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 1906 : gen_ker(GEN x, long deplin, void *E, const struct bb_field *ff)
124 : {
125 1906 : pari_sp av0 = avma, av, tetpil;
126 : GEN y, c, d;
127 : long i, j, k, r, t, n, m;
128 :
129 1906 : n=lg(x)-1; if (!n) return cgetg(1,t_MAT);
130 1906 : m=nbrows(x); r=0;
131 1906 : x = RgM_shallowcopy(x);
132 1906 : c = zero_zv(m);
133 1906 : d=new_chunk(n+1);
134 1906 : av=avma;
135 6897 : for (k=1; k<=n; k++)
136 : {
137 15211 : for (j=1; j<=m; j++)
138 13052 : if (!c[j])
139 : {
140 9266 : gcoeff(x,j,k) = ff->red(E, gcoeff(x,j,k));
141 9266 : if (!ff->equal0(gcoeff(x,j,k))) break;
142 : }
143 5026 : if (j>m)
144 : {
145 2159 : if (deplin)
146 : {
147 35 : GEN c = cgetg(n+1, t_COL), g0 = ff->s(E,0), g1=ff->s(E,1);
148 98 : for (i=1; i<k; i++) gel(c,i) = ff->red(E, gcoeff(x,d[i],k));
149 63 : gel(c,k) = g1; for (i=k+1; i<=n; i++) gel(c,i) = g0;
150 35 : return gerepileupto(av0, c);
151 : }
152 2124 : r++; d[k]=0;
153 5101 : for(j=1; j<k; j++)
154 2977 : if (d[j]) gcoeff(x,d[j],k) = gclone(gcoeff(x,d[j],k));
155 : }
156 : else
157 : {
158 2867 : GEN piv = ff->neg(E,ff->inv(E,gcoeff(x,j,k)));
159 2867 : c[j] = k; d[k] = j;
160 2867 : gcoeff(x,j,k) = ff->s(E,-1);
161 6888 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = ff->red(E,ff->mul(E,piv,gcoeff(x,j,i)));
162 16275 : for (t=1; t<=m; t++)
163 : {
164 13408 : if (t==j) continue;
165 :
166 10541 : piv = ff->red(E,gcoeff(x,t,k));
167 10541 : if (ff->equal0(piv)) continue;
168 :
169 3105 : gcoeff(x,t,k) = ff->s(E,0);
170 7570 : for (i=k+1; i<=n; i++)
171 4465 : gcoeff(x,t,i) = ff->red(E, ff->add(E, gcoeff(x,t,i),
172 4465 : ff->mul(E,piv,gcoeff(x,j,i))));
173 3105 : if (gc_needed(av,1))
174 0 : gen_gerepile_gauss_ker(x,k,t,av,E,ff->red);
175 : }
176 : }
177 : }
178 1871 : if (deplin) return gc_NULL(av0);
179 :
180 1843 : tetpil=avma; y=cgetg(r+1,t_MAT);
181 3967 : for (j=k=1; j<=r; j++,k++)
182 : {
183 2124 : GEN C = cgetg(n+1,t_COL);
184 2124 : GEN g0 = ff->s(E,0), g1 = ff->s(E,1);
185 4092 : gel(y,j) = C; while (d[k]) k++;
186 5101 : for (i=1; i<k; i++)
187 2977 : if (d[i])
188 : {
189 2427 : GEN p1=gcoeff(x,d[i],k);
190 2427 : gel(C,i) = ff->red(E,p1); gunclone(p1);
191 : }
192 : else
193 550 : gel(C,i) = g0;
194 2969 : gel(C,k) = g1; for (i=k+1; i<=n; i++) gel(C,i) = g0;
195 : }
196 1843 : return gerepile(av0,tetpil,y);
197 : }
198 :
199 : GEN
200 1891 : gen_Gauss_pivot(GEN x, long *rr, void *E, const struct bb_field *ff)
201 : {
202 : pari_sp av;
203 : GEN c, d;
204 1891 : long i, j, k, r, t, m, n = lg(x)-1;
205 :
206 1891 : if (!n) { *rr = 0; return NULL; }
207 :
208 1891 : m=nbrows(x); r=0;
209 1891 : d = cgetg(n+1, t_VECSMALL);
210 1891 : x = RgM_shallowcopy(x);
211 1891 : c = zero_zv(m);
212 1891 : av=avma;
213 6830 : for (k=1; k<=n; k++)
214 : {
215 13759 : for (j=1; j<=m; j++)
216 13151 : if (!c[j])
217 : {
218 9256 : gcoeff(x,j,k) = ff->red(E,gcoeff(x,j,k));
219 9256 : if (!ff->equal0(gcoeff(x,j,k))) break;
220 : }
221 4939 : if (j>m) { r++; d[k]=0; }
222 : else
223 : {
224 4331 : GEN piv = ff->neg(E,ff->inv(E,gcoeff(x,j,k)));
225 4331 : GEN g0 = ff->s(E,0);
226 4331 : c[j] = k; d[k] = j;
227 9032 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = ff->red(E,ff->mul(E,piv,gcoeff(x,j,i)));
228 26263 : for (t=1; t<=m; t++)
229 : {
230 21932 : if (c[t]) continue; /* already a pivot on that line */
231 :
232 13493 : piv = ff->red(E,gcoeff(x,t,k));
233 13493 : if (ff->equal0(piv)) continue;
234 5272 : gcoeff(x,t,k) = g0;
235 9673 : for (i=k+1; i<=n; i++)
236 4401 : gcoeff(x,t,i) = ff->red(E, ff->add(E,gcoeff(x,t,i), ff->mul(E,piv,gcoeff(x,j,i))));
237 5272 : if (gc_needed(av,1))
238 0 : gerepile_gauss(x,k,t,av,j,c);
239 : }
240 13363 : for (i=k; i<=n; i++) gcoeff(x,j,i) = g0; /* dummy */
241 : }
242 : }
243 1891 : *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 152260 : _gen_addmul(GEN b, long k, long i, GEN m, void *E, const struct bb_field *ff)
292 : {
293 152260 : gel(b,i) = ff->red(E,gel(b,i));
294 152260 : gel(b,k) = ff->add(E,gel(b,k), ff->mul(E,m, gel(b,i)));
295 152260 : }
296 :
297 : static GEN
298 61178 : _gen_get_col(GEN a, GEN b, long li, void *E, const struct bb_field *ff)
299 : {
300 61178 : GEN u = cgetg(li+1,t_COL);
301 61178 : pari_sp av = avma;
302 : long i, j;
303 :
304 61178 : gel(u,li) = gerepileupto(av, ff->red(E,ff->mul(E,gel(b,li), gcoeff(a,li,li))));
305 314277 : for (i=li-1; i>0; i--)
306 : {
307 253099 : pari_sp av = avma;
308 253099 : GEN m = gel(b,i);
309 1005070 : 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 253099 : m = ff->red(E, m);
311 253099 : gel(u,i) = gerepileupto(av, ff->red(E,ff->mul(E,m, gcoeff(a,i,i))));
312 : }
313 61178 : return u;
314 : }
315 :
316 : GEN
317 13745 : gen_Gauss(GEN a, GEN b, void *E, const struct bb_field *ff)
318 : {
319 : long i, j, k, li, bco, aco;
320 13745 : GEN u, g0 = ff->s(E,0);
321 13745 : pari_sp av = avma;
322 13745 : a = RgM_shallowcopy(a);
323 13745 : b = RgM_shallowcopy(b);
324 13745 : aco = lg(a)-1; bco = lg(b)-1; li = nbrows(a);
325 59932 : for (i=1; i<=aco; i++)
326 : {
327 : GEN invpiv;
328 72138 : for (k = i; k <= li; k++)
329 : {
330 72082 : GEN piv = ff->red(E,gcoeff(a,k,i));
331 72082 : if (!ff->equal0(piv)) { gcoeff(a,k,i) = ff->inv(E,piv); break; }
332 12206 : gcoeff(a,k,i) = g0;
333 : }
334 : /* found a pivot on line k */
335 59932 : if (k > li) return NULL;
336 59876 : if (k != i)
337 : { /* swap lines so that k = i */
338 50677 : for (j=i; j<=aco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
339 67853 : for (j=1; j<=bco; j++) swap(gcoeff(b,i,j), gcoeff(b,k,j));
340 : }
341 59876 : if (i == aco) break;
342 :
343 46187 : invpiv = gcoeff(a,i,i); /* 1/piv mod p */
344 173552 : for (k=i+1; k<=li; k++)
345 : {
346 127365 : GEN m = ff->red(E,gcoeff(a,k,i)); gcoeff(a,k,i) = g0;
347 127365 : if (ff->equal0(m)) continue;
348 :
349 18719 : m = ff->red(E,ff->neg(E,ff->mul(E,m, invpiv)));
350 75014 : for (j=i+1; j<=aco; j++) _gen_addmul(gel(a,j),k,i,m,E,ff);
351 114684 : for (j=1 ; j<=bco; j++) _gen_addmul(gel(b,j),k,i,m,E,ff);
352 : }
353 46187 : 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 13689 : if(DEBUGLEVEL>4) err_printf("Solving the triangular system\n");
361 13689 : u = cgetg(bco+1,t_MAT);
362 74867 : for (j=1; j<=bco; j++) gel(u,j) = _gen_get_col(a, gel(b,j), aco, E, ff);
363 13689 : return u;
364 : }
365 :
366 : /* compatible t_MAT * t_COL, lgA = lg(A) = lg(B) > 1, l = lgcols(A) */
367 : static GEN
368 631002 : gen_matcolmul_i(GEN A, GEN B, ulong lgA, ulong l,
369 : void *E, const struct bb_field *ff)
370 : {
371 631002 : GEN C = cgetg(l, t_COL);
372 : ulong i;
373 4152844 : for (i = 1; i < l; i++) {
374 3521842 : pari_sp av = avma;
375 3521842 : GEN e = ff->mul(E, gcoeff(A, i, 1), gel(B, 1));
376 : ulong k;
377 15208558 : for(k = 2; k < lgA; k++)
378 11686716 : e = ff->add(E, e, ff->mul(E, gcoeff(A, i, k), gel(B, k)));
379 3521842 : gel(C, i) = gerepileupto(av, ff->red(E, e));
380 : }
381 631002 : return C;
382 : }
383 :
384 : GEN
385 199558 : gen_matcolmul(GEN A, GEN B, void *E, const struct bb_field *ff)
386 : {
387 199558 : ulong lgA = lg(A);
388 199558 : if (lgA != (ulong)lg(B))
389 0 : pari_err_OP("operation 'gen_matcolmul'", A, B);
390 199558 : if (lgA == 1)
391 0 : return cgetg(1, t_COL);
392 199558 : return gen_matcolmul_i(A, B, lgA, lgcols(A), E, ff);
393 : }
394 :
395 : static GEN
396 83789 : 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 83789 : GEN C = cgetg(lb, t_MAT);
401 515233 : for(j = 1; j < lb; j++)
402 431444 : gel(C, j) = gen_matcolmul_i(A, gel(B, j), la, l, E, ff);
403 83789 : 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 83789 : gen_matmul_i(GEN A, GEN B, long l, long la, long lb,
587 : void *E, const struct bb_field *ff)
588 : {
589 83789 : if (l <= gen_matmul_sw_bound
590 7 : || la <= gen_matmul_sw_bound
591 0 : || lb <= gen_matmul_sw_bound)
592 83789 : 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 83789 : gen_matmul(GEN A, GEN B, void *E, const struct bb_field *ff)
599 : {
600 83789 : ulong lgA, lgB = lg(B);
601 83789 : if (lgB == 1)
602 0 : return cgetg(1, t_MAT);
603 83789 : lgA = lg(A);
604 83789 : if (lgA != (ulong)lgcols(B))
605 0 : pari_err_OP("operation 'gen_matmul'", A, B);
606 83789 : if (lgA == 1)
607 0 : return zeromat(0, lgB - 1);
608 83789 : return gen_matmul_i(A, B, lgcols(A), lgA, lgB, E, ff);
609 : }
610 :
611 : static GEN
612 19333 : gen_colneg(GEN A, void *E, const struct bb_field *ff)
613 : {
614 : long i, l;
615 19333 : GEN B = cgetg_copy(A, &l);
616 74092 : for (i = 1; i < l; i++)
617 54759 : gel(B, i) = ff->neg(E, gel(A, i));
618 19333 : return B;
619 : }
620 :
621 : static GEN
622 4008 : gen_matneg(GEN A, void *E, const struct bb_field *ff)
623 : {
624 : long i, l;
625 4008 : GEN B = cgetg_copy(A, &l);
626 23271 : for (i = 1; i < l; i++)
627 19263 : gel(B, i) = gen_colneg(gel(A, i), E, ff);
628 4008 : return B;
629 : }
630 :
631 : static GEN
632 354731 : gen_colscalmul(GEN A, GEN b, void *E, const struct bb_field *ff)
633 : {
634 : long i, l;
635 354731 : GEN B = cgetg_copy(A, &l);
636 826456 : for (i = 1; i < l; i++)
637 471725 : gel(B, i) = ff->red(E, ff->mul(E, gel(A, i), b));
638 354731 : return B;
639 : }
640 :
641 : static GEN
642 54448 : gen_matscalmul(GEN A, GEN b, void *E, const struct bb_field *ff)
643 : {
644 : long i, l;
645 54448 : GEN B = cgetg_copy(A, &l);
646 409179 : for (i = 1; i < l; i++)
647 354731 : gel(B, i) = gen_colscalmul(gel(A, i), b, E, ff);
648 54448 : return B;
649 : }
650 :
651 : static GEN
652 681972 : gen_colsub(GEN A, GEN C, void *E, const struct bb_field *ff)
653 : {
654 : long i, l;
655 681972 : GEN B = cgetg_copy(A, &l);
656 2432697 : for (i = 1; i < l; i++)
657 1750725 : gel(B, i) = ff->add(E, gel(A, i), ff->neg(E, gel(C, i)));
658 681972 : return B;
659 : }
660 :
661 : static GEN
662 78597 : gen_matsub(GEN A, GEN C, void *E, const struct bb_field *ff)
663 : {
664 : long i, l;
665 78597 : GEN B = cgetg_copy(A, &l);
666 760569 : for (i = 1; i < l; i++)
667 681972 : gel(B, i) = gen_colsub(gel(A, i), gel(C, i), E, ff);
668 78597 : return B;
669 : }
670 :
671 : static GEN
672 46755 : gen_zerocol(long n, void* data, const struct bb_field *R)
673 : {
674 46755 : GEN C = cgetg(n+1,t_COL), zero = R->s(data, 0);
675 : long i;
676 282231 : for (i=1; i<=n; i++) gel(C,i) = zero;
677 46755 : return C;
678 : }
679 :
680 : static GEN
681 14906 : gen_zeromat(long m, long n, void* data, const struct bb_field *R)
682 : {
683 14906 : GEN M = cgetg(n+1,t_MAT);
684 : long i;
685 61661 : for (i=1; i<=n; i++) gel(M,i) = gen_zerocol(m, data, R);
686 14906 : return M;
687 : }
688 :
689 : static GEN
690 154 : gen_colei(long n, long i, void *E, const struct bb_field *S)
691 : {
692 154 : GEN y = cgetg(n+1,t_COL), _0, _1;
693 : long j;
694 154 : if (n < 0) pari_err_DOMAIN("gen_colei", "dimension","<",gen_0,stoi(n));
695 154 : _0 = S->s(E,0);
696 154 : _1 = S->s(E,1);
697 2422 : for (j=1; j<=n; j++)
698 2268 : gel(y, j) = i==j ? _1: _0;
699 154 : 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 352266 : image_from_pivot(GEN x, GEN d, long r)
785 : {
786 : GEN y;
787 : long j, k;
788 :
789 352266 : if (!d) return gcopy(x);
790 : /* d left on stack for efficiency */
791 348587 : r = lg(x)-1 - r; /* = dim Im(x) */
792 348587 : y = cgetg(r+1,t_MAT);
793 2020123 : for (j=k=1; j<=r; k++)
794 1671540 : if (d[k]) gel(y,j++) = gcopy(gel(x,k));
795 348583 : return y;
796 : }
797 :
798 : /* r = dim Ker x, n = nbrows(x) */
799 : static GEN
800 261107 : get_suppl(GEN x, GEN d, long n, long r, GEN(*ei)(long,long))
801 : {
802 : pari_sp av;
803 : GEN y, c;
804 261107 : long j, k, rx = lg(x)-1; /* != 0 due to init_suppl() */
805 :
806 261107 : if (rx == n && r == 0) return gcopy(x);
807 191481 : y = cgetg(n+1, t_MAT);
808 191483 : 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 810963 : for (k = j = 1; j<=rx; j++)
812 619479 : if (d[j]) { c[ d[j] ] = 1; gel(y,k++) = gel(x,j); }
813 1139263 : for (j=1; j<=n; j++)
814 947779 : if (!c[j]) gel(y,k++) = (GEN)j; /* HACK */
815 191484 : set_avma(av);
816 :
817 191484 : rx -= r;
818 810892 : for (j=1; j<=rx; j++) gel(y,j) = gcopy(gel(y,j));
819 519848 : for ( ; j<=n; j++) gel(y,j) = ei(n, y[j]);
820 191485 : return y;
821 : }
822 :
823 : /* n = dim x, r = dim Ker(x), d from gauss_pivot */
824 : static GEN
825 1929902 : indexrank0(long n, long r, GEN d)
826 : {
827 1929902 : GEN p1, p2, res = cgetg(3,t_VEC);
828 : long i, j;
829 :
830 1929902 : r = n - r; /* now r = dim Im(x) */
831 1929902 : p1 = cgetg(r+1,t_VECSMALL); gel(res,1) = p1;
832 1929899 : p2 = cgetg(r+1,t_VECSMALL); gel(res,2) = p2;
833 1929898 : if (d)
834 : {
835 7720152 : for (i=0,j=1; j<=n; j++)
836 5791619 : if (d[j]) { i++; p1[i] = d[j]; p2[i] = j; }
837 1928533 : vecsmall_sort(p1);
838 : }
839 1929903 : 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 17571 : indexcompl(GEN v, long n)
867 : {
868 17571 : long i, j, k, m = lg(v) - 1;
869 17571 : GEN w = cgetg(n - m + 1, t_VECSMALL);
870 165614 : for (i = j = k = 1; i <= n; i++)
871 148043 : if (j <= m && v[j] == i) j++; else w[k++] = i;
872 17571 : return w;
873 : }
874 :
875 : static GEN
876 4085 : gen_solve_upper_1(GEN U, GEN B, void *E, const struct bb_field *ff)
877 4085 : { return gen_matscalmul(B, ff->inv(E, gcoeff(U, 1, 1)), E, ff); }
878 :
879 : static GEN
880 2256 : gen_rsolve_upper_2(GEN U, GEN B, void *E, const struct bb_field *ff)
881 : {
882 2256 : GEN a = gcoeff(U, 1, 1), b = gcoeff(U, 1, 2), d = gcoeff(U, 2, 2);
883 2256 : GEN D = ff->red(E, ff->mul(E, a, d)), Dinv = ff->inv(E, D);
884 2256 : GEN ainv = ff->red(E, ff->mul(E, d, Dinv));
885 2256 : GEN dinv = ff->red(E, ff->mul(E, a, Dinv));
886 2256 : GEN B1 = rowslice(B, 1, 1);
887 2256 : GEN B2 = rowslice(B, 2, 2);
