, D);
5529 0 : if (q != gen_1) D = lcmii(D, q);
5530 : }
5531 12650 : if (DEBUGLEVEL >=4)
5532 0 : timer_printf(&ti,"ZM_det: Dixon %ld/%ld bits",expi(D),expi(h));
5533 : /* determinant is a multiple of D */
5534 12650 : if (is_pm1(D)) D = NULL;
5535 12650 : if (D) h = diviiexact(h, D);
5536 12650 : worker = snm_closure(is_entry("_ZM_det_worker"), mkvec(M));
5537 12650 : H = gen_crt("ZM_det", worker, &S, D, expi(h)+1, 0, &mod,
5538 : ZV_chinese, NULL);
5539 12650 : if (D) H = Fp_div(H, D, mod);
5540 12650 : H = Fp_center(H, mod, shifti(mod,-1));
5541 12650 : if (D) H = mulii(H, D);
5542 12650 : return gerepileuptoint(av, H);
5543 : }
5544 :
5545 : static GEN
5546 1519 : RgM_det_FpM(GEN a, GEN p)
5547 : {
5548 1519 : pari_sp av = avma;
5549 : ulong pp, d;
5550 1519 : a = RgM_Fp_init(a,p,&pp);
5551 1519 : switch(pp)
5552 : {
5553 70 : case 0: return gerepileupto(av, Fp_to_mod(FpM_det(a,p),p)); break;
5554 14 : case 2: d = F2m_det_sp(a); break;
5555 1435 : default:d = Flm_det_sp(a, pp); break;
5556 : }
5557 1449 : set_avma(av); return mkintmodu(d, pp);
5558 : }
5559 :
5560 : static GEN
5561 42 : RgM_det_FqM(GEN x, GEN pol, GEN p)
5562 : {
5563 42 : pari_sp av = avma;
5564 42 : GEN b, T = RgX_to_FpX(pol, p);
5565 42 : if (signe(T) == 0) pari_err_OP("%",x,pol);
5566 42 : b = FqM_det(RgM_to_FqM(x, T, p), T, p);
5567 42 : if (!b) return gc_NULL(av);
5568 42 : return gerepilecopy(av, mkpolmod(FpX_to_mod(b, p), FpX_to_mod(T, p)));
5569 : }
5570 :
5571 : #define code(t1,t2) ((t1 << 6) | t2)
5572 : static GEN
5573 36078 : RgM_det_fast(GEN x)
5574 : {
5575 : GEN p, pol;
5576 : long pa;
5577 36078 : long t = RgM_type(x, &p,&pol,&pa);
5578 36078 : switch(t)
5579 : {
5580 175 : case t_INT: return ZM_det(x);
5581 196 : case t_FRAC: return QM_det(x);
5582 63 : case t_FFELT: return FFM_det(x, pol);
5583 1519 : case t_INTMOD: return RgM_det_FpM(x, p);
5584 42 : case code(t_POLMOD, t_INTMOD):
5585 42 : return RgM_det_FqM(x, pol, p);
5586 34083 : default: return NULL;
5587 : }
5588 : }
5589 : #undef code
5590 :
5591 : static long
5592 238 : det_init_max(long n)
5593 : {
5594 238 : if (n > 100) return 0;
5595 238 : if (n > 50) return 1;
5596 238 : if (n > 30) return 4;
5597 238 : return 7;
5598 : }
5599 :
5600 : GEN
5601 354045 : det(GEN a)
5602 : {
5603 354045 : long n = lg(a)-1;
5604 : double B;
5605 : GEN data, b;
5606 : pivot_fun pivot;
5607 :
5608 354045 : if (typ(a)!=t_MAT) pari_err_TYPE("det",a);
5609 354045 : if (!n) return gen_1;
5610 354003 : if (n != nbrows(a)) pari_err_DIM("det");
5611 353996 : if (n == 1) return gcopy(gcoeff(a,1,1));
5612 71098 : if (n == 2) return RgM_det2(a);
5613 36078 : b = RgM_det_fast(a);
5614 36078 : if (b) return b;
5615 34083 : pivot = get_pivot_fun(a, a, &data);
5616 34083 : if (pivot != gauss_get_pivot_NZ) return det_simple_gauss(a, data, pivot);
5617 238 : B = (double)n;
5618 238 : return det_develop(a, det_init_max(n), B*B*B);
5619 : }
5620 :
5621 : GEN
5622 196 : QM_det(GEN M)
5623 : {
5624 196 : pari_sp av = avma;
5625 196 : GEN cM, pM = Q_primitive_part(M, &cM);
5626 196 : GEN b = ZM_det(pM);
5627 196 : if (cM) b = gmul(b, gpowgs(cM, lg(M)-1));
5628 196 : return gerepileupto(av, b);
5629 : }
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