888 2256 : GEN X2 = gen_matscalmul(B2, dinv, E, ff);
889 2256 : GEN X1 = gen_matscalmul(gen_matsub(B1, gen_matscalmul(X2, b, E, ff), E, ff),
890 : ainv, E, ff);
891 2256 : return vconcat(X1, X2);
892 : }
893 :
894 : /* solve U*X = B, U upper triangular and invertible */
895 : static GEN
896 5840 : gen_rsolve_upper(GEN U, GEN B, void *E, const struct bb_field *ff,
897 : GEN (*mul)(void *E, GEN a, GEN))
898 : {
899 5840 : long n = lg(U) - 1, n1;
900 : GEN U2, U11, U12, U22, B1, B2, X1, X2, X;
901 5840 : pari_sp av = avma;
902 :
903 5840 : if (n == 0) return B;
904 5840 : if (n == 1) return gen_solve_upper_1(U, B, E, ff);
905 4914 : if (n == 2) return gen_rsolve_upper_2(U, B, E, ff);
906 2658 : n1 = (n + 1)/2;
907 2658 : U2 = vecslice(U, n1 + 1, n);
908 2658 : U11 = matslice(U, 1,n1, 1,n1);
909 2658 : U12 = rowslice(U2, 1, n1);
910 2658 : U22 = rowslice(U2, n1 + 1, n);
911 2658 : B1 = rowslice(B, 1, n1);
912 2658 : B2 = rowslice(B, n1 + 1, n);
913 2658 : X2 = gen_rsolve_upper(U22, B2, E, ff, mul);
914 2658 : B1 = gen_matsub(B1, mul(E, U12, X2), E, ff);
915 2658 : if (gc_needed(av, 1)) gerepileall(av, 3, &B1, &U11, &X2);
916 2658 : X1 = gen_rsolve_upper(U11, B1, E, ff, mul);
917 2658 : X = vconcat(X1, X2);
918 2658 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
919 2658 : return X;
920 : }
921 :
922 : static GEN
923 6180 : gen_lsolve_upper_2(GEN U, GEN B, void *E, const struct bb_field *ff)
924 : {
925 6180 : GEN a = gcoeff(U, 1, 1), b = gcoeff(U, 1, 2), d = gcoeff(U, 2, 2);
926 6180 : GEN D = ff->red(E, ff->mul(E, a, d)), Dinv = ff->inv(E, D);
927 6180 : GEN ainv = ff->red(E, ff->mul(E, d, Dinv)), dinv = ff->red(E, ff->mul(E, a, Dinv));
928 6180 : GEN B1 = vecslice(B, 1, 1);
929 6180 : GEN B2 = vecslice(B, 2, 2);
930 6180 : GEN X1 = gen_matscalmul(B1, ainv, E, ff);
931 6180 : GEN X2 = gen_matscalmul(gen_matsub(B2, gen_matscalmul(X1, b, E, ff), E, ff), dinv, E, ff);
932 6180 : return shallowconcat(X1, X2);
933 : }
934 :
935 : /* solve X*U = B, U upper triangular and invertible */
936 : static GEN
937 14256 : gen_lsolve_upper(GEN U, GEN B, void *E, const struct bb_field *ff,
938 : GEN (*mul)(void *E, GEN a, GEN))
939 : {
940 14256 : long n = lg(U) - 1, n1;
941 : GEN U2, U11, U12, U22, B1, B2, X1, X2, X;
942 14256 : pari_sp av = avma;
943 :
944 14256 : if (n == 0) return B;
945 14256 : if (n == 1) return gen_solve_upper_1(U, B, E, ff);
946 11097 : if (n == 2) return gen_lsolve_upper_2(U, B, E, ff);
947 4917 : n1 = (n + 1)/2;
948 4917 : U2 = vecslice(U, n1 + 1, n);
949 4917 : U11 = matslice(U, 1,n1, 1,n1);
950 4917 : U12 = rowslice(U2, 1, n1);
951 4917 : U22 = rowslice(U2, n1 + 1, n);
952 4917 : B1 = vecslice(B, 1, n1);
953 4917 : B2 = vecslice(B, n1 + 1, n);
954 4917 : X1 = gen_lsolve_upper(U11, B1, E, ff, mul);
955 4917 : B2 = gen_matsub(B2, mul(E, X1, U12), E, ff);
956 4917 : if (gc_needed(av, 1)) gerepileall(av, 3, &B2, &U22, &X1);
957 4917 : X2 = gen_lsolve_upper(U22, B2, E, ff, mul);
958 4917 : X = shallowconcat(X1, X2);
959 4917 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
960 4917 : return X;
961 : }
962 :
963 : static GEN
964 17190 : gen_rsolve_lower_unit_2(GEN L, GEN A, void *E, const struct bb_field *ff)
965 : {
966 17190 : GEN X1 = rowslice(A, 1, 1);
967 17190 : GEN X2 = gen_matsub(rowslice(A, 2, 2), gen_matscalmul(X1, gcoeff(L, 2, 1), E, ff), E, ff);
968 17190 : 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 40212 : 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 40212 : long m = lg(L) - 1, m1, n;
978 : GEN L1, L11, L21, L22, A1, A2, X1, X2, X;
979 40212 : pari_sp av = avma;
980 :
981 40212 : if (m == 0) return zeromat(0, lg(A) - 1);
982 40212 : if (m == 1) return rowslice(A, 1, 1);
983 31789 : if (m == 2) return gen_rsolve_lower_unit_2(L, A, E, ff);
984 14599 : m1 = (m + 1)/2;
985 14599 : n = nbrows(L);
986 14599 : L1 = vecslice(L, 1, m1);
987 14599 : L11 = rowslice(L1, 1, m1);
988 14599 : L21 = rowslice(L1, m1 + 1, n);
989 14599 : A1 = rowslice(A, 1, m1);
990 14599 : X1 = gen_rsolve_lower_unit(L11, A1, E, ff, mul);
991 14599 : A2 = rowslice(A, m1 + 1, n);
992 14599 : A2 = gen_matsub(A2, mul(E, L21, X1), E, ff);
993 14599 : if (gc_needed(av, 1)) gerepileall(av, 2, &A2, &X1);
994 14599 : L22 = matslice(L, m1+1,n, m1+1,m);
995 14599 : X2 = gen_rsolve_lower_unit(L22, A2, E, ff, mul);
996 14599 : X = vconcat(X1, X2);
997 14599 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
998 14599 : return X;
999 : }
1000 :
1001 : static GEN
1002 7865 : gen_lsolve_lower_unit_2(GEN L, GEN A, void *E, const struct bb_field *ff)
1003 : {
1004 7865 : GEN X2 = vecslice(A, 2, 2);
1005 7865 : GEN X1 = gen_matsub(vecslice(A, 1, 1),
1006 7865 : gen_matscalmul(X2, gcoeff(L, 2, 1), E, ff), E, ff);
1007 7865 : 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 20125 : 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 20125 : long m = lg(L) - 1, m1;
1017 : GEN L1, L2, L11, L21, L22, A1, A2, X1, X2, X;
1018 20125 : pari_sp av = avma;
1019 :
1020 20125 : if (m <= 1) return A;
1021 15927 : if (m == 2) return gen_lsolve_lower_unit_2(L, A, E, ff);
1022 8062 : m1 = (m + 1)/2;
1023 8062 : L2 = vecslice(L, m1 + 1, m);
1024 8062 : L22 = rowslice(L2, m1 + 1, m);
1025 8062 : A2 = vecslice(A, m1 + 1, m);
1026 8062 : X2 = gen_lsolve_lower_unit(L22, A2, E, ff, mul);
1027 8062 : if (gc_needed(av, 1)) X2 = gerepilecopy(av, X2);
1028 8062 : L1 = vecslice(L, 1, m1);
1029 8062 : L21 = rowslice(L1, m1 + 1, m);
1030 8062 : A1 = vecslice(A, 1, m1);
1031 8062 : A1 = gen_matsub(A1, mul(E, X2, L21), E, ff);
1032 8062 : L11 = rowslice(L1, 1, m1);
1033 8062 : if (gc_needed(av, 1)) gerepileall(av, 3, &A1, &L11, &X2);
1034 8062 : X1 = gen_lsolve_lower_unit(L11, A1, E, ff, mul);
1035 8062 : X = shallowconcat(X1, X2);
1036 8062 : if (gc_needed(av, 1)) X = gerepilecopy(av, X);
1037 8062 : return X;
1038 : }
1039 :
1040 : /* destroy A */
1041 : static long
1042 22408 : gen_CUP_basecase(GEN A, GEN *R, GEN *C, GEN *U, GEN *P, void *E, const struct bb_field *ff)
1043 : {
1044 22408 : long i, j, k, m = nbrows(A), n = lg(A) - 1, pr, pc;
1045 : pari_sp av;
1046 : GEN u, v;
1047 :
1048 22408 : if (P) *P = identity_perm(n);
1049 22408 : *R = cgetg(m + 1, t_VECSMALL);
1050 22408 : av = avma;
1051 58418 : for (j = 1, pr = 0; j <= n; j++)
1052 : {
1053 134099 : for (pr++, pc = 0; pr <= m; pr++)
1054 : {
1055 656510 : for (k = j; k <= n; k++)
1056 : {
1057 539883 : v = ff->red(E, gcoeff(A, pr, k));
1058 539883 : gcoeff(A, pr, k) = v;
1059 539883 : if (!pc && !ff->equal0(v)) pc = k;
1060 : }
1061 116627 : if (pc) break;
1062 : }
1063 53482 : if (!pc) break;
1064 36010 : (*R)[j] = pr;
1065 36010 : if (pc != j)
1066 : {
1067 5087 : swap(gel(A, j), gel(A, pc));
1068 5087 : if (P) lswap((*P)[j], (*P)[pc]);
1069 : }
1070 36010 : u = ff->inv(E, gcoeff(A, pr, j));
1071 179244 : for (i = pr + 1; i <= m; i++)
1072 : {
1073 143234 : v = ff->red(E, ff->mul(E, gcoeff(A, i, j), u));
1074 143234 : gcoeff(A, i, j) = v;
1075 143234 : v = ff->neg(E, v);
1076 462720 : for (k = j + 1; k <= n; k++)
1077 319486 : gcoeff(A, i, k) = ff->add(E, gcoeff(A, i, k),
1078 319486 : ff->red(E, ff->mul(E, gcoeff(A, pr, k), v)));
1079 : }
1080 36010 : if (gc_needed(av, 2)) A = gerepilecopy(av, A);
1081 : }
1082 22408 : setlg(*R, j);
1083 22408 : *C = vecslice(A, 1, j - 1);
1084 22408 : if (U) *U = rowpermute(A, *R);
1085 22408 : return j - 1;
1086 : }
1087 :
1088 : static const long gen_CUP_LIMIT = 5;
1089 :
1090 : static long
1091 11480 : 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 11480 : 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 11480 : pari_sp av = avma;
1098 :
1099 11480 : if (m < gen_CUP_LIMIT || n < gen_CUP_LIMIT)
1100 : /* destroy A; not called at the outermost recursion level */
1101 6573 : return gen_CUP_basecase(A, R, C, U, P, E, ff);
1102 4907 : m1 = (minss(m, n) + 1)/2;
1103 4907 : A1 = rowslice(A, 1, m1);
1104 4907 : A2 = rowslice(A, m1 + 1, m);
1105 4907 : r1 = gen_CUP(A1, &R1, &C1, &U1, &P1, E, ff, mul);
1106 4907 : if (r1 == 0)
1107 : {
1108 485 : r2 = gen_CUP(A2, &R2, &C2, &U2, &P2, E, ff, mul);
1109 485 : *R = cgetg(r2 + 1, t_VECSMALL);
1110 790 : for (i = 1; i <= r2; i++) (*R)[i] = R2[i] + m1;
1111 485 : *C = vconcat(gen_zeromat(m1, r2, E, ff), C2);
1112 485 : *U = U2;
1113 485 : *P = P2;
1114 485 : r = r2;
1115 : }
1116 : else
1117 : {
1118 4422 : U11 = vecslice(U1, 1, r1);
1119 4422 : U12 = vecslice(U1, r1 + 1, n);
1120 4422 : T21 = vecslicepermute(A2, P1, 1, r1);
1121 4422 : T22 = vecslicepermute(A2, P1, r1 + 1, n);
1122 4422 : C21 = gen_lsolve_upper(U11, T21, E, ff, mul);
1123 4422 : if (gc_needed(av, 1))
1124 0 : gerepileall(av, 7, &R1, &C1, &P1, &U11, &U12, &T22, &C21);
1125 4422 : B2 = gen_matsub(T22, mul(E, C21, U12), E, ff);
1126 4422 : r2 = gen_CUP(B2, &R2, &C2, &U2, &P2, E, ff, mul);
1127 4422 : r = r1 + r2;
1128 4422 : *R = cgetg(r + 1, t_VECSMALL);
1129 19941 : for (i = 1; i <= r1; i++) (*R)[i] = R1[i];
1130 21029 : for ( ; i <= r; i++) (*R)[i] = R2[i - r1] + m1;
1131 4422 : *C = shallowconcat(vconcat(C1, C21),
1132 : vconcat(gen_zeromat(m1, r2, E, ff), C2));
1133 4422 : *U = shallowconcat(vconcat(U11, gen_zeromat(r2, r1, E, ff)),
1134 : vconcat(vecpermute(U12, P2), U2));
1135 :
1136 4422 : *P = cgetg(n + 1, t_VECSMALL);
1137 19941 : for (i = 1; i <= r1; i++) (*P)[i] = P1[i];
1138 50825 : for ( ; i <= n; i++) (*P)[i] = P1[P2[i - r1] + r1];
1139 : }
1140 4907 : if (gc_needed(av, 1)) gerepileall(av, 4, R, C, U, P);
1141 4907 : return r;
1142 : }
1143 :
1144 : /* column echelon form */
1145 : static long
1146 27851 : gen_echelon(GEN A, GEN *R, GEN *C, void *E, const struct bb_field *ff,
1147 : GEN (*mul)(void*, GEN, GEN))
1148 : {
1149 27851 : 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 27851 : pari_sp av = avma;
1153 :
1154 27851 : if (m < gen_CUP_LIMIT || n < gen_CUP_LIMIT)
1155 15835 : return gen_CUP_basecase(shallowcopy(A), R, C, NULL, NULL, E, ff);
1156 :
1157 12016 : n1 = (n + 1)/2;
1158 12016 : A1 = vecslice(A, 1, n1);
1159 12016 : A2 = vecslice(A, n1 + 1, n);
1160 12016 : r1 = gen_echelon(A1, &R1, &C1, E, ff, mul);
1161 12016 : if (!r1) return gen_echelon(A2, R, C, E, ff, mul);
1162 10693 : if (r1 == m) { *R = R1; *C = C1; return r1; }
1163 10448 : R1c = indexcompl(R1, m);
1164 10448 : C11 = rowpermute(C1, R1);
1165 10448 : C21 = rowpermute(C1, R1c);
1166 10448 : A12 = rowpermute(A2, R1);
1167 10448 : A22 = rowpermute(A2, R1c);
1168 10448 : M12 = gen_rsolve_lower_unit(C11, A12, E, ff, mul);
1169 10448 : B2 = gen_matsub(A22, mul(E, C21, M12), E, ff);
1170 10448 : r2 = gen_echelon(B2, &R2, &C2, E, ff, mul);
1171 10448 : if (!r2) { *R = R1; *C = C1; r = r1; }
1172 : else
1173 : {
1174 5556 : R2 = perm_mul(R1c, R2);
1175 5556 : C2 = rowpermute(vconcat(gen_zeromat(r1, r2, E, ff), C2),
1176 : perm_inv(vecsmall_concat(R1, R1c)));
1177 5556 : r = r1 + r2;
1178 5556 : *R = cgetg(r + 1, t_VECSMALL);
1179 5556 : *C = cgetg(r + 1, t_MAT);
1180 39722 : for (j = j1 = j2 = 1; j <= r; j++)
1181 34166 : if (j2 > r2 || (j1 <= r1 && R1[j1] < R2[j2]))
1182 : {
1183 20094 : gel(*C, j) = gel(C1, j1);
1184 20094 : (*R)[j] = R1[j1++];
1185 : }
1186 : else
1187 : {
1188 14072 : gel(*C, j) = gel(C2, j2);
1189 14072 : (*R)[j] = R2[j2++];
1190 : }
1191 : }
1192 10448 : if (gc_needed(av, 1)) gerepileall(av, 2, R, C);
1193 10448 : return r;
1194 : }
1195 :
1196 : static GEN
1197 904 : 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 904 : long i, n = lg(x) - 1, r;
1202 904 : GEN R, C, U, P, d = zero_zv(n);
1203 904 : av = avma;
1204 904 : r = gen_CUP(x, &R, &C, &U, &P, E, ff, mul);
1205 6921 : for(i = 1; i <= r; i++)
1206 6017 : d[P[i]] = R[i];
1207 904 : set_avma(av);
1208 904 : *rr = n - r;
1209 904 : 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 3931 : gen_ker_echelon(GEN x, void *E, const struct bb_field *ff,
1268 : GEN (*mul)(void*, GEN, GEN))
1269 : {
1270 3931 : pari_sp av = avma;
1271 : GEN R, Rc, C, C1, C2, S, K;
1272 3931 : long n = lg(x) - 1, r;
1273 3931 : r = gen_echelon(shallowtrans(x), &R, &C, E, ff, mul);
1274 3931 : Rc = indexcompl(R, n);
1275 3931 : C1 = rowpermute(C, R);
1276 3931 : C2 = rowpermute(C, Rc);
1277 3931 : S = gen_lsolve_lower_unit(C1, C2, E, ff, mul);
1278 3931 : K = vecpermute(shallowconcat(gen_matneg(S, E, ff), gen_matid(n - r, E, ff)),
1279 : perm_inv(vecsmall_concat(R, Rc)));
1280 3931 : K = shallowtrans(K);
1281 3931 : return gerepilecopy(av, K);
1282 : }
1283 :
1284 : static GEN
1285 105 : gen_deplin_echelon(GEN x, void *E, const struct bb_field *ff,
1286 : GEN (*mul)(void*, GEN, GEN))
1287 : {
1288 105 : pari_sp av = avma;
1289 : GEN R, Rc, C, C1, C2, s, v;
1290 105 : long i, n = lg(x) - 1, r;
1291 105 : r = gen_echelon(shallowtrans(x), &R, &C, E, ff, mul);
1292 105 : if (r == n) return gc_NULL(av);
1293 70 : Rc = indexcompl(R, n);
1294 70 : i = Rc[1];
1295 70 : C1 = rowpermute(C, R);
1296 70 : C2 = rowslice(C, i, i);
1297 70 : s = row(gen_lsolve_lower_unit(C1, C2, E, ff, mul), 1);
1298 70 : settyp(s, t_COL);
1299 70 : v = vecpermute(shallowconcat(gen_colneg(s, E, ff), gen_colei(n - r, 1, E, ff)),
1300 : perm_inv(vecsmall_concat(R, Rc)));
1301 70 : return gerepilecopy(av, v);
1302 : }
1303 :
1304 : static GEN
1305 559 : 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 559 : long n = lg(a) - 1, r;
1310 559 : if (nbrows(a) < n || (r = gen_CUP(a, &R, &C, &U, &P, E, ff, mul)) < n)
1311 56 : return NULL;
1312 503 : Y = gen_rsolve_lower_unit(rowpermute(C, R), rowpermute(b, R), E, ff, mul);
1313 503 : return rowpermute(gen_rsolve_upper(U, Y, E, ff, mul), perm_inv(P));
1314 : }
1315 :
1316 : static GEN
1317 3960 : gen_gauss(GEN a, GEN b, void *E, const struct bb_field *ff,
1318 : GEN (*mul)(void*, GEN, GEN))
1319 : {
1320 3960 : if (lg(a) - 1 >= gen_CUP_LIMIT)
1321 559 : return gen_gauss_CUP(a, b, E, ff, mul);
1322 3401 : return gen_Gauss(a, b, E, ff);
1323 : }
1324 :
1325 : static GEN
1326 5865 : gen_ker_i(GEN x, long deplin, void *E, const struct bb_field *ff,
1327 : GEN (*mul)(void*, GEN, GEN)) {
1328 5865 : if (lg(x) - 1 >= gen_CUP_LIMIT && nbrows(x) >= gen_CUP_LIMIT)
1329 4036 : return deplin? gen_deplin_echelon(x, E, ff, mul): gen_ker_echelon(x, E, ff, mul);
1330 1829 : 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 71 : gen_matcolinvimage_i(GEN A, GEN y, void *E, const struct bb_field *ff,
1348 : GEN (*mul)(void*, GEN, GEN))
1349 : {
1350 71 : pari_sp av = avma;
1351 71 : long i, l = lg(A);
1352 : GEN M, x, t;
1353 :
1354 71 : M = gen_ker_i(shallowconcat(A, y), 0, E, ff, mul);
1355 71 : i = lg(M) - 1;
1356 71 : if (!i) return gc_NULL(av);
1357 :
1358 71 : x = gel(M, i);
1359 71 : t = gel(x, l);
1360 71 : if (ff->equal0(t)) return gc_NULL(av);
1361 :
1362 50 : t = ff->neg(E, ff->inv(E, t));
1363 50 : setlg(x, l);
1364 178 : for (i = 1; i < l; i++)
1365 128 : gel(x, i) = ff->red(E, ff->mul(E, t, gel(x, i)));
1366 50 : 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 2788 : gen_pivots(GEN x, long *rr, void *E, const struct bb_field *ff,
1381 : GEN (*mul)(void*, GEN, GEN))
1382 : {
1383 2788 : if (lg(x) - 1 >= gen_CUP_LIMIT && nbrows(x) >= gen_CUP_LIMIT)
1384 904 : return gen_pivots_CUP(x, rr, E, ff, mul);
1385 1884 : 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 63 : FqM_FqC_mul(GEN A, GEN B, GEN T, GEN p) {
1441 : void *E;
1442 63 : const struct bb_field *ff = get_Fq_field(&E, T, p);
1443 63 : 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 157824 : FlxqM_mul(GEN A, GEN B, GEN T, ulong p) {
1455 : void *E;
1456 : const struct bb_field *ff;
1457 157824 : long n = lg(A) - 1;
1458 :
1459 157824 : if (n == 0)
1460 0 : return cgetg(1, t_MAT);
1461 157824 : if (n > 1)
1462 89013 : return FlxqM_mul_Kronecker(A, B, T, p);
1463 68811 : ff = get_Flxq_field(&E, T, p);
1464 68811 : return gen_matmul(A, B, E, ff);
1465 : }
1466 :
1467 : GEN
1468 66409 : FqM_mul(GEN A, GEN B, GEN T, GEN p) {
1469 : void *E;
1470 66409 : long n = lg(A) - 1;
1471 : const struct bb_field *ff;
1472 66409 : if (n == 0)
1473 0 : return cgetg(1, t_MAT);
1474 66409 : if (n > 1)
1475 62790 : 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 16079 : _FlxqM_mul(void *E, GEN A, GEN B)
1498 : {
1499 16079 : struct _Flxq *D = (struct _Flxq*)E;
1500 16079 : return FlxqM_mul(A, B, D->T, D->p);
1501 : }
1502 :
1503 : static GEN
1504 22741 : _FpM_mul(void *E, GEN A, GEN B)
1505 22741 : { 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 1228333 : FpM_init(GEN a, GEN p, ulong *pp)
1521 : {
1522 1228333 : if (lgefint(p) == 3)
1523 : {
1524 1223144 : *pp = uel(p,2);
1525 1223144 : return (*pp==2)? ZM_to_F2m(a): ZM_to_Flm(a, *pp);
1526 : }
1527 5189 : *pp = 0; return a;
1528 : }
1529 : static GEN
1530 1278566 : FpM_init3(GEN a, GEN p, ulong *pp)
1531 : {
1532 1278566 : if (lgefint(p) == 3)
1533 : {
1534 1275990 : *pp = uel(p,2);
1535 1275990 : switch(*pp)
1536 : {
1537 701571 : case 2: return ZM_to_F2m(a);
1538 154225 : case 3: return ZM_to_F3m(a);
1539 420194 : default:return ZM_to_Flm(a, *pp);
1540 : }
1541 : }
1542 2576 : *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 : return gc_utoi(av, 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 1214 : FpM_gauss_pivot_gen(GEN x, GEN p, long *rr)
1617 : {
1618 : void *E;
1619 1214 : const struct bb_field *S = get_Fp_field(&E,p);
1620 1214 : return gen_pivots(x, rr, E, S, _FpM_mul);
1621 : }
1622 :
1623 : static GEN
1624 602048 : FpM_gauss_pivot(GEN x, GEN p, long *rr)
1625 : {
1626 : ulong pp;
1627 602048 : if (lg(x)==1) { *rr = 0; return NULL; }
1628 599020 : x = FpM_init(x, p, &pp);
1629 599031 : switch(pp)
1630 : {
1631 1214 : case 0: return FpM_gauss_pivot_gen(x, p, rr);
1632 345311 : case 2: return F2m_gauss_pivot(x, rr);
1633 252506 : 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 1427 : FlxqM_gauss_pivot(GEN x, GEN T, ulong p, long *rr) {
1647 : void *E;
1648 1427 : const struct bb_field *S = get_Flxq_field(&E, T, p);
1649 1427 : 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 1504 : FqM_gauss_pivot(GEN x, GEN T, GEN p, long *rr)
1661 : {
1662 1504 : if (lg(x)==1) { *rr = 0; return NULL; }
1663 1504 : if (!T) return FpM_gauss_pivot(x, p, rr);
1664 1504 : if (lgefint(p) == 3)
1665 : {
1666 1399 : pari_sp av = avma;
1667 1399 : ulong pp = uel(p,2);
1668 1399 : GEN Tp = ZXT_to_FlxT(T, pp);
1669 1399 : GEN d = FlxqM_gauss_pivot(ZXM_to_FlxM(x, pp, get_Flx_var(Tp)), Tp, pp, rr);
1670 1399 : return d ? gerepileuptoleaf(av, d): d;
1671 : }
1672 105 : return FqM_gauss_pivot_gen(x, T, p, rr);
1673 : }
1674 :
1675 : GEN
1676 312428 : FpM_image(GEN x, GEN p)
1677 : {
1678 : long r;
1679 312428 : GEN d = FpM_gauss_pivot(x,p,&r); /* d left on stack for efficiency */
1680 312429 : return image_from_pivot(x,d,r);
1681 : }
1682 :
1683 : GEN
1684 39739 : Flm_image(GEN x, ulong p)
1685 : {
1686 : long r;
1687 39739 : GEN d = Flm_pivots(x, p, &r, 0); /* d left on stack for efficiency */
1688 39739 : 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 8 : FpM_FpC_invimage_gen(GEN A, GEN y, GEN p)
1813 : {
1814 : void *E;
1815 8 : const struct bb_field *ff = get_Fp_field(&E, p);
1816 8 : return gen_matcolinvimage_i(A, y, E, ff, _FpM_mul);
1817 : }
1818 :
1819 : GEN
1820 296309 : FpM_FpC_invimage(GEN A, GEN x, GEN p)
1821 : {
1822 296309 : pari_sp av = avma;
1823 : ulong pp;
1824 : GEN y;
1825 :
1826 296309 : A = FpM_init(A, p, &pp);
1827 296319 : switch(pp)
1828 : {
1829 8 : case 0: return FpM_FpC_invimage_gen(A, x, p);
1830 193063 : case 2:
1831 193063 : y = F2m_F2c_invimage(A, ZV_to_F2v(x));
1832 193062 : if (!y) return y;
1833 193062 : y = F2c_to_ZC(y);
1834 193062 : return gerepileupto(av, y);
1835 103248 : default:
1836 103248 : y = Flm_Flc_invimage(A, ZV_to_Flv(x, pp), pp);
1837 103248 : if (!y) return y;
1838 103248 : y = Flc_to_ZC(y);
1839 103248 : 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 2646 : FpM_ker_gen(GEN x, GEN p, long deplin)
1866 : {
1867 : void *E;
1868 2646 : const struct bb_field *S = get_Fp_field(&E,p);
1869 2646 : return gen_ker_i(x, deplin, E, S, _FpM_mul);
1870 : }
1871 : static GEN
1872 1278559 : FpM_ker_i(GEN x, GEN p, long deplin)
1873 : {
1874 1278559 : pari_sp av = avma;
1875 : ulong pp;
1876 : GEN y;
1877 :
1878 1278559 : if (lg(x)==1) return cgetg(1,t_MAT);
1879 1278559 : x = FpM_init3(x, p, &pp);
1880 1278608 : switch(pp)
1881 : {
1882 2576 : case 0: return FpM_ker_gen(x,p,deplin);
1883 701610 : case 2:
1884 701610 : y = F2m_ker_sp(x, deplin);
1885 701598 : if (!y) return gc_NULL(av);
1886 701612 : y = deplin? F2c_to_ZC(y): F2m_to_ZM(y);
1887 701609 : return gerepileupto(av, y);
1888 154226 : case 3:
1889 154226 : y = F3m_ker_sp(x, deplin);
1890 154226 : if (!y) return gc_NULL(av);
1891 154226 : y = deplin? F3c_to_ZC(y): F3m_to_ZM(y);
1892 154226 : return gerepileupto(av, y);
1893 420196 : default:
1894 420196 : y = Flm_ker_sp(x, pp, deplin);
1895 420196 : if (!y) return gc_NULL(av);
1896 420196 : y = deplin? Flc_to_ZC(y): Flm_to_ZM(y);
1897 420196 : return gerepileupto(av, y);
1898 : }
1899 : }
1900 :
1901 : GEN
1902 817197 : 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 2987 : FlxqM_ker_i(GEN x, GEN T, ulong p, long deplin) {
1923 : void *E;
1924 2987 : const struct bb_field *S = get_Flxq_field(&E, T, p);
1925 2987 : return gen_ker_i(x, deplin, E, S, _FlxqM_mul);
1926 : }
1927 :
1928 : GEN
1929 28 : FlxqM_ker(GEN x, GEN T, ulong p)
1930 : {
1931 28 : 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 10540 : FqM_ker_i(GEN x, GEN T, GEN p, long deplin)
1943 : {
1944 10540 : if (!T) return FpM_ker_i(x,p,deplin);
1945 3050 : if (lg(x)==1) return cgetg(1,t_MAT);
1946 :
1947 3050 : if (lgefint(p)==3)
1948 : {
1949 2924 : pari_sp av = avma;
1950 2924 : ulong l = p[2];
1951 2924 : GEN Tl = ZXT_to_FlxT(T,l);
1952 2924 : GEN Ml = ZXM_to_FlxM(x, l, get_Flx_var(Tl));
1953 2924 : GEN K = FlxqM_ker_i(Ml, Tl, l, deplin);
1954 2924 : if (!deplin) K = FlxM_to_ZXM(K);
1955 28 : else if (!K) return gc_NULL(av);
1956 21 : else K = FlxC_to_ZXC(K);
1957 2917 : return gerepileupto(av, K);
1958 : }
1959 126 : return FqM_ker_gen(x, T, p, deplin);
1960 : }
1961 :
1962 : GEN
1963 10456 : FqM_ker(GEN x, GEN T, GEN p) { return FqM_ker_i(x,T,p,0); }
1964 :
1965 : GEN
1966 453894 : FpM_deplin(GEN x, GEN p) { return FpM_ker_i(x,p,1); }
1967 :
1968 : GEN
1969 14 : F2xqM_deplin(GEN x, GEN T)
1970 : {
1971 14 : return F2xqM_ker_i(x, T, 1);
1972 : }
1973 :
1974 : GEN
1975 35 : FlxqM_deplin(GEN x, GEN T, ulong p)
1976 : {
1977 35 : return FlxqM_ker_i(x, T, p, 1);
1978 : }
1979 :
1980 : GEN
1981 84 : FqM_deplin(GEN x, GEN T, GEN p) { return FqM_ker_i(x,T,p,1); }
1982 :
1983 : static GEN
1984 3652 : FpM_gauss_gen(GEN a, GEN b, GEN p)
1985 : {
1986 : void *E;
1987 3652 : const struct bb_field *S = get_Fp_field(&E,p);
1988 3652 : return gen_gauss(a,b, E, S, _FpM_mul);
1989 : }
1990 : /* a an FpM, lg(a)>1; b an FpM or NULL (replace by identity) */
1991 : static GEN
1992 329094 : FpM_gauss_i(GEN a, GEN b, GEN p, ulong *pp)
1993 : {
1994 329094 : long n = nbrows(a);
1995 329092 : a = FpM_init(a,p,pp);
1996 329093 : switch(*pp)
1997 : {
1998 3652 : case 0:
1999 3652 : if (!b) b = matid(n);
2000 3652 : return FpM_gauss_gen(a,b,p);
2001 224730 : case 2:
2002 224730 : if (b) b = ZM_to_F2m(b); else b = matid_F2m(n);
2003 224727 : return F2m_gauss_sp(a,b);
2004 100711 : default:
2005 100711 : if (b) b = ZM_to_Flm(b, *pp); else b = matid_Flm(n);
2006 100710 : return Flm_gauss_sp(a,b, NULL, *pp);
2007 : }
2008 : }
2009 : GEN
2010 35 : FpM_gauss(GEN a, GEN b, GEN p)
2011 : {
2012 35 : pari_sp av = avma;
2013 : ulong pp;
2014 : GEN u;
2015 35 : if (lg(a) == 1 || lg(b)==1) return cgetg(1, t_MAT);
2016 35 : u = FpM_gauss_i(a, b, p, &pp);
2017 35 : if (!u) return gc_NULL(av);
2018 28 : switch(pp)
2019 : {
2020 28 : case 0: return gerepilecopy(av, u);
2021 0 : case 2: u = F2m_to_ZM(u); break;
2022 0 : default: u = Flm_to_ZM(u); break;
2023 : }
2024 0 : return gerepileupto(av, u);
2025 : }
2026 :
2027 : static GEN
2028 84 : F2xqM_gauss_gen(GEN a, GEN b, GEN T)
2029 : {
2030 : void *E;
2031 84 : const struct bb_field *S = get_F2xq_field(&E, T);
2032 84 : return gen_gauss(a, b, E, S, _F2xqM_mul);
2033 : }
2034 :
2035 : GEN
2036 21 : F2xqM_gauss(GEN a, GEN b, GEN T)
2037 : {
2038 21 : pari_sp av = avma;
2039 21 : long n = lg(a)-1;
2040 : GEN u;
2041 21 : if (!n || lg(b)==1) { set_avma(av); return cgetg(1, t_MAT); }
2042 21 : u = F2xqM_gauss_gen(a, b, T);
2043 21 : if (!u) return gc_NULL(av);
2044 14 : return gerepilecopy(av, u);
2045 : }
2046 :
2047 : static GEN
2048 91 : FlxqM_gauss_i(GEN a, GEN b, GEN T, ulong p) {
2049 : void *E;
2050 91 : const struct bb_field *S = get_Flxq_field(&E, T, p);
2051 91 : return gen_gauss(a, b, E, S, _FlxqM_mul);
2052 : }
2053 :
2054 : GEN
2055 21 : FlxqM_gauss(GEN a, GEN b, GEN T, ulong p)
2056 : {
2057 21 : pari_sp av = avma;
2058 21 : long n = lg(a)-1;
2059 : GEN u;
2060 21 : if (!n || lg(b)==1) { set_avma(av); return cgetg(1, t_MAT); }
2061 21 : u = FlxqM_gauss_i(a, b, T, p);
2062 21 : if (!u) return gc_NULL(av);
2063 14 : return gerepilecopy(av, u);
2064 : }
2065 :
2066 : static GEN
2067 133 : FqM_gauss_gen(GEN a, GEN b, GEN T, GEN p)
2068 : {
2069 : void *E;
2070 133 : const struct bb_field *S = get_Fq_field(&E,T,p);
2071 133 : return gen_gauss(a,b,E,S,_FqM_mul);
2072 : }
2073 : GEN
2074 21 : FqM_gauss(GEN a, GEN b, GEN T, GEN p)
2075 : {
2076 21 : pari_sp av = avma;
2077 : GEN u;
2078 : long n;
2079 21 : if (!T) return FpM_gauss(a,b,p);
2080 21 : n = lg(a)-1; if (!n || lg(b)==1) return cgetg(1, t_MAT);
2081 21 : u = FqM_gauss_gen(a,b,T,p);
2082 21 : if (!u) return gc_NULL(av);
2083 14 : return gerepilecopy(av, u);
2084 : }
2085 :
2086 : GEN
2087 14 : FpM_FpC_gauss(GEN a, GEN b, GEN p)
2088 : {
2089 14 : pari_sp av = avma;
2090 : ulong pp;
2091 : GEN u;
2092 14 : if (lg(a) == 1) return cgetg(1, t_COL);
2093 14 : u = FpM_gauss_i(a, mkmat(b), p, &pp);
2094 14 : if (!u) return gc_NULL(av);
2095 14 : switch(pp)
2096 : {
2097 14 : case 0: return gerepilecopy(av, gel(u,1));
2098 0 : case 2: u = F2c_to_ZC(gel(u,1)); break;
2099 0 : default: u = Flc_to_ZC(gel(u,1)); break;
2100 : }
2101 0 : return gerepileupto(av, u);
2102 : }
2103 :
2104 : GEN
2105 28 : F2xqM_F2xqC_gauss(GEN a, GEN b, GEN T)
2106 : {
2107 28 : pari_sp av = avma;
2108 : GEN u;
2109 28 : if (lg(a) == 1) return cgetg(1, t_COL);
2110 28 : u = F2xqM_gauss_gen(a, mkmat(b), T);
2111 28 : if (!u) return gc_NULL(av);
2112 14 : return gerepilecopy(av, gel(u,1));
2113 : }
2114 :
2115 : GEN
2116 14 : FlxqM_FlxqC_gauss(GEN a, GEN b, GEN T, ulong p)
2117 : {
2118 14 : pari_sp av = avma;
2119 : GEN u;
2120 14 : if (lg(a) == 1) return cgetg(1, t_COL);
2121 14 : u = FlxqM_gauss_i(a, mkmat(b), T, p);
2122 14 : if (!u) return gc_NULL(av);
2123 7 : return gerepilecopy(av, gel(u,1));
2124 : }
2125 :
2126 : GEN
2127 14 : FqM_FqC_gauss(GEN a, GEN b, GEN T, GEN p)
2128 : {
2129 14 : pari_sp av = avma;
2130 : GEN u;
2131 14 : if (!T) return FpM_FpC_gauss(a,b,p);
2132 14 : if (lg(a) == 1) return cgetg(1, t_COL);
2133 14 : u = FqM_gauss_gen(a,mkmat(b),T,p);
2134 14 : if (!u) return gc_NULL(av);
2135 7 : return gerepilecopy(av, gel(u,1));
2136 : }
2137 :
2138 : GEN
2139 329045 : FpM_inv(GEN a, GEN p)
2140 : {
2141 329045 : pari_sp av = avma;
2142 : ulong pp;
2143 : GEN u;
2144 329045 : if (lg(a) == 1) return cgetg(1, t_MAT);
2145 329045 : u = FpM_gauss_i(a, NULL, p, &pp);
2146 329047 : if (!u) return gc_NULL(av);
2147 329033 : switch(pp)
2148 : {
2149 3596 : case 0: return gerepilecopy(av, u);
2150 224727 : case 2: u = F2m_to_ZM(u); break;
2151 100710 : default: u = Flm_to_ZM(u); break;
2152 : }
2153 325438 : return gerepileupto(av, u);
2154 : }
2155 :
2156 : GEN
2157 35 : F2xqM_inv(GEN a, GEN T)
2158 : {
2159 35 : pari_sp av = avma;
2160 : GEN u;
2161 35 : if (lg(a) == 1) { set_avma(av); return cgetg(1, t_MAT); }
2162 35 : u = F2xqM_gauss_gen(a, matid_F2xqM(nbrows(a),T), T);
2163 35 : if (!u) return gc_NULL(av);
2164 28 : return gerepilecopy(av, u);
2165 : }
2166 :
2167 : GEN
2168 56 : FlxqM_inv(GEN a, GEN T, ulong p)
2169 : {
2170 56 : pari_sp av = avma;
2171 : GEN u;
2172 56 : if (lg(a) == 1) { set_avma(av); return cgetg(1, t_MAT); }
2173 56 : u = FlxqM_gauss_i(a, matid_FlxqM(nbrows(a),T,p), T,p);
2174 56 : if (!u) return gc_NULL(av);
2175 42 : return gerepilecopy(av, u);
2176 : }
2177 :
2178 : GEN
2179 98 : FqM_inv(GEN a, GEN T, GEN p)
2180 : {
2181 98 : pari_sp av = avma;
2182 : GEN u;
2183 98 : if (!T) return FpM_inv(a,p);
2184 98 : if (lg(a) == 1) return cgetg(1, t_MAT);
2185 98 : u = FqM_gauss_gen(a,matid(nbrows(a)),T,p);
2186 98 : if (!u) return gc_NULL(av);
2187 70 : return gerepilecopy(av, u);
2188 : }
2189 :
2190 : GEN
2191 263832 : FpM_intersect_i(GEN x, GEN y, GEN p)
2192 : {
2193 263832 : long j, lx = lg(x);
2194 : GEN z;
2195 :
2196 263832 : if (lx == 1 || lg(y) == 1) return cgetg(1,t_MAT);
2197 263832 : if (lgefint(p) == 3)
2198 : {
2199 263832 : ulong pp = p[2];
2200 263832 : return Flm_to_ZM(Flm_intersect_i(ZM_to_Flm(x,pp), ZM_to_Flm(y,pp), pp));
2201 : }
2202 0 : z = FpM_ker(shallowconcat(x,y), p);
2203 0 : for (j=lg(z)-1; j; j--) setlg(gel(z,j),lx);
2204 0 : return FpM_mul(x,z,p);
2205 : }
2206 : GEN
2207 0 : FpM_intersect(GEN x, GEN y, GEN p)
2208 : {
2209 0 : pari_sp av = avma;
2210 : GEN z;
2211 0 : if (lgefint(p) == 3)
2212 : {
2213 0 : ulong pp = p[2];
2214 0 : z = Flm_to_ZM(Flm_image(Flm_intersect_i(ZM_to_Flm(x,pp), ZM_to_Flm(y,pp), pp), pp));
2215 : }
2216 : else
2217 0 : z = FpM_image(FpM_intersect_i(x,y,p), p);
2218 0 : return gerepileupto(av, z);
2219 : }
2220 :
2221 : static void
2222 261101 : init_suppl(GEN x)
2223 : {
2224 261101 : if (lg(x) == 1) pari_err_IMPL("suppl [empty matrix]");
2225 : /* HACK: avoid overwriting d from gauss_pivot() after set_avma(av) */
2226 261101 : (void)new_chunk(lgcols(x) * 2);
2227 261100 : }
2228 :
2229 : GEN
2230 259534 : FpM_suppl(GEN x, GEN p)
2231 : {
2232 : GEN d;
2233 : long r;
2234 259534 : init_suppl(x); d = FpM_gauss_pivot(x,p, &r);
2235 259540 : return get_suppl(x,d,nbrows(x),r,&col_ei);
2236 : }
2237 :
2238 : GEN
2239 14 : F2m_suppl(GEN x)
2240 : {
2241 : GEN d;
2242 : long r;
2243 14 : init_suppl(x); d = F2m_gauss_pivot(F2m_copy(x), &r);
2244 14 : return get_suppl(x,d,mael(x,1,1),r,&F2v_ei);
2245 : }
2246 :
2247 : GEN
2248 105 : Flm_suppl(GEN x, ulong p)
2249 : {
2250 : GEN d;
2251 : long r;
2252 105 : init_suppl(x); d = Flm_pivots(x, p, &r, 0);
2253 105 : return get_suppl(x,d,nbrows(x),r,&vecsmall_ei);
2254 : }
2255 :
2256 : GEN
2257 7 : F2xqM_suppl(GEN x, GEN T)
2258 : {
2259 : void *E;
2260 7 : const struct bb_field *S = get_F2xq_field(&E, T);
2261 7 : return gen_suppl(x, E, S, _F2xqM_mul);
2262 : }
2263 :
2264 : GEN
2265 14 : FlxqM_suppl(GEN x, GEN T, ulong p)
2266 : {
2267 : void *E;
2268 14 : const struct bb_field *S = get_Flxq_field(&E, T, p);
2269 14 : return gen_suppl(x, E, S, _FlxqM_mul);
2270 : }
2271 :
2272 : GEN
2273 4976 : FqM_suppl(GEN x, GEN T, GEN p)
2274 : {
2275 4976 : pari_sp av = avma;
2276 : GEN d;
2277 : long r;
2278 :
2279 4976 : if (!T) return FpM_suppl(x,p);
2280 1378 : init_suppl(x);
2281 1378 : d = FqM_gauss_pivot(x,T,p,&r);
2282 1378 : set_avma(av); return get_suppl(x,d,nbrows(x),r,&col_ei);
2283 : }
2284 :
2285 : static void
2286 1910360 : init_indexrank(GEN x) {
2287 1910360 : (void)new_chunk(3 + 2*lg(x)); /* HACK */
2288 1910359 : }
2289 :
2290 : GEN
2291 30035 : FpM_indexrank(GEN x, GEN p) {
2292 30035 : pari_sp av = avma;
2293 : long r;
2294 : GEN d;
2295 30035 : init_indexrank(x);
2296 30035 : d = FpM_gauss_pivot(x,p,&r);
2297 30035 : set_avma(av); return indexrank0(lg(x)-1, r, d);
2298 : }
2299 :
2300 : GEN
2301 53024 : Flm_indexrank(GEN x, ulong p) {
2302 53024 : pari_sp av = avma;
2303 : long r;
2304 : GEN d;
2305 53024 : init_indexrank(x);
2306 53024 : d = Flm_pivots(x, p, &r, 0);
2307 53025 : set_avma(av); return indexrank0(lg(x)-1, r, d);
2308 : }
2309 :
2310 : GEN
2311 60 : F2m_indexrank(GEN x) {
2312 60 : pari_sp av = avma;
2313 : long r;
2314 : GEN d;
2315 60 : init_indexrank(x);
2316 60 : d = F2m_gauss_pivot(F2m_copy(x),&r);
2317 60 : set_avma(av); return indexrank0(lg(x)-1, r, d);
2318 : }
2319 :
2320 : GEN
2321 7 : F2xqM_indexrank(GEN x, GEN T) {
2322 7 : pari_sp av = avma;
2323 : long r;
2324 : GEN d;
2325 7 : init_indexrank(x);
2326 7 : d = F2xqM_gauss_pivot(x, T, &r);
2327 7 : set_avma(av); return indexrank0(lg(x) - 1, r, d);
2328 : }
2329 :
2330 : GEN
2331 7 : FlxqM_indexrank(GEN x, GEN T, ulong p) {
2332 7 : pari_sp av = avma;
2333 : long r;
2334 : GEN d;
2335 7 : init_indexrank(x);
2336 7 : d = FlxqM_gauss_pivot(x, T, p, &r);
2337 7 : set_avma(av); return indexrank0(lg(x) - 1, r, d);
2338 : }
2339 :
2340 : GEN
2341 7 : FqM_indexrank(GEN x, GEN T, GEN p) {
2342 7 : pari_sp av = avma;
2343 : long r;
2344 : GEN d;
2345 7 : init_indexrank(x);
2346 7 : d = FqM_gauss_pivot(x, T, p, &r);
2347 7 : set_avma(av); return indexrank0(lg(x) - 1, r, d);
2348 : }
2349 :
2350 : /*******************************************************************/
2351 : /* */
2352 : /* Solve A*X=B (Gauss pivot) */
2353 : /* */
2354 : /*******************************************************************/
2355 : /* x a column, x0 same column in the original input matrix (for reference),
2356 : * c list of pivots so far */
2357 : static long
2358 1919933 : gauss_get_pivot_max(GEN X, GEN X0, long ix, GEN c)
2359 : {
2360 1919933 : GEN p, r, x = gel(X,ix), x0 = gel(X0,ix);
2361 1919933 : long i, k = 0, ex = - (long)HIGHEXPOBIT, lx = lg(x);
2362 1919933 : if (c)
2363 : {
2364 324334 : for (i=1; i<lx; i++)
2365 187257 : if (!c[i])
2366 : {
2367 89946 : long e = gexpo(gel(x,i));
2368 89946 : if (e > ex) { ex = e; k = i; }
2369 : }
2370 : }
2371 : else
2372 : {
2373 6438244 : for (i=ix; i<lx; i++)
2374 : {
2375 4655384 : long e = gexpo(gel(x,i));
2376 4655388 : if (e > ex) { ex = e; k = i; }
2377 : }
2378 : }
2379 1919937 : if (!k) return lx;
2380 1851341 : p = gel(x,k);
2381 1851341 : r = gel(x0,k); if (isrationalzero(r)) r = x0;
2382 1851347 : return cx_approx0(p, r)? lx: k;
2383 : }
2384 : static long
2385 200735 : gauss_get_pivot_padic(GEN X, GEN p, long ix, GEN c)
2386 : {
2387 200735 : GEN x = gel(X, ix);
2388 200735 : long i, k = 0, ex = (long)HIGHVALPBIT, lx = lg(x);
2389 200735 : if (c)
2390 : {
2391 504 : for (i=1; i<lx; i++)
2392 378 : if (!c[i] && !gequal0(gel(x,i)))
2393 : {
2394 245 : long e = gvaluation(gel(x,i), p);
2395 245 : if (e < ex) { ex = e; k = i; }
2396 : }
2397 : }
2398 : else
2399 : {
2400 1715332 : for (i=ix; i<lx; i++)
2401 1514723 : if (!gequal0(gel(x,i)))
2402 : {
2403 1143526 : long e = gvaluation(gel(x,i), p);
2404 1143526 : if (e < ex) { ex = e; k = i; }
2405 : }
2406 : }
2407 200735 : return k? k: lx;
2408 : }
2409 : static long
2410 4501 : gauss_get_pivot_NZ(GEN X, GEN x0/*unused*/, long ix, GEN c)
2411 : {
2412 4501 : GEN x = gel(X, ix);
2413 4501 : long i, lx = lg(x);
2414 : (void)x0;
2415 4501 : if (c)
2416 : {
2417 12775 : for (i=1; i<lx; i++)
2418 11795 : if (!c[i] && !gequal0(gel(x,i))) return i;
2419 : }
2420 : else
2421 : {
2422 2380 : for (i=ix; i<lx; i++)
2423 2366 : if (!gequal0(gel(x,i))) return i;
2424 : }
2425 994 : return lx;
2426 : }
2427 :
2428 : /* Return pivot seeking function appropriate for the domain of the RgM x
2429 : * (first non zero pivot, maximal pivot...)
2430 : * x0 is a reference point used when guessing whether x[i,j] ~ 0
2431 : * (iff x[i,j] << x0[i,j]); typical case: mateigen, Gauss pivot on x - vp.Id,
2432 : * but use original x when deciding whether a prospective pivot is nonzero */
2433 : static pivot_fun
2434 600926 : get_pivot_fun(GEN x, GEN x0, GEN *data)
2435 : {
2436 600926 : long i, j, hx, lx = lg(x);
2437 600926 : int res = t_INT;
2438 600926 : GEN p = NULL;
2439 :
2440 600926 : *data = NULL;
2441 600926 : if (lx == 1) return &gauss_get_pivot_NZ;
2442 600891 : hx = lgcols(x);
2443 2785535 : for (j=1; j<lx; j++)
2444 : {
2445 2184686 : GEN xj = gel(x,j);
2446 12821593 : for (i=1; i<hx; i++)
2447 : {
2448 10636949 : GEN c = gel(xj,i);
2449 10636949 : switch(typ(c))
2450 : {
2451 5690261 : case t_REAL:
2452 5690261 : res = t_REAL;
2453 5690261 : break;
2454 3640 : case t_COMPLEX:
2455 3640 : if (typ(gel(c,1)) == t_REAL || typ(gel(c,2)) == t_REAL) res = t_REAL;
2456 3640 : break;
2457 3766580 : case t_INT: case t_INTMOD: case t_FRAC: case t_FFELT: case t_QUAD:
2458 : case t_POLMOD: /* exact types */
2459 3766580 : break;
2460 1176426 : case t_PADIC:
2461 1176426 : p = gel(c,2);
2462 1176426 : res = t_PADIC;
2463 1176426 : break;
2464 42 : default: return &gauss_get_pivot_NZ;
2465 : }
2466 : }
2467 : }
2468 600849 : switch(res)
2469 : {
2470 572674 : case t_REAL: *data = x0; return &gauss_get_pivot_max;
2471 26830 : case t_PADIC: *data = p; return &gauss_get_pivot_padic;
2472 1345 : default: return &gauss_get_pivot_NZ;
2473 : }
2474 : }
2475 :
2476 : static GEN
2477 675170 : get_col(GEN a, GEN b, GEN p, long li)
2478 : {
2479 675170 : GEN u = cgetg(li+1,t_COL);
2480 : long i, j;
2481 :
2482 675170 : gel(u,li) = gdiv(gel(b,li), p);
2483 2960841 : for (i=li-1; i>0; i--)
2484 : {
2485 2285675 : pari_sp av = avma;
2486 2285675 : GEN m = gel(b,i);
2487 10613276 : for (j=i+1; j<=li; j++) m = gsub(m, gmul(gcoeff(a,i,j), gel(u,j)));
2488 2285654 : gel(u,i) = gerepileupto(av, gdiv(m, gcoeff(a,i,i)));
2489 : }
2490 675166 : return u;
2491 : }
2492 :
2493 : /* bk -= m * bi */
2494 : static void
2495 10541567 : _submul(GEN b, long k, long i, GEN m)
2496 : {
2497 10541567 : gel(b,k) = gsub(gel(b,k), gmul(m, gel(b,i)));
2498 10541506 : }
2499 : static int
2500 2218365 : init_gauss(GEN a, GEN *b, long *aco, long *li, int *iscol)
2501 : {
2502 2218365 : *iscol = *b ? (typ(*b) == t_COL): 0;
2503 2218365 : *aco = lg(a) - 1;
2504 2218365 : if (!*aco) /* a empty */
2505 : {
2506 70 : if (*b && lg(*b) != 1) pari_err_DIM("gauss");
2507 70 : *li = 0; return 0;
2508 : }
2509 2218295 : *li = nbrows(a);
2510 2218295 : if (*li < *aco) pari_err_INV("gauss [no left inverse]", a);
2511 2218294 : if (*b)
2512 : {
2513 2137877 : switch(typ(*b))
2514 : {
2515 125144 : case t_MAT:
2516 125144 : if (lg(*b) == 1) return 0;
2517 125144 : *b = RgM_shallowcopy(*b);
2518 125144 : break;
2519 2012734 : case t_COL:
2520 2012734 : *b = mkmat( leafcopy(*b) );
2521 2012735 : break;
2522 0 : default: pari_err_TYPE("gauss",*b);
2523 : }
2524 2137879 : if (nbrows(*b) != *li) pari_err_DIM("gauss");
2525 : }
2526 : else
2527 80417 : *b = matid(*li);
2528 2218292 : return 1;
2529 : }
2530 :
2531 : static GEN
2532 112 : RgM_inv_FpM(GEN a, GEN p)
2533 : {
2534 : ulong pp;
2535 112 : a = RgM_Fp_init(a, p, &pp);
2536 112 : switch(pp)
2537 : {
2538 35 : case 0:
2539 35 : a = FpM_inv(a,p);
2540 35 : if (a) a = FpM_to_mod(a, p);
2541 35 : break;
2542 35 : case 2:
2543 35 : a = F2m_inv(a);
2544 35 : if (a) a = F2m_to_mod(a);
2545 35 : break;
2546 42 : default:
2547 42 : a = Flm_inv_sp(a, NULL, pp);
2548 42 : if (a) a = Flm_to_mod(a, pp);
2549 : }
2550 112 : return a;
2551 : }
2552 :
2553 : static GEN
2554 42 : RgM_inv_FqM(GEN x, GEN pol, GEN p)
2555 : {
2556 42 : pari_sp av = avma;
2557 42 : GEN b, T = RgX_to_FpX(pol, p);
2558 42 : if (signe(T) == 0) pari_err_OP("^",x,gen_m1);
2559 42 : b = FqM_inv(RgM_to_FqM(x, T, p), T, p);
2560 42 : if (!b) return gc_NULL(av);
2561 28 : return gerepileupto(av, FqM_to_mod(b, T, p));
2562 : }
2563 :
2564 : #define code(t1,t2) ((t1 << 6) | t2)
2565 : static GEN
2566 204625 : RgM_inv_fast(GEN x)
2567 : {
2568 : GEN p, pol;
2569 : long pa;
2570 204625 : long t = RgM_type(x, &p,&pol,&pa);
2571 204628 : switch(t)
2572 : {
2573 45689 : case t_INT: /* Fall back */
2574 45689 : case t_FRAC: return QM_inv(x);
2575 147 : case t_FFELT: return FFM_inv(x, pol);
2576 112 : case t_INTMOD: return RgM_inv_FpM(x, p);
2577 41 : case code(t_POLMOD, t_INTMOD):
2578 41 : return RgM_inv_FqM(x, pol, p);
2579 158639 : default: return gen_0;
2580 : }
2581 : }
2582 : #undef code
2583 :
2584 : static GEN
2585 49 : RgM_RgC_solve_FpC(GEN a, GEN b, GEN p)
2586 : {
2587 49 : pari_sp av = avma;
2588 : ulong pp;
2589 49 : a = RgM_Fp_init(a, p, &pp);
2590 49 : switch(pp)
2591 : {
2592 14 : case 0:
2593 14 : b = RgC_to_FpC(b, p);
2594 14 : a = FpM_FpC_gauss(a,b,p);
2595 14 : return a ? gerepileupto(av, FpC_to_mod(a, p)): NULL;
2596 14 : case 2:
2597 14 : b = RgV_to_F2v(b);
2598 14 : a = F2m_F2c_gauss(a,b);
2599 14 : return a ? gerepileupto(av, F2c_to_mod(a)): NULL;
2600 21 : default:
2601 21 : b = RgV_to_Flv(b, pp);
2602 21 : a = Flm_Flc_gauss(a, b, pp);
2603 21 : return a ? gerepileupto(av, Flc_to_mod(a, pp)): NULL;
2604 : }
2605 : }
2606 :
2607 : static GEN
2608 98 : RgM_solve_FpM(GEN a, GEN b, GEN p)
2609 : {
2610 98 : pari_sp av = avma;
2611 : ulong pp;
2612 98 : a = RgM_Fp_init(a, p, &pp);
2613 98 : switch(pp)
2614 : {
2615 35 : case 0:
2616 35 : b = RgM_to_FpM(b, p);
2617 35 : a = FpM_gauss(a,b,p);
2618 35 : return a ? gerepileupto(av, FpM_to_mod(a, p)): NULL;
2619 21 : case 2:
2620 21 : b = RgM_to_F2m(b);
2621 21 : a = F2m_gauss(a,b);
2622 21 : return a ? gerepileupto(av, F2m_to_mod(a)): NULL;
2623 42 : default:
2624 42 : b = RgM_to_Flm(b, pp);
2625 42 : a = Flm_gauss(a,b,pp);
2626 42 : return a ? gerepileupto(av, Flm_to_mod(a, pp)): NULL;
2627 : }
2628 : }
2629 :
2630 : /* Gaussan Elimination. If a is square, return a^(-1)*b;
2631 : * if a has more rows than columns and b is NULL, return c such that c a = Id.
2632 : * a is a (not necessarily square) matrix
2633 : * b is a matrix or column vector, NULL meaning: take the identity matrix,
2634 : * effectively returning the inverse of a
2635 : * If a and b are empty, the result is the empty matrix.
2636 : *
2637 : * li: number of rows of a and b
2638 : * aco: number of columns of a
2639 : * bco: number of columns of b (if matrix)
2640 : */
2641 : static GEN
2642 1374412 : RgM_solve_basecase(GEN a, GEN b)
2643 : {
2644 1374412 : pari_sp av = avma;
2645 : long i, j, k, li, bco, aco;
2646 : int iscol;
2647 : pivot_fun pivot;
2648 : GEN p, u, data;
2649 :
2650 1374412 : set_avma(av);
2651 :
2652 1374411 : if (lg(a)-1 == 2 && nbrows(a) == 2) {
2653 : /* 2x2 matrix, start by inverting a */
2654 892475 : GEN u = gcoeff(a,1,1), v = gcoeff(a,1,2);
2655 892475 : GEN w = gcoeff(a,2,1), x = gcoeff(a,2,2);
2656 892475 : GEN D = gsub(gmul(u,x), gmul(v,w)), ainv;
2657 892475 : if (gequal0(D)) return NULL;
2658 892475 : ainv = mkmat2(mkcol2(x, gneg(w)), mkcol2(gneg(v), u));
2659 892476 : ainv = gmul(ainv, ginv(D));
2660 892473 : if (b) ainv = gmul(ainv, b);
2661 892471 : return gerepileupto(av, ainv);
2662 : }
2663 :
2664 481936 : if (!init_gauss(a, &b, &aco, &li, &iscol)) return cgetg(1, iscol?t_COL:t_MAT);
2665 481938 : pivot = get_pivot_fun(a, a, &data);
2666 481938 : a = RgM_shallowcopy(a);
2667 481938 : bco = lg(b)-1;
2668 481938 : if(DEBUGLEVEL>4) err_printf("Entering gauss\n");
2669 :
2670 481938 : p = NULL; /* gcc -Wall */
2671 1700742 : for (i=1; i<=aco; i++)
2672 : {
2673 : /* k is the line where we find the pivot */
2674 1700739 : k = pivot(a, data, i, NULL);
2675 1700745 : if (k > li) return NULL;
2676 1700730 : if (k != i)
2677 : { /* exchange the lines s.t. k = i */
2678 1239538 : for (j=i; j<=aco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
2679 938540 : for (j=1; j<=bco; j++) swap(gcoeff(b,i,j), gcoeff(b,k,j));
2680 : }
2681 1700730 : p = gcoeff(a,i,i);
2682 1700730 : if (i == aco) break;
2683 :
2684 3909602 : for (k=i+1; k<=li; k++)
2685 : {
2686 2690808 : GEN m = gcoeff(a,k,i);
2687 2690808 : if (!gequal0(m))
2688 : {
2689 2042641 : m = gdiv(m,p);
2690 8371761 : for (j=i+1; j<=aco; j++) _submul(gel(a,j),k,i,m);
2691 6255239 : for (j=1; j<=bco; j++) _submul(gel(b,j),k,i,m);
2692 : }
2693 : }
2694 1218794 : if (gc_needed(av,1))
2695 : {
2696 12 : if(DEBUGMEM>1) pari_warn(warnmem,"gauss. i=%ld",i);
2697 12 : gerepileall(av,2, &a,&b);
2698 : }
2699 : }
2700 :
2701 481926 : if(DEBUGLEVEL>4) err_printf("Solving the triangular system\n");
2702 481926 : u = cgetg(bco+1,t_MAT);
2703 1157089 : for (j=1; j<=bco; j++) gel(u,j) = get_col(a,gel(b,j),p,aco);
2704 481920 : return gerepilecopy(av, iscol? gel(u,1): u);
2705 : }
2706 :
2707 : static GEN
2708 1182275 : RgM_RgC_solve_fast(GEN x, GEN y)
2709 : {
2710 : GEN p, pol;
2711 : long pa;
2712 1182275 : long t = RgM_RgC_type(x, y, &p,&pol,&pa);
2713 1182274 : switch(t)
2714 : {
2715 14629 : case t_INT: return ZM_gauss(x, y);
2716 175 : case t_FRAC: return QM_gauss(x, y);
2717 49 : case t_INTMOD: return RgM_RgC_solve_FpC(x, y, p);
2718 56 : case t_FFELT: return FFM_FFC_gauss(x, y, pol);
2719 1167365 : default: return gen_0;
2720 : }
2721 : }
2722 :
2723 : static GEN
2724 48657 : RgM_solve_fast(GEN x, GEN y)
2725 : {
2726 : GEN p, pol;
2727 : long pa;
2728 48657 : long t = RgM_type2(x, y, &p,&pol,&pa);
2729 48657 : switch(t)
2730 : {
2731 70 : case t_INT: return ZM_gauss(x, y);
2732 14 : case t_FRAC: return QM_gauss(x, y);
2733 98 : case t_INTMOD: return RgM_solve_FpM(x, y, p);
2734 63 : case t_FFELT: return FFM_gauss(x, y, pol);
2735 48412 : default: return gen_0;
2736 : }
2737 : }
2738 :
2739 : GEN
2740 1230932 : RgM_solve(GEN a, GEN b)
2741 : {
2742 1230932 : pari_sp av = avma;
2743 : GEN u;
2744 1230932 : if (!b) return RgM_inv(a);
2745 1230932 : u = typ(b)==t_MAT ? RgM_solve_fast(a, b): RgM_RgC_solve_fast(a, b);
2746 1230931 : if (!u) return gc_NULL(av);
2747 1230833 : if (u != gen_0) return u;
2748 1215777 : return RgM_solve_basecase(a, b);
2749 : }
2750 :
2751 : GEN
2752 21 : RgM_div(GEN a, GEN b)
2753 : {
2754 21 : pari_sp av = avma;
2755 21 : return gerepilecopy(av, shallowtrans(RgM_solve(shallowtrans(b),shallowtrans(a))));
2756 : }
2757 :
2758 : GEN
2759 204625 : RgM_inv(GEN a)
2760 : {
2761 204625 : GEN b = RgM_inv_fast(a);
2762 204615 : return b==gen_0? RgM_solve_basecase(a, NULL): b;
2763 : }
2764 :
2765 : /* assume dim A >= 1, A invertible + upper triangular */
2766 : static GEN
2767 3256378 : RgM_inv_upper_ind(GEN A, long index)
2768 : {
2769 3256378 : long n = lg(A)-1, i = index, j;
2770 3256378 : GEN u = zerocol(n);
2771 3256374 : gel(u,i) = ginv(gcoeff(A,i,i));
2772 6521457 : for (i--; i>0; i--)
2773 : {
2774 3265074 : pari_sp av = avma;
2775 3265074 : GEN m = gneg(gmul(gcoeff(A,i,i+1),gel(u,i+1))); /* j = i+1 */
2776 13955216 : for (j=i+2; j<=n; j++) m = gsub(m, gmul(gcoeff(A,i,j),gel(u,j)));
2777 3265060 : gel(u,i) = gerepileupto(av, gdiv(m, gcoeff(A,i,i)));
2778 : }
2779 3256383 : return u;
2780 : }
2781 : GEN
2782 1633434 : RgM_inv_upper(GEN A)
2783 : {
2784 : long i, l;
2785 1633434 : GEN B = cgetg_copy(A, &l);
2786 4889807 : for (i = 1; i < l; i++) gel(B,i) = RgM_inv_upper_ind(A, i);
2787 1633432 : return B;
2788 : }
2789 :
2790 : static GEN
2791 4509177 : split_realimag_col(GEN z, long r1, long r2)
2792 : {
2793 4509177 : long i, ru = r1+r2;
2794 4509177 : GEN x = cgetg(ru+r2+1,t_COL), y = x + r2;
2795 12506444 : for (i=1; i<=r1; i++) {
2796 7997268 : GEN a = gel(z,i);
2797 7997268 : if (typ(a) == t_COMPLEX) a = gel(a,1); /* paranoia: a should be real */
2798 7997268 : gel(x,i) = a;
2799 : }
2800 7198005 : for ( ; i<=ru; i++) {
2801 2688829 : GEN b, a = gel(z,i);
2802 2688829 : if (typ(a) == t_COMPLEX) { b = gel(a,2); a = gel(a,1); } else b = gen_0;
2803 2688829 : gel(x,i) = a;
2804 2688829 : gel(y,i) = b;
2805 : }
2806 4509176 : return x;
2807 : }
2808 : GEN
2809 2565253 : split_realimag(GEN x, long r1, long r2)
2810 : {
2811 : long i,l; GEN y;
2812 2565253 : if (typ(x) == t_COL) return split_realimag_col(x,r1,r2);
2813 1276776 : y = cgetg_copy(x, &l);
2814 4497475 : for (i=1; i<l; i++) gel(y,i) = split_realimag_col(gel(x,i), r1, r2);
2815 1276775 : return y;
2816 : }
2817 :
2818 : /* assume M = (r1+r2) x (r1+2r2) matrix and y compatible vector or matrix
2819 : * r1 first lines of M,y are real. Solve the system obtained by splitting
2820 : * real and imaginary parts. */
2821 : GEN
2822 1214644 : RgM_solve_realimag(GEN M, GEN y)
2823 : {
2824 1214644 : long l = lg(M), r2 = l - lgcols(M), r1 = l-1 - 2*r2;
2825 1214644 : return RgM_solve(split_realimag(M, r1,r2),
2826 : split_realimag(y, r1,r2));
2827 : }
2828 :
2829 : GEN
2830 434 : gauss(GEN a, GEN b)
2831 : {
2832 : GEN z;
2833 434 : long t = typ(b);
2834 434 : if (typ(a)!=t_MAT) pari_err_TYPE("gauss",a);
2835 434 : if (t!=t_COL && t!=t_MAT) pari_err_TYPE("gauss",b);
2836 434 : z = RgM_solve(a,b);
2837 434 : if (!z) pari_err_INV("gauss",a);
2838 329 : return z;
2839 : }
2840 :
2841 : /* #C = n, C[z[i]] = K[i], complete by 0s */
2842 : static GEN
2843 14 : RgC_inflate(GEN K, GEN z, long n)
2844 : {
2845 14 : GEN c = zerocol(n);
2846 14 : long j, l = lg(K);
2847 28 : for (j = 1; j < l; j++) gel(c, z[j]) = gel(K, j);
2848 14 : return c;
2849 : }
2850 : /* in place: C[i] *= cB / v[i] */
2851 : static void
2852 6384 : QC_normalize(GEN C, GEN v, GEN cB)
2853 : {
2854 6384 : long l = lg(C), i;
2855 47929 : for (i = 1; i < l; i++)
2856 : {
2857 41545 : GEN c = cB, k = gel(C,i), d = gel(v,i);
2858 41545 : if (d)
2859 : {
2860 24691 : if (isintzero(d)) { gel(C,i) = gen_0; continue; }
2861 24691 : c = div_content(c, d);
2862 : }
2863 41545 : gel(C,i) = c? gmul(k,c): k;
2864 : }
2865 6384 : }
2866 :
2867 : /* same as above, M rational; if flag = 1, call indexrank and return 1 sol */
2868 : GEN
2869 6377 : QM_gauss_i(GEN M, GEN B, long flag)
2870 : {
2871 6377 : pari_sp av = avma;
2872 : long i, l, n;
2873 6377 : int col = typ(B) == t_COL;
2874 6377 : GEN K, cB, N = cgetg_copy(M, &l), v = cgetg(l, t_VEC), z2 = NULL;
2875 :
2876 47950 : for (i = 1; i < l; i++)
2877 41573 : gel(N,i) = Q_primitive_part(gel(M,i), &gel(v,i));
2878 6377 : if (flag)
2879 : {
2880 329 : GEN z = ZM_indexrank(N), z1 = gel(z,1);
2881 329 : z2 = gel(z,2);
2882 329 : N = shallowmatextract(N, z1, z2);
2883 329 : B = col? vecpermute(B,z1): rowpermute(B,z1);
2884 329 : if (lg(z2) == l) z2 = NULL; else v = vecpermute(v, z2);
2885 : }
2886 6377 : B = Q_primitive_part(B, &cB);
2887 6377 : K = ZM_gauss(N, B); if (!K) return gc_NULL(av);
2888 6377 : n = l - 1;
2889 6377 : if (col)
2890 : {
2891 6349 : QC_normalize(K, v, cB);
2892 6349 : if (z2) K = RgC_inflate(K, z2, n);
2893 : }
2894 : else
2895 : {
2896 28 : long lK = lg(K);
2897 63 : for (i = 1; i < lK; i++)
2898 : {
2899 35 : QC_normalize(gel(K,i), v, cB);
2900 35 : if (z2) gel(K,i) = RgC_inflate(gel(K,i), z2, n);
2901 : }
2902 : }
2903 6377 : return gerepilecopy(av, K);
2904 : }
2905 : GEN
2906 6048 : QM_gauss(GEN M, GEN B) { return QM_gauss_i(M, B, 0); }
2907 :
2908 : static GEN
2909 785156 : ZM_inv_slice(GEN A, GEN P, GEN *mod)
2910 : {
2911 785156 : pari_sp av = avma;
2912 785156 : long i, n = lg(P)-1;
2913 : GEN H, T;
2914 785156 : if (n == 1)
2915 : {
2916 749801 : ulong p = uel(P,1);
2917 749801 : GEN Hp, a = ZM_to_Flm(A, p);
2918 749802 : Hp = Flm_adjoint(a, p);
2919 749802 : Hp = gerepileupto(av, Flm_to_ZM(Hp));
2920 749804 : *mod = utoipos(p); return Hp;
2921 : }
2922 35355 : T = ZV_producttree(P);
2923 35356 : A = ZM_nv_mod_tree(A, P, T);
2924 35357 : H = cgetg(n+1, t_VEC);
2925 205400 : for(i=1; i <= n; i++)
2926 170043 : gel(H,i) = Flm_adjoint(gel(A, i), uel(P,i));
2927 35357 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P,T));
2928 35357 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2929 : }
2930 :
2931 : static GEN
2932 709236 : RgM_true_Hadamard(GEN a)
2933 : {
2934 709236 : pari_sp av = avma;
2935 709236 : long n = lg(a)-1, i;
2936 : GEN B;
2937 709236 : if (n == 0) return gen_1;
2938 709236 : a = RgM_gtofp(a, LOWDEFAULTPREC);
2939 709233 : B = gnorml2(gel(a,1));
2940 2908818 : for (i = 2; i <= n; i++) B = gmul(B, gnorml2(gel(a,i)));
2941 709232 : return gerepileuptoint(av, ceil_safe(sqrtr(B)));
2942 : }
2943 :
2944 : GEN
2945 785159 : ZM_inv_worker(GEN P, GEN A)
2946 : {
2947 785159 : GEN V = cgetg(3, t_VEC);
2948 785156 : gel(V,1) = ZM_inv_slice(A, P, &gel(V,2));
2949 785161 : return V;
2950 : }
2951 :
2952 : static GEN
2953 41244 : ZM_inv0(GEN A, GEN *pden)
2954 : {
2955 41244 : if (pden) *pden = gen_1;
2956 41244 : (void)A; return cgetg(1, t_MAT);
2957 : }
2958 : static GEN
2959 643250 : ZM_inv1(GEN A, GEN *pden)
2960 : {
2961 643250 : GEN a = gcoeff(A,1,1);
2962 643250 : long s = signe(a);
2963 643250 : if (!s) return NULL;
2964 643250 : if (pden) *pden = absi(a);
2965 643250 : retmkmat(mkcol(s == 1? gen_1: gen_m1));
2966 : }
2967 : static GEN
2968 741526 : ZM_inv2(GEN A, GEN *pden)
2969 : {
2970 : GEN a, b, c, d, D, cA;
2971 : long s;
2972 741526 : A = Q_primitive_part(A, &cA);
2973 741521 : a = gcoeff(A,1,1); b = gcoeff(A,1,2);
2974 741521 : c = gcoeff(A,2,1); d = gcoeff(A,2,2);
2975 741521 : D = subii(mulii(a,d), mulii(b,c)); /* left on stack */
2976 741513 : s = signe(D);
2977 741513 : if (!s) return NULL;
2978 741499 : if (s < 0) D = negi(D);
2979 741500 : if (pden) *pden = mul_denom(D, cA);
2980 741502 : if (s > 0)
2981 700501 : retmkmat2(mkcol2(icopy(d), negi(c)), mkcol2(negi(b), icopy(a)));
2982 : else
2983 41001 : retmkmat2(mkcol2(negi(d), icopy(c)), mkcol2(icopy(b), negi(a)));
2984 : }
2985 :
2986 : /* to be used when denom(M^(-1)) << det(M) and a sharp multiple is
2987 : * not available. Return H primitive such that M*H = den*Id */
2988 : GEN
2989 0 : ZM_inv_ratlift(GEN M, GEN *pden)
2990 : {
2991 0 : pari_sp av2, av = avma;
2992 : GEN Hp, q, H;
2993 : ulong p;
2994 0 : long m = lg(M)-1;
2995 : forprime_t S;
2996 : pari_timer ti;
2997 :
2998 0 : if (m == 0) return ZM_inv0(M,pden);
2999 0 : if (m == 1 && nbrows(M)==1) return ZM_inv1(M,pden);
3000 0 : if (m == 2 && nbrows(M)==2) return ZM_inv2(M,pden);
3001 :
3002 0 : if (DEBUGLEVEL>5) timer_start(&ti);
3003 0 : init_modular_big(&S);
3004 0 : av2 = avma;
3005 0 : H = NULL;
3006 0 : while ((p = u_forprime_next(&S)))
3007 : {
3008 : GEN Mp, B, Hr;
3009 0 : Mp = ZM_to_Flm(M,p);
3010 0 : Hp = Flm_inv_sp(Mp, NULL, p);
3011 0 : if (!Hp) continue;
3012 0 : if (!H)
3013 : {
3014 0 : H = ZM_init_CRT(Hp, p);
3015 0 : q = utoipos(p);
3016 : }
3017 : else
3018 0 : ZM_incremental_CRT(&H, Hp, &q, p);
3019 0 : B = sqrti(shifti(q,-1));
3020 0 : Hr = FpM_ratlift(H,q,B,B,NULL);
3021 0 : if (DEBUGLEVEL>5)
3022 0 : timer_printf(&ti,"ZM_inv mod %lu (ratlift=%ld)", p,!!Hr);
3023 0 : if (Hr) {/* DONE ? */
3024 0 : GEN Hl = Q_remove_denom(Hr, pden);
3025 0 : if (ZM_isscalar(ZM_mul(Hl, M), *pden)) { H = Hl; break; }
3026 : }
3027 :
3028 0 : if (gc_needed(av,2))
3029 : {
3030 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZM_inv_ratlift");
3031 0 : gerepileall(av2, 2, &H, &q);
3032 : }
3033 : }
3034 0 : if (!*pden) *pden = gen_1;
3035 0 : return gc_all(av, 2, &H, pden);
3036 : }
3037 :
3038 : GEN
3039 73637 : FpM_ratlift_worker(GEN A, GEN mod, GEN B)
3040 : {
3041 : long l, i;
3042 73637 : GEN H = cgetg_copy(A, &l);
3043 154238 : for (i = 1; i < l; i++)
3044 : {
3045 80603 : GEN c = FpC_ratlift(gel(A,i), mod, B, B, NULL);
3046 80606 : gel(H,i) = c? c: gen_0;
3047 : }
3048 73635 : return H;
3049 : }
3050 : static int
3051 754985 : can_ratlift(GEN x, GEN mod, GEN B)
3052 : {
3053 754985 : pari_sp av = avma;
3054 : GEN a, b;
3055 754985 : return gc_bool(av, Fp_ratlift(x, mod, B, B, &a,&b));
3056 : }
3057 : static GEN
3058 2720692 : FpM_ratlift_parallel(GEN A, GEN mod, GEN B)
3059 : {
3060 2720692 : pari_sp av = avma;
3061 : GEN worker;
3062 2720692 : long i, l = lg(A), m = mt_nbthreads();
3063 2720690 : int test = !!B;
3064 :
3065 2720690 : if (l == 1 || lgcols(A) == 1) return gcopy(A);
3066 2720691 : if (!B) B = sqrti(shifti(mod,-1));
3067 2720637 : if (m == 1 || l == 2 || lgcols(A) < 10)
3068 : {
3069 2713225 : A = FpM_ratlift(A, mod, B, B, NULL);
3070 2713274 : return A? A: gc_NULL(av);
3071 : }
3072 : /* test one coefficient first */
3073 7412 : if (test && !can_ratlift(gcoeff(A,1,1), mod, B)) return gc_NULL(av);
3074 7293 : worker = snm_closure(is_entry("_FpM_ratlift_worker"), mkvec2(mod,B));
3075 7293 : A = gen_parapply_slice(worker, A, m);
3076 79434 : for (i = 1; i < l; i++) if (typ(gel(A,i)) != t_COL) return gc_NULL(av);
3077 6242 : return A;
3078 : }
3079 :
3080 : static GEN
3081 747900 : ZM_adj_ratlift(GEN A, GEN H, GEN mod, GEN T)
3082 : {
3083 747900 : pari_sp av = avma;
3084 : GEN B, D, g;
3085 747900 : D = ZMrow_ZC_mul(H, gel(A,1), 1);
3086 747899 : if (T) D = mulii(T, D);
3087 747899 : g = gcdii(D, mod);
3088 747894 : if (!equali1(g))
3089 : {
3090 14 : mod = diviiexact(mod, g);
3091 14 : H = FpM_red(H, mod);
3092 : }
3093 747894 : D = Fp_inv(Fp_red(D, mod), mod);
3094 : /* test 1 coeff first */
3095 747897 : B = sqrti(shifti(mod,-1));
3096 747895 : if (!can_ratlift(Fp_mul(D, gcoeff(A,1,1), mod), mod, B)) return gc_NULL(av);
3097 726876 : H = FpM_Fp_mul(H, D, mod);
3098 726870 : H = FpM_ratlift_parallel(H, mod, B);
3099 726876 : return H? H: gc_NULL(av);
3100 : }
3101 :
3102 : /* if (T) return T A^(-1) in Mn(Q), else B in Mn(Z) such that A B = den*Id */
3103 : static GEN
3104 2135260 : ZM_inv_i(GEN A, GEN *pden, GEN T)
3105 : {
3106 2135260 : pari_sp av = avma;
3107 2135260 : long m = lg(A)-1, n, k1 = 1, k2;
3108 2135260 : GEN H = NULL, D, H1 = NULL, mod1 = NULL, worker;
3109 : ulong bnd, mask;
3110 : forprime_t S;
3111 : pari_timer ti;
3112 :
3113 2135260 : if (m == 0) return ZM_inv0(A,pden);
3114 2094016 : if (pden) *pden = gen_1;
3115 2094016 : if (nbrows(A) < m) return NULL;
3116 2094009 : if (m == 1 && nbrows(A)==1 && !T) return ZM_inv1(A,pden);
3117 1450759 : if (m == 2 && nbrows(A)==2 && !T) return ZM_inv2(A,pden);
3118 :
3119 709232 : if (DEBUGLEVEL>=5) timer_start(&ti);
3120 709232 : init_modular_big(&S);
3121 709236 : bnd = expi(RgM_true_Hadamard(A));
3122 709236 : worker = snm_closure(is_entry("_ZM_inv_worker"), mkvec(A));
3123 709238 : gen_inccrt("ZM_inv_r", worker, NULL, k1, 0, &S, &H1, &mod1, nmV_chinese_center, FpM_center);
3124 709237 : n = (bnd+1)/expu(S.p)+1;
3125 709237 : if (DEBUGLEVEL>=5) timer_printf(&ti,"inv (%ld/%ld primes)", k1, n);
3126 709237 : mask = quadratic_prec_mask(n);
3127 709236 : for (k2 = 0;;)
3128 66665 : {
3129 : GEN Hr;
3130 775901 : if (k2 > 0)
3131 : {
3132 59464 : gen_inccrt("ZM_inv_r", worker, NULL, k2, 0, &S, &H1, &mod1,nmV_chinese_center,FpM_center);
3133 59465 : k1 += k2;
3134 59465 : if (DEBUGLEVEL>=5) timer_printf(&ti,"CRT (%ld/%ld primes)", k1, n);
3135 : }
3136 775903 : if (mask == 1) break;
3137 747901 : k2 = (mask&1UL) ? k1-1: k1;
3138 747901 : mask >>= 1;
3139 :
3140 747901 : Hr = ZM_adj_ratlift(A, H1, mod1, T);
3141 747899 : if (DEBUGLEVEL>=5) timer_printf(&ti,"ratlift (%ld/%ld primes)", k1, n);
3142 747899 : if (Hr) {/* DONE ? */
3143 685091 : GEN Hl = Q_primpart(Hr), R = ZM_mul(Hl, A), d = gcoeff(R,1,1);
3144 685090 : if (gsigne(d) < 0) { d = gneg(d); Hl = ZM_neg(Hl); }
3145 685090 : if (DEBUGLEVEL>=5) timer_printf(&ti,"mult (%ld/%ld primes)", k1, n);
3146 685090 : if (equali1(d))
3147 : {
3148 592949 : if (ZM_isidentity(R)) { H = Hl; break; }
3149 : }
3150 92141 : else if (ZM_isscalar(R, d))
3151 : {
3152 88287 : if (T) T = gdiv(T,d);
3153 84863 : else if (pden) *pden = d;
3154 88287 : H = Hl; break;
3155 : }
3156 : }
3157 : }
3158 709237 : if (!H)
3159 : {
3160 : GEN d;
3161 28002 : H = H1;
3162 28002 : D = ZMrow_ZC_mul(H, gel(A,1), 1);
3163 28002 : if (signe(D)==0) pari_err_INV("ZM_inv", A);
3164 28002 : if (T) T = gdiv(T, D);
3165 : else
3166 : {
3167 27049 : d = gcdii(Q_content_safe(H), D);
3168 27049 : if (signe(D) < 0) d = negi(d);
3169 27049 : if (!equali1(d))
3170 : {
3171 15260 : H = ZM_Z_divexact(H, d);
3172 15260 : D = diviiexact(D, d);
3173 : }
3174 27049 : if (pden) *pden = D;
3175 : }
3176 : }
3177 709237 : if (T && !isint1(T)) H = ZM_Q_mul(H, T);
3178 709237 : return gc_all(av, pden? 2: 1, &H, pden);
3179 : }
3180 : GEN
3181 2074505 : ZM_inv(GEN A, GEN *pden) { return ZM_inv_i(A, pden, NULL); }
3182 :
3183 : /* same as above, M rational */
3184 : GEN
3185 60755 : QM_inv(GEN M)
3186 : {
3187 60755 : pari_sp av = avma;
3188 : GEN den, dM, K;
3189 60755 : M = Q_remove_denom(M, &dM);
3190 60755 : K = ZM_inv_i(M, &den, dM);
3191 60755 : if (!K) return gc_NULL(av);
3192 60734 : if (den && !equali1(den)) K = ZM_Q_mul(K, ginv(den));
3193 60720 : return gerepileupto(av, K);
3194 : }
3195 :
3196 : static GEN
3197 105223 : ZM_ker_filter(GEN A, GEN P)
3198 : {
3199 105223 : long i, j, l = lg(A), n = 1, d = lg(gmael(A,1,1));
3200 105223 : GEN B, Q, D = gmael(A,1,2);
3201 215185 : for (i=2; i<l; i++)
3202 : {
3203 109962 : GEN Di = gmael(A,i,2);
3204 109962 : long di = lg(gmael(A,i,1));
3205 109962 : int c = vecsmall_lexcmp(D, Di);
3206 109962 : if (di==d && c==0) n++;
3207 45588 : else if (d > di || (di==d && c>0))
3208 37680 : { n = 1; d = di; D = Di; }
3209 : }
3210 105223 : B = cgetg(n+1, t_VEC);
3211 105223 : Q = cgetg(n+1, typ(P));
3212 320408 : for (i=1, j=1; i<l; i++)
3213 : {
3214 215185 : if (lg(gmael(A,i,1))==d && vecsmall_lexcmp(D, gmael(A,i,2))==0)
3215 : {
3216 169597 : gel(B,j) = gmael(A,i,1);
3217 169597 : Q[j] = P[i];
3218 169597 : j++;
3219 : }
3220 : }
3221 105223 : return mkvec3(B,Q,D);
3222 : }
3223 :
3224 : static GEN
3225 69559 : ZM_ker_chinese(GEN A, GEN P, GEN *mod)
3226 : {
3227 69559 : GEN BQD = ZM_ker_filter(A, P);
3228 69559 : return mkvec2(nmV_chinese_center(gel(BQD,1), gel(BQD,2), mod), gel(BQD,3));
3229 : }
3230 :
3231 : static GEN
3232 129591 : ZM_ker_slice(GEN A, GEN P, GEN *mod)
3233 : {
3234 129591 : pari_sp av = avma;
3235 129591 : long i, n = lg(P)-1;
3236 : GEN BQD, D, H, T, Q;
3237 129591 : if (n == 1)
3238 : {
3239 93924 : ulong p = uel(P,1);
3240 93924 : GEN K = Flm_ker_sp(ZM_to_Flm(A, p), p, 2);
3241 93924 : *mod = utoipos(p); return mkvec2(Flm_to_ZM(gel(K,1)), gel(K,2));
3242 : }
3243 35667 : T = ZV_producttree(P);
3244 35667 : A = ZM_nv_mod_tree(A, P, T);
3245 35666 : H = cgetg(n+1, t_VEC);
3246 111498 : for(i=1 ; i <= n; i++)
3247 75834 : gel(H,i) = Flm_ker_sp(gel(A, i), P[i], 2);
3248 35664 : BQD = ZM_ker_filter(H, P); Q = gel(BQD,2);
3249 35664 : if (lg(Q) != lg(P)) T = ZV_producttree(Q);
3250 35664 : H = nmV_chinese_center_tree_seq(gel(BQD,1), Q, T, ZV_chinesetree(Q,T));
3251 35663 : *mod = gmael(T, lg(T)-1, 1);
3252 35663 : D = gel(BQD, 3);
3253 35663 : gerepileall(av, 3, &H, &D, mod);
3254 35666 : return mkvec2(H,D);
3255 : }
3256 :
3257 : GEN
3258 129591 : ZM_ker_worker(GEN P, GEN A)
3259 : {
3260 129591 : GEN V = cgetg(3, t_VEC);
3261 129591 : gel(V,1) = ZM_ker_slice(A, P, &gel(V,2));
3262 129590 : return V;
3263 : }
3264 :
3265 : /* assume lg(A) > 1 */
3266 : static GEN
3267 62856 : ZM_ker_i(GEN A)
3268 : {
3269 : pari_sp av;
3270 62856 : long k, m = lg(A)-1;
3271 62856 : GEN HD = NULL, mod = gen_1, worker;
3272 : forprime_t S;
3273 :
3274 62856 : if (m >= 2*nbrows(A))
3275 : {
3276 3059 : GEN v = ZM_indexrank(A), y = gel(v,2), z = indexcompl(y, m);
3277 : GEN B, A1, A1i, d;
3278 3059 : A = rowpermute(A, gel(v,1)); /* same kernel */
3279 3059 : A1 = vecpermute(A, y); /* maximal rank submatrix */
3280 3059 : B = vecpermute(A, z);
3281 3059 : A1i = ZM_inv(A1, &d);
3282 3059 : if (!d) d = gen_1;
3283 3059 : B = vconcat(ZM_mul(ZM_neg(A1i), B), scalarmat_shallow(d, lg(B)-1));
3284 3059 : if (!gequal(y, identity_perm(lg(y)-1)))
3285 692 : B = rowpermute(B, perm_inv(shallowconcat(y,z)));
3286 3059 : return vec_Q_primpart(B);
3287 : }
3288 59797 : init_modular_big(&S);
3289 59797 : worker = snm_closure(is_entry("_ZM_ker_worker"), mkvec(A));
3290 59797 : av = avma;
3291 59797 : for (k = 1;; k <<= 1)
3292 65346 : {
3293 : pari_timer ti;
3294 : GEN H, Hr;
3295 125143 : gen_inccrt_i("ZM_ker", worker, NULL, (k+1)>>1, 0,
3296 : &S, &HD, &mod, ZM_ker_chinese, NULL);
3297 125143 : gerepileall(av, 2, &HD, &mod);
3298 140693 : H = gel(HD, 1); if (lg(H) == 1) return H;
3299 80896 : if (DEBUGLEVEL >= 4) timer_start(&ti);
3300 80896 : Hr = FpM_ratlift_parallel(H, mod, NULL);
3301 80896 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"ZM_ker: ratlift (%ld)",!!Hr);
3302 80896 : if (Hr)
3303 : {
3304 : GEN MH;
3305 70146 : Hr = vec_Q_primpart(Hr);
3306 70146 : MH = ZM_mul(A, Hr);
3307 70146 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"ZM_ker: QM_mul");
3308 70146 : if (ZM_equal0(MH)) return Hr;
3309 : }
3310 : }
3311 : }
3312 :
3313 : GEN
3314 47106 : ZM_ker(GEN M)
3315 : {
3316 47106 : pari_sp av = avma;
3317 47106 : long l = lg(M)-1;
3318 47106 : if (l==0) return cgetg(1, t_MAT);
3319 47106 : if (lgcols(M)==1) return matid(l);
3320 47106 : return gerepilecopy(av, ZM_ker_i(M));
3321 : }
3322 :
3323 : static GEN
3324 1986234 : ZM_gauss_slice(GEN A, GEN B, GEN P, GEN *mod)
3325 : {
3326 1986234 : pari_sp av = avma;
3327 1986234 : long i, n = lg(P)-1;
3328 : GEN H, T;
3329 1986234 : if (n == 1)
3330 : {
3331 1947211 : ulong p = uel(P,1);
3332 1947211 : GEN Hp = Flm_gauss(ZM_to_Flm(A, p) , ZM_to_Flm(B, p) ,p);
3333 1947210 : if (!Hp) { *mod=gen_1; return zeromat(lg(A)-1,lg(B)-1); }
3334 1947210 : Hp = gerepileupto(av, Flm_to_ZM(Hp));
3335 1947207 : *mod = utoipos(p); return Hp;
3336 : }
3337 39023 : T = ZV_producttree(P);
3338 39023 : A = ZM_nv_mod_tree(A, P, T);
3339 39023 : B = ZM_nv_mod_tree(B, P, T);
3340 39023 : H = cgetg(n+1, t_VEC);
3341 224397 : for(i=1; i <= n; i++)
3342 : {
3343 185374 : GEN Hi = Flm_gauss(gel(A, i), gel(B,i), uel(P,i));
3344 185374 : gel(H,i) = Hi ? Hi: zero_Flm(lg(A)-1,lg(B)-1);
3345 185374 : if (!Hi) uel(P,i)=1;
3346 : }
3347 39023 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P,T));
3348 39023 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
3349 : }
3350 :
3351 : GEN
3352 1986237 : ZM_gauss_worker(GEN P, GEN A, GEN B)
3353 : {
3354 1986237 : GEN V = cgetg(3, t_VEC);
3355 1986234 : gel(V,1) = ZM_gauss_slice(A, B, P, &gel(V,2));
3356 1986230 : return V;
3357 : }
3358 :
3359 : /* assume lg(A) > 1 */
3360 : static GEN
3361 1736427 : ZM_gauss_i(GEN A, GEN B)
3362 : {
3363 : pari_sp av;
3364 : long k, m, ncol;
3365 : int iscol;
3366 1736427 : GEN y, y1, y2, Hr, H = NULL, mod = gen_1, worker;
3367 : forprime_t S;
3368 1736427 : if (!init_gauss(A, &B, &m, &ncol, &iscol)) return cgetg(1, iscol?t_COL:t_MAT);
3369 1736356 : init_modular_big(&S);
3370 1736356 : y = ZM_indexrank(A); y1 = gel(y,1); y2 = gel(y,2);
3371 1736365 : if (lg(y2)-1 != m) return NULL;
3372 1736344 : A = rowpermute(A, y1);
3373 1736341 : B = rowpermute(B, y1);
3374 : /* a is square and invertible */
3375 1736338 : ncol = lg(B);
3376 1736338 : worker = snm_closure(is_entry("_ZM_gauss_worker"), mkvec2(A,B));
3377 1736346 : av = avma;
3378 1736346 : for (k = 1;; k <<= 1)
3379 176587 : {
3380 : pari_timer ti;
3381 1912933 : gen_inccrt_i("ZM_gauss", worker, NULL, (k+1)>>1 , m,
3382 : &S, &H, &mod, nmV_chinese_center, FpM_center);
3383 1912905 : gerepileall(av, 2, &H, &mod);
3384 1912933 : if (DEBUGLEVEL >= 4) timer_start(&ti);
3385 1912933 : Hr = FpM_ratlift_parallel(H, mod, NULL);
3386 1912916 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"ZM_gauss: ratlift (%ld)",!!Hr);
3387 1912921 : if (Hr)
3388 : {
3389 : GEN MH, c;
3390 1779853 : MH = ZM_mul(A, Q_remove_denom(Hr, &c));
3391 1779841 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"ZM_ker: QM_mul");
3392 1779850 : if (ZM_equal(MH, c ? ZM_Z_mul(B, c): B)) break;
3393 : }
3394 : }
3395 1736328 : return iscol ? gel(Hr, 1): Hr;
3396 : }
3397 :
3398 : GEN
3399 1736427 : ZM_gauss(GEN A, GEN B)
3400 : {
3401 1736427 : pari_sp av = avma;
3402 1736427 : GEN C = ZM_gauss_i(A,B);
3403 1736414 : return C ? gerepilecopy(av, C): NULL;
3404 : }
3405 :
3406 : GEN
3407 16569 : QM_ker(GEN M)
3408 : {
3409 16569 : pari_sp av = avma;
3410 16569 : long l = lg(M)-1;
3411 16569 : if (l==0) return cgetg(1, t_MAT);
3412 16527 : if (lgcols(M)==1) return matid(l);
3413 15666 : return gerepilecopy(av, ZM_ker_i(row_Q_primpart(M)));
3414 : }
3415 :
3416 : /* x a ZM. Return a multiple of the determinant of the lattice generated by
3417 : * the columns of x. From Algorithm 2.2.6 in GTM138 */
3418 : GEN
3419 47549 : detint(GEN A)
3420 : {
3421 47549 : if (typ(A) != t_MAT) pari_err_TYPE("detint",A);
3422 47549 : RgM_check_ZM(A, "detint");
3423 47549 : return ZM_detmult(A);
3424 : }
3425 : GEN
3426 157266 : ZM_detmult(GEN A)
3427 : {
3428 157266 : pari_sp av1, av = avma;
3429 : GEN B, c, v, piv;
3430 157266 : long rg, i, j, k, m, n = lg(A) - 1;
3431 :
3432 157266 : if (!n) return gen_1;
3433 157266 : m = nbrows(A);
3434 157266 : if (n < m) return gen_0;
3435 157189 : c = zero_zv(m);
3436 157189 : av1 = avma;
3437 157189 : B = zeromatcopy(m,m);
3438 157189 : v = cgetg(m+1, t_COL);
3439 157187 : piv = gen_1; rg = 0;
3440 687009 : for (k=1; k<=n; k++)
3441 : {
3442 686995 : GEN pivprec = piv;
3443 686995 : long t = 0;
3444 4934046 : for (i=1; i<=m; i++)
3445 : {
3446 4247050 : pari_sp av2 = avma;
3447 : GEN vi;
3448 4247050 : if (c[i]) continue;
3449 :
3450 2467273 : vi = mulii(piv, gcoeff(A,i,k));
3451 22519258 : for (j=1; j<=m; j++)
3452 20051941 : if (c[j]) vi = addii(vi, mulii(gcoeff(B,j,i),gcoeff(A,j,k)));
3453 2467317 : if (!t && signe(vi)) t = i;
3454 2467317 : gel(v,i) = gerepileuptoint(av2, vi);
3455 : }
3456 686996 : if (!t) continue;
3457 : /* at this point c[t] = 0 */
3458 :
3459 686905 : if (++rg >= m) { /* full rank; mostly done */
3460 157175 : GEN det = gel(v,t); /* last on stack */
3461 157175 : if (++k > n)
3462 157036 : det = absi(det);
3463 : else
3464 : {
3465 : /* improve further; at this point c[i] is set for all i != t */
3466 139 : gcoeff(B,t,t) = piv; v = centermod(gel(B,t), det);
3467 446 : for ( ; k<=n; k++)
3468 307 : det = gcdii(det, ZV_dotproduct(v, gel(A,k)));
3469 : }
3470 157175 : return gerepileuptoint(av, det);
3471 : }
3472 :
3473 529730 : piv = gel(v,t);
3474 4089382 : for (i=1; i<=m; i++)
3475 : {
3476 : GEN mvi;
3477 3559652 : if (c[i] || i == t) continue;
3478 :
3479 1779826 : gcoeff(B,t,i) = mvi = negi(gel(v,i));
3480 17581302 : for (j=1; j<=m; j++)
3481 15801476 : if (c[j]) /* implies j != t */
3482 : {
3483 4080608 : pari_sp av2 = avma;
3484 4080608 : GEN z = addii(mulii(gcoeff(B,j,i), piv), mulii(gcoeff(B,j,t), mvi));
3485 4080608 : if (rg > 1) z = diviiexact(z, pivprec);
3486 4080608 : gcoeff(B,j,i) = gerepileuptoint(av2, z);
3487 : }
3488 : }
3489 529730 : c[t] = k;
3490 529730 : if (gc_needed(av,1))
3491 : {
3492 0 : if(DEBUGMEM>1) pari_warn(warnmem,"detint. k=%ld",k);
3493 0 : gerepileall(av1, 2, &piv,&B); v = zerovec(m);
3494 : }
3495 : }
3496 14 : return gc_const(av, gen_0);
3497 : }
3498 :
3499 : /* Reduce x modulo (invertible) y */
3500 : GEN
3501 14678 : closemodinvertible(GEN x, GEN y)
3502 : {
3503 14678 : return gmul(y, ground(RgM_solve(y,x)));
3504 : }
3505 : GEN
3506 7 : reducemodinvertible(GEN x, GEN y)
3507 : {
3508 7 : return gsub(x, closemodinvertible(x,y));
3509 : }
3510 : GEN
3511 0 : reducemodlll(GEN x,GEN y)
3512 : {
3513 0 : return reducemodinvertible(x, ZM_lll(y, 0.75, LLL_INPLACE));
3514 : }
3515 :
3516 : /*******************************************************************/
3517 : /* */
3518 : /* KERNEL of an m x n matrix */
3519 : /* return n - rk(x) linearly independent vectors */
3520 : /* */
3521 : /*******************************************************************/
3522 : static GEN
3523 28 : RgM_deplin_i(GEN x0)
3524 : {
3525 28 : pari_sp av = avma, av2;
3526 28 : long i, j, k, nl, nc = lg(x0)-1;
3527 : GEN D, x, y, c, l, d, ck;
3528 :
3529 28 : if (!nc) return NULL;
3530 28 : nl = nbrows(x0);
3531 28 : c = zero_zv(nl);
3532 28 : l = cgetg(nc+1, t_VECSMALL); /* not initialized */
3533 28 : av2 = avma;
3534 28 : x = RgM_shallowcopy(x0);
3535 28 : d = const_vec(nl, gen_1); /* pivot list */
3536 28 : ck = NULL; /* gcc -Wall */
3537 98 : for (k=1; k<=nc; k++)
3538 : {
3539 91 : ck = gel(x,k);
3540 196 : for (j=1; j<k; j++)
3541 : {
3542 105 : GEN cj = gel(x,j), piv = gel(d,j), q = gel(ck,l[j]);
3543 420 : for (i=1; i<=nl; i++)
3544 315 : if (i!=l[j]) gel(ck,i) = gsub(gmul(piv, gel(ck,i)), gmul(q, gel(cj,i)));
3545 : }
3546 :
3547 91 : i = gauss_get_pivot_NZ(x, NULL, k, c);
3548 91 : if (i > nl) break;
3549 70 : if (gc_needed(av,1))
3550 : {
3551 0 : if (DEBUGMEM>1) pari_warn(warnmem,"deplin k = %ld/%ld",k,nc);
3552 0 : gerepileall(av2, 2, &x, &d);
3553 0 : ck = gel(x,k);
3554 : }
3555 70 : gel(d,k) = gel(ck,i);
3556 70 : c[i] = k; l[k] = i; /* pivot d[k] in x[i,k] */
3557 : }
3558 28 : if (k > nc) return gc_NULL(av);
3559 21 : if (k == 1) { set_avma(av); return scalarcol_shallow(gen_1,nc); }
3560 21 : y = cgetg(nc+1,t_COL);
3561 21 : gel(y,1) = gcopy(gel(ck, l[1]));
3562 49 : for (D=gel(d,1),j=2; j<k; j++)
3563 : {
3564 28 : gel(y,j) = gmul(gel(ck, l[j]), D);
3565 28 : D = gmul(D, gel(d,j));
3566 : }
3567 21 : gel(y,j) = gneg(D);
3568 21 : for (j++; j<=nc; j++) gel(y,j) = gen_0;
3569 21 : y = primitive_part(y, &c);
3570 21 : return c? gerepileupto(av, y): gerepilecopy(av, y);
3571 : }
3572 : static GEN
3573 0 : RgV_deplin(GEN v)
3574 : {
3575 0 : pari_sp av = avma;
3576 0 : long n = lg(v)-1;
3577 0 : GEN y, p = NULL;
3578 0 : if (n <= 1)
3579 : {
3580 0 : if (n == 1 && gequal0(gel(v,1))) return mkcol(gen_1);
3581 0 : return cgetg(1, t_COL);
3582 : }
3583 0 : if (gequal0(gel(v,1))) return scalarcol_shallow(gen_1, n);
3584 0 : v = primpart(mkvec2(gel(v,1),gel(v,2)));
3585 0 : if (RgV_is_FpV(v, &p) && p) v = centerlift(v);
3586 0 : y = zerocol(n);
3587 0 : gel(y,1) = gneg(gel(v,2));
3588 0 : gel(y,2) = gcopy(gel(v,1));
3589 0 : return gerepileupto(av, y);
3590 :
3591 : }
3592 :
3593 : static GEN
3594 105 : RgM_deplin_FpM(GEN x, GEN p)
3595 : {
3596 105 : pari_sp av = avma;
3597 : ulong pp;
3598 105 : x = RgM_Fp_init3(x, p, &pp);
3599 105 : switch(pp)
3600 : {
3601 35 : case 0:
3602 35 : x = FpM_ker_gen(x,p,1);
3603 35 : if (!x) return gc_NULL(av);
3604 21 : x = FpC_center(x,p,shifti(p,-1));
3605 21 : break;
3606 14 : case 2:
3607 14 : x = F2m_ker_sp(x,1);
3608 14 : if (!x) return gc_NULL(av);
3609 7 : x = F2c_to_ZC(x); break;
3610 0 : case 3:
3611 0 : x = F3m_ker_sp(x,1);
3612 0 : if (!x) return gc_NULL(av);
3613 0 : x = F3c_to_ZC(x); break;
3614 56 : default:
3615 56 : x = Flm_ker_sp(x,pp,1);
3616 56 : if (!x) return gc_NULL(av);
3617 35 : x = Flv_center(x, pp, pp>>1);
3618 35 : x = zc_to_ZC(x);
3619 35 : break;
3620 : }
3621 63 : return gerepileupto(av, x);
3622 : }
3623 :
3624 : /* FIXME: implement direct modular ZM_deplin ? */
3625 : static GEN
3626 119 : QM_deplin(GEN M)
3627 : {
3628 119 : pari_sp av = avma;
3629 119 : long l = lg(M)-1;
3630 : GEN k;
3631 119 : if (l==0) return NULL;
3632 84 : if (lgcols(M)==1) return col_ei(l, 1);
3633 84 : k = ZM_ker_i(row_Q_primpart(M));
3634 84 : if (lg(k)== 1) return gc_NULL(av);
3635 70 : return gerepilecopy(av, gel(k,1));
3636 : }
3637 :
3638 : static GEN
3639 49 : RgM_deplin_FqM(GEN x, GEN pol, GEN p)
3640 : {
3641 49 : pari_sp av = avma;
3642 49 : GEN b, T = RgX_to_FpX(pol, p);
3643 49 : if (signe(T) == 0) pari_err_OP("deplin",x,pol);
3644 49 : b = FqM_deplin(RgM_to_FqM(x, T, p), T, p);
3645 49 : if (!b) return gc_NULL(av);
3646 35 : return gerepileupto(av, b);
3647 : }
3648 :
3649 : #define code(t1,t2) ((t1 << 6) | t2)
3650 : static GEN
3651 385 : RgM_deplin_fast(GEN x)
3652 : {
3653 : GEN p, pol;
3654 : long pa;
3655 385 : long t = RgM_type(x, &p,&pol,&pa);
3656 385 : switch(t)
3657 : {
3658 119 : case t_INT: /* fall through */
3659 119 : case t_FRAC: return QM_deplin(x);
3660 84 : case t_FFELT: return FFM_deplin(x, pol);
3661 105 : case t_INTMOD: return RgM_deplin_FpM(x, p);
3662 49 : case code(t_POLMOD, t_INTMOD):
3663 49 : return RgM_deplin_FqM(x, pol, p);
3664 28 : default: return gen_0;
3665 : }
3666 : }
3667 : #undef code
3668 :
3669 : static GEN
3670 385 : RgM_deplin(GEN x)
3671 : {
3672 385 : GEN z = RgM_deplin_fast(x);
3673 385 : if (z!= gen_0) return z;
3674 28 : return RgM_deplin_i(x);
3675 : }
3676 :
3677 : GEN
3678 385 : deplin(GEN x)
3679 : {
3680 385 : switch(typ(x))
3681 : {
3682 385 : case t_MAT:
3683 : {
3684 385 : GEN z = RgM_deplin(x);
3685 385 : if (z) return z;
3686 147 : return cgetg(1, t_COL);
3687 : }
3688 0 : case t_VEC: return RgV_deplin(x);
3689 0 : default: pari_err_TYPE("deplin",x);
3690 : }
3691 : return NULL;/*LCOV_EXCL_LINE*/
3692 : }
3693 :
3694 : /*******************************************************************/
3695 : /* */
3696 : /* GAUSS REDUCTION OF MATRICES (m lines x n cols) */
3697 : /* (kernel, image, complementary image, rank) */
3698 : /* */
3699 : /*******************************************************************/
3700 : /* return the transform of x under a standard Gauss pivot.
3701 : * x0 is a reference point when guessing whether x[i,j] ~ 0
3702 : * (iff x[i,j] << x0[i,j])
3703 : * Set r = dim ker(x). d[k] contains the index of the first nonzero pivot
3704 : * in column k */
3705 : static GEN
3706 1271 : gauss_pivot_ker(GEN x, GEN x0, GEN *dd, long *rr)
3707 : {
3708 : GEN c, d, p, data;
3709 : pari_sp av;
3710 : long i, j, k, r, t, n, m;
3711 : pivot_fun pivot;
3712 :
3713 1271 : n=lg(x)-1; if (!n) { *dd=NULL; *rr=0; return cgetg(1,t_MAT); }
3714 1271 : m=nbrows(x); r=0;
3715 1271 : pivot = get_pivot_fun(x, x0, &data);
3716 1271 : x = RgM_shallowcopy(x);
3717 1271 : c = zero_zv(m);
3718 1271 : d = cgetg(n+1,t_VECSMALL);
3719 1271 : av=avma;
3720 7475 : for (k=1; k<=n; k++)
3721 : {
3722 6204 : j = pivot(x, data, k, c);
3723 6204 : if (j > m)
3724 : {
3725 1463 : r++; d[k]=0;
3726 6496 : for(j=1; j<k; j++)
3727 5033 : if (d[j]) gcoeff(x,d[j],k) = gclone(gcoeff(x,d[j],k));
3728 : }
3729 : else
3730 : { /* pivot for column k on row j */
3731 4741 : c[j]=k; d[k]=j; p = gdiv(gen_m1,gcoeff(x,j,k));
3732 4741 : gcoeff(x,j,k) = gen_m1;
3733 : /* x[j,] /= - x[j,k] */
3734 24169 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = gmul(p,gcoeff(x,j,i));
3735 42136 : for (t=1; t<=m; t++)
3736 37395 : if (t!=j)
3737 : { /* x[t,] -= 1 / x[j,k] x[j,] */
3738 32654 : p = gcoeff(x,t,k); gcoeff(x,t,k) = gen_0;
3739 32654 : if (gequal0(p)) continue;
3740 86920 : for (i=k+1; i<=n; i++)
3741 69463 : gcoeff(x,t,i) = gadd(gcoeff(x,t,i),gmul(p,gcoeff(x,j,i)));
3742 17457 : if (gc_needed(av,1)) gerepile_gauss_ker(x,k,t,av);
3743 : }
3744 : }
3745 : }
3746 1271 : *dd=d; *rr=r; return x;
3747 : }
3748 :
3749 : /* r = dim ker(x).
3750 : * Returns d:
3751 : * d[k] != 0 contains the index of a nonzero pivot in column k
3752 : * d[k] == 0 if column k is a linear combination of the (k-1) first ones */
3753 : GEN
3754 149525 : RgM_pivots(GEN x0, GEN data, long *rr, pivot_fun pivot)
3755 : {
3756 : GEN x, c, d, p;
3757 149525 : long i, j, k, r, t, m, n = lg(x0)-1;
3758 : pari_sp av;
3759 :
3760 149525 : if (RgM_is_ZM(x0)) return ZM_pivots(x0, rr);
3761 137056 : if (!n) { *rr = 0; return NULL; }
3762 :
3763 137056 : d = cgetg(n+1, t_VECSMALL);
3764 137056 : x = RgM_shallowcopy(x0);
3765 137056 : m = nbrows(x); r = 0;
3766 137056 : c = zero_zv(m);
3767 137127 : av = avma;
3768 3413706 : for (k=1; k<=n; k++)
3769 : {
3770 3276652 : j = pivot(x, data, k, c);
3771 3276663 : if (j > m) { r++; d[k] = 0; }
3772 : else
3773 : {
3774 270258 : c[j] = k; d[k] = j; p = gdiv(gen_m1, gcoeff(x,j,k));
3775 9270860 : for (i=k+1; i<=n; i++) gcoeff(x,j,i) = gmul(p,gcoeff(x,j,i));
3776 :
3777 1049935 : for (t=1; t<=m; t++)
3778 779761 : if (!c[t]) /* no pivot on that line yet */
3779 : {
3780 293202 : p = gcoeff(x,t,k); gcoeff(x,t,k) = gen_0;
3781 19379370 : for (i=k+1; i<=n; i++)
3782 19086169 : gcoeff(x,t,i) = gadd(gcoeff(x,t,i), gmul(p, gcoeff(x,j,i)));
3783 293201 : if (gc_needed(av,1)) gerepile_gauss(x,k,t,av,j,c);
3784 : }
3785 9543291 : for (i=k; i<=n; i++) gcoeff(x,j,i) = gen_0; /* dummy */
3786 : }
3787 : }
3788 137054 : *rr = r; return gc_const((pari_sp)d, d);
3789 : }
3790 :
3791 : static long
3792 4228749 : ZM_count_0_cols(GEN M)
3793 : {
3794 4228749 : long i, l = lg(M), n = 0;
3795 18240378 : for (i = 1; i < l; i++)
3796 14011644 : if (ZV_equal0(gel(M,i))) n++;
3797 4228734 : return n;
3798 : }
3799 :
3800 : static void indexrank_all(long m, long n, long r, GEN d, GEN *prow, GEN *pcol);
3801 : /* As RgM_pivots, integer entries. Set *rr = dim Ker M0 */
3802 : GEN
3803 4245864 : ZM_pivots(GEN M0, long *rr)
3804 : {
3805 4245864 : GEN d, dbest = NULL;
3806 : long m, mm, n, nn, i, imax, rmin, rbest, zc;
3807 4245864 : int beenthere = 0;
3808 4245864 : pari_sp av, av0 = avma;
3809 : forprime_t S;
3810 :
3811 4245864 : rbest = n = lg(M0)-1;
3812 4245864 : if (n == 0) { *rr = 0; return NULL; }
3813 4228749 : zc = ZM_count_0_cols(M0);
3814 4228735 : if (n == zc) { *rr = zc; return zero_zv(n); }
3815 :
3816 4228609 : m = nbrows(M0);
3817 4228606 : rmin = maxss(zc, n-m);
3818 4228603 : init_modular_small(&S);
3819 4228592 : if (n <= m) { nn = n; mm = m; } else { nn = m; mm = n; }
3820 4228592 : imax = (nn < 16)? 1: (nn < 64)? 2: 3; /* heuristic */
3821 :
3822 : for(;;)
3823 0 : {
3824 : GEN row, col, M, KM, IM, RHS, X, cX;
3825 : long rk;
3826 4256425 : for (av = avma, i = 0;; set_avma(av), i++)
3827 27838 : {
3828 4256425 : ulong p = u_forprime_next(&S);
3829 : long rp;
3830 4256428 : if (!p) pari_err_OVERFLOW("ZM_pivots [ran out of primes]");
3831 4256428 : d = Flm_pivots(ZM_to_Flm(M0, p), p, &rp, 1);
3832 4256455 : if (rp == rmin) { rbest = rp; goto END; } /* maximal rank, return */
3833 53345 : if (rp < rbest) { /* save best r so far */
3834 25530 : rbest = rp;
3835 25530 : guncloneNULL(dbest);
3836 25530 : dbest = gclone(d);
3837 25530 : if (beenthere) break;
3838 : }
3839 53345 : if (!beenthere && i >= imax) break;
3840 : }
3841 25507 : beenthere = 1;
3842 : /* Dubious case: there is (probably) a non trivial kernel */
3843 25507 : indexrank_all(m,n, rbest, dbest, &row, &col);
3844 25507 : M = rowpermute(vecpermute(M0, col), row);
3845 25507 : rk = n - rbest; /* (probable) dimension of image */
3846 25507 : if (n > m) M = shallowtrans(M);
3847 25507 : IM = vecslice(M,1,rk);
3848 25507 : KM = vecslice(M,rk+1, nn);
3849 25507 : M = rowslice(IM, 1,rk); /* square maximal rank */
3850 25507 : X = ZM_gauss(M, rowslice(KM, 1,rk));
3851 25507 : RHS = rowslice(KM,rk+1,mm);
3852 25507 : M = rowslice(IM,rk+1,mm);
3853 25507 : X = Q_remove_denom(X, &cX);
3854 25507 : if (cX) RHS = ZM_Z_mul(RHS, cX);
3855 25507 : if (ZM_equal(ZM_mul(M, X), RHS)) { d = vecsmall_copy(dbest); goto END; }
3856 0 : set_avma(av);
3857 : }
3858 4228617 : END:
3859 4228617 : *rr = rbest; guncloneNULL(dbest);
3860 4228616 : return gerepileuptoleaf(av0, d);
3861 : }
3862 :
3863 : /* set *pr = dim Ker x */
3864 : static GEN
3865 59071 : gauss_pivot(GEN x, long *pr) {
3866 : GEN data;
3867 59071 : pivot_fun pivot = get_pivot_fun(x, x, &data);
3868 59071 : return RgM_pivots(x, data, pr, pivot);
3869 : }
3870 :
3871 : /* compute ker(x), x0 is a reference point when guessing whether x[i,j] ~ 0
3872 : * (iff x[i,j] << x0[i,j]) */
3873 : static GEN
3874 1271 : ker_aux(GEN x, GEN x0)
3875 : {
3876 1271 : pari_sp av = avma;
3877 : GEN d,y;
3878 : long i,j,k,r,n;
3879 :
3880 1271 : x = gauss_pivot_ker(x,x0,&d,&r);
3881 1271 : if (!r) { set_avma(av); return cgetg(1,t_MAT); }
3882 1211 : n = lg(x)-1; y=cgetg(r+1,t_MAT);
3883 2674 : for (j=k=1; j<=r; j++,k++)
3884 : {
3885 1463 : GEN p = cgetg(n+1,t_COL);
3886 :
3887 5586 : gel(y,j) = p; while (d[k]) k++;
3888 6496 : for (i=1; i<k; i++)
3889 5033 : if (d[i])
3890 : {
3891 4641 : GEN p1=gcoeff(x,d[i],k);
3892 4641 : gel(p,i) = gcopy(p1); gunclone(p1);
3893 : }
3894 : else
3895 392 : gel(p,i) = gen_0;
3896 2541 : gel(p,k) = gen_1; for (i=k+1; i<=n; i++) gel(p,i) = gen_0;
3897 : }
3898 1211 : return gerepileupto(av,y);
3899 : }
3900 :
3901 : static GEN
3902 84 : RgM_ker_FpM(GEN x, GEN p)
3903 : {
3904 84 : pari_sp av = avma;
3905 : ulong pp;
3906 84 : x = RgM_Fp_init3(x, p, &pp);
3907 84 : switch(pp)
3908 : {
3909 35 : case 0: x = FpM_to_mod(FpM_ker_gen(x,p,0),p); break;
3910 7 : case 2: x = F2m_to_mod(F2m_ker_sp(x,0)); break;
3911 7 : case 3: x = F3m_to_mod(F3m_ker_sp(x,0)); break;
3912 35 : default:x = Flm_to_mod(Flm_ker_sp(x,pp,0), pp); break;
3913 : }
3914 84 : return gerepileupto(av, x);
3915 : }
3916 :
3917 : static GEN
3918 91 : RgM_ker_FqM(GEN x, GEN pol, GEN p)
3919 : {
3920 91 : pari_sp av = avma;
3921 91 : GEN b, T = RgX_to_FpX(pol, p);
3922 91 : if (signe(T) == 0) pari_err_OP("ker",x,pol);
3923 84 : b = FqM_ker(RgM_to_FqM(x, T, p), T, p);
3924 84 : return gerepileupto(av, FqM_to_mod(b, T, p));
3925 : }
3926 :
3927 : #define code(t1,t2) ((t1 << 6) | t2)
3928 : static GEN
3929 8603 : RgM_ker_fast(GEN x)
3930 : {
3931 : GEN p, pol;
3932 : long pa;
3933 8603 : long t = RgM_type(x, &p,&pol,&pa);
3934 8603 : switch(t)
3935 : {
3936 7469 : case t_INT: /* fall through */
3937 7469 : case t_FRAC: return QM_ker(x);
3938 77 : case t_FFELT: return FFM_ker(x, pol);
3939 84 : case t_INTMOD: return RgM_ker_FpM(x, p);
3940 91 : case code(t_POLMOD, t_INTMOD):
3941 91 : return RgM_ker_FqM(x, pol, p);
3942 882 : default: return NULL;
3943 : }
3944 : }
3945 : #undef code
3946 :
3947 : GEN
3948 8603 : ker(GEN x)
3949 : {
3950 8603 : GEN b = RgM_ker_fast(x);
3951 8596 : if (b) return b;
3952 882 : return ker_aux(x,x);
3953 : }
3954 :
3955 : GEN
3956 46221 : matker0(GEN x,long flag)
3957 : {
3958 46221 : if (typ(x)!=t_MAT) pari_err_TYPE("matker",x);
3959 46221 : if (!flag) return ker(x);
3960 45934 : RgM_check_ZM(x, "matker");
3961 45934 : return ZM_ker(x);
3962 : }
3963 :
3964 : static GEN
3965 63 : RgM_image_FpM(GEN x, GEN p)
3966 : {
3967 63 : pari_sp av = avma;
3968 : ulong pp;
3969 63 : x = RgM_Fp_init(x, p, &pp);
3970 63 : switch(pp)
3971 : {
3972 28 : case 0: x = FpM_to_mod(FpM_image(x,p),p); break;
3973 7 : case 2: x = F2m_to_mod(F2m_image(x)); break;
3974 28 : default:x = Flm_to_mod(Flm_image(x,pp), pp); break;
3975 : }
3976 63 : return gerepileupto(av, x);
3977 : }
3978 :
3979 : static GEN
3980 35 : RgM_image_FqM(GEN x, GEN pol, GEN p)
3981 : {
3982 35 : pari_sp av = avma;
3983 35 : GEN b, T = RgX_to_FpX(pol, p);
3984 35 : if (signe(T) == 0) pari_err_OP("image",x,pol);
3985 28 : b = FqM_image(RgM_to_FqM(x, T, p), T, p);
3986 28 : return gerepileupto(av, FqM_to_mod(b, T, p));
3987 : }
3988 :
3989 : GEN
3990 5971 : QM_image_shallow(GEN A)
3991 : {
3992 5971 : A = vec_Q_primpart(A);
3993 5971 : return vecpermute(A, ZM_indeximage(A));
3994 : }
3995 : GEN
3996 5201 : QM_image(GEN A)
3997 : {
3998 5201 : pari_sp av = avma;
3999 5201 : return gerepilecopy(av, QM_image_shallow(A));
4000 : }
4001 :
4002 : #define code(t1,t2) ((t1 << 6) | t2)
4003 : static GEN
4004 5362 : RgM_image_fast(GEN x)
4005 : {
4006 : GEN p, pol;
4007 : long pa;
4008 5362 : long t = RgM_type(x, &p,&pol,&pa);
4009 5362 : switch(t)
4010 : {
4011 5201 : case t_INT: /* fall through */
4012 5201 : case t_FRAC: return QM_image(x);
4013 49 : case t_FFELT: return FFM_image(x, pol);
4014 63 : case t_INTMOD: return RgM_image_FpM(x, p);
4015 35 : case code(t_POLMOD, t_INTMOD):
4016 35 : return RgM_image_FqM(x, pol, p);
4017 14 : default: return NULL;
4018 : }
4019 : }
4020 : #undef code
4021 :
4022 : GEN
4023 5362 : image(GEN x)
4024 : {
4025 : GEN d, M;
4026 : long r;
4027 :
4028 5362 : if (typ(x)!=t_MAT) pari_err_TYPE("matimage",x);
4029 5362 : M = RgM_image_fast(x);
4030 5355 : if (M) return M;
4031 14 : d = gauss_pivot(x,&r); /* d left on stack for efficiency */
4032 14 : return image_from_pivot(x,d,r);
4033 : }
4034 :
4035 : static GEN
4036 84 : imagecompl_aux(GEN x, GEN(*PIVOT)(GEN,long*))
4037 : {
4038 84 : pari_sp av = avma;
4039 : GEN d,y;
4040 : long j,i,r;
4041 :
4042 84 : if (typ(x)!=t_MAT) pari_err_TYPE("imagecompl",x);
4043 84 : (void)new_chunk(lg(x) * 4 + 1); /* HACK */
4044 84 : d = PIVOT(x,&r); /* if (!d) then r = 0 */
4045 84 : set_avma(av); y = cgetg(r+1,t_VECSMALL);
4046 126 : for (i=j=1; j<=r; i++)
4047 42 : if (!d[i]) y[j++] = i;
4048 84 : return y;
4049 : }
4050 : GEN
4051 84 : imagecompl(GEN x) { return imagecompl_aux(x, &gauss_pivot); }
4052 : GEN
4053 0 : ZM_imagecompl(GEN x) { return imagecompl_aux(x, &ZM_pivots); }
4054 :
4055 : static GEN
4056 28 : RgM_RgC_invimage_FpC(GEN A, GEN y, GEN p)
4057 : {
4058 28 : pari_sp av = avma;
4059 : ulong pp;
4060 : GEN x;
4061 28 : A = RgM_Fp_init(A,p,&pp);
4062 28 : switch(pp)
4063 : {
4064 7 : case 0:
4065 7 : y = RgC_to_FpC(y,p);
4066 7 : x = FpM_FpC_invimage(A, y, p);
4067 7 : return x ? gerepileupto(av, FpC_to_mod(x,p)): NULL;
4068 7 : case 2:
4069 7 : y = RgV_to_F2v(y);
4070 7 : x = F2m_F2c_invimage(A, y);
4071 7 : return x ? gerepileupto(av, F2c_to_mod(x)): NULL;
4072 14 : default:
4073 14 : y = RgV_to_Flv(y,pp);
4074 14 : x = Flm_Flc_invimage(A, y, pp);
4075 14 : return x ? gerepileupto(av, Flc_to_mod(x,pp)): NULL;
4076 : }
4077 : }
4078 :
4079 : static GEN
4080 2170 : RgM_RgC_invimage_fast(GEN x, GEN y)
4081 : {
4082 : GEN p, pol;
4083 : long pa;
4084 2170 : long t = RgM_RgC_type(x, y, &p,&pol,&pa);
4085 2170 : switch(t)
4086 : {
4087 28 : case t_INTMOD: return RgM_RgC_invimage_FpC(x, y, p);
4088 63 : case t_FFELT: return FFM_FFC_invimage(x, y, pol);
4089 2079 : default: return gen_0;
4090 : }
4091 : }
4092 :
4093 : GEN
4094 2275 : RgM_RgC_invimage(GEN A, GEN y)
4095 : {
4096 2275 : pari_sp av = avma;
4097 2275 : long i, l = lg(A);
4098 : GEN M, x, t;
4099 2275 : if (l==1) return NULL;
4100 2170 : if (lg(y) != lgcols(A)) pari_err_DIM("inverseimage");
4101 2170 : M = RgM_RgC_invimage_fast(A, y);
4102 2170 : if (!M) return gc_NULL(av);
4103 2149 : if (M != gen_0) return M;
4104 2079 : M = ker(shallowconcat(A, y));
4105 2079 : i = lg(M)-1;
4106 2079 : if (!i) return gc_NULL(av);
4107 :
4108 1820 : x = gel(M,i); t = gel(x,l);
4109 1820 : if (gequal0(t)) return gc_NULL(av);
4110 :
4111 1785 : t = gneg_i(t); setlg(x,l);
4112 1785 : return gerepileupto(av, RgC_Rg_div(x, t));
4113 : }
4114 :
4115 : /* Return X such that m X = v (t_COL or t_MAT), resp. an empty t_COL / t_MAT
4116 : * if no solution exist */
4117 : GEN
4118 2436 : inverseimage(GEN m, GEN v)
4119 : {
4120 : GEN y;
4121 2436 : if (typ(m)!=t_MAT) pari_err_TYPE("inverseimage",m);
4122 2436 : switch(typ(v))
4123 : {
4124 2198 : case t_COL:
4125 2198 : y = RgM_RgC_invimage(m,v);
4126 2198 : return y? y: cgetg(1,t_COL);
4127 238 : case t_MAT:
4128 238 : y = RgM_invimage(m, v);
4129 238 : return y? y: cgetg(1,t_MAT);
4130 : }
4131 0 : pari_err_TYPE("inverseimage",v);
4132 : return NULL;/*LCOV_EXCL_LINE*/
4133 : }
4134 :
4135 : static GEN
4136 84 : RgM_invimage_FpM(GEN A, GEN B, GEN p)
4137 : {
4138 84 : pari_sp av = avma;
4139 : ulong pp;
4140 : GEN x;
4141 84 : A = RgM_Fp_init(A,p,&pp);
4142 84 : switch(pp)
4143 : {
4144 35 : case 0:
4145 35 : B = RgM_to_FpM(B,p);
4146 35 : x = FpM_invimage_gen(A, B, p);
4147 35 : return x ? gerepileupto(av, FpM_to_mod(x, p)): x;
4148 7 : case 2:
4149 7 : B = RgM_to_F2m(B);
4150 7 : x = F2m_invimage_i(A, B);
4151 7 : return x ? gerepileupto(av, F2m_to_mod(x)): x;
4152 42 : default:
4153 42 : B = RgM_to_Flm(B,pp);
4154 42 : x = Flm_invimage_i(A, B, pp);
4155 42 : return x ? gerepileupto(av, Flm_to_mod(x, pp)): x;
4156 : }
4157 : }
4158 :
4159 : static GEN
4160 364 : RgM_invimage_fast(GEN x, GEN y)
4161 : {
4162 : GEN p, pol;
4163 : long pa;
4164 364 : long t = RgM_type2(x, y, &p,&pol,&pa);
4165 364 : switch(t)
4166 : {
4167 84 : case t_INTMOD: return RgM_invimage_FpM(x, y, p);
4168 105 : case t_FFELT: return FFM_invimage(x, y, pol);
4169 175 : default: return gen_0;
4170 : }
4171 : }
4172 :
4173 : /* find Z such that A Z = B. Return NULL if no solution */
4174 : GEN
4175 364 : RgM_invimage(GEN A, GEN B)
4176 : {
4177 364 : pari_sp av = avma;
4178 : GEN d, x, X, Y;
4179 364 : long i, j, nY, nA = lg(A)-1, nB = lg(B)-1;
4180 364 : X = RgM_invimage_fast(A, B);
4181 364 : if (!X) return gc_NULL(av);
4182 252 : if (X != gen_0) return X;
4183 175 : x = ker(shallowconcat(RgM_neg(A), B));
4184 : /* AX = BY, Y in strict upper echelon form with pivots = 1.
4185 : * We must find T such that Y T = Id_nB then X T = Z. This exists iff
4186 : * Y has at least nB columns and full rank */
4187 175 : nY = lg(x)-1;
4188 175 : if (nY < nB) return gc_NULL(av);
4189 161 : Y = rowslice(x, nA+1, nA+nB); /* nB rows */
4190 161 : d = cgetg(nB+1, t_VECSMALL);
4191 721 : for (i = nB, j = nY; i >= 1; i--, j--)
4192 : {
4193 805 : for (; j>=1; j--)
4194 756 : if (!gequal0(gcoeff(Y,i,j))) { d[i] = j; break; }
4195 609 : if (!j) return gc_NULL(av);
4196 : }
4197 : /* reduce to the case Y square, upper triangular with 1s on diagonal */
4198 112 : Y = vecpermute(Y, d);
4199 112 : x = vecpermute(x, d);
4200 112 : X = rowslice(x, 1, nA);
4201 112 : return gerepileupto(av, RgM_mul(X, RgM_inv_upper(Y)));
4202 : }
4203 :
4204 : static GEN
4205 70 : RgM_suppl_FpM(GEN x, GEN p)
4206 : {
4207 70 : pari_sp av = avma;
4208 : ulong pp;
4209 70 : x = RgM_Fp_init(x, p, &pp);
4210 70 : switch(pp)
4211 : {
4212 21 : case 0: x = FpM_to_mod(FpM_suppl(x,p), p); break;
4213 14 : case 2: x = F2m_to_mod(F2m_suppl(x)); break;
4214 35 : default:x = Flm_to_mod(Flm_suppl(x,pp), pp); break;
4215 : }
4216 70 : return gerepileupto(av, x);
4217 : }
4218 :
4219 : static GEN
4220 175 : RgM_suppl_fast(GEN x)
4221 : {
4222 : GEN p, pol;
4223 : long pa;
4224 175 : long t = RgM_type(x,&p,&pol,&pa);
4225 175 : switch(t)
4226 : {
4227 70 : case t_INTMOD: return RgM_suppl_FpM(x, p);
4228 35 : case t_FFELT: return FFM_suppl(x, pol);
4229 70 : default: return NULL;
4230 : }
4231 : }
4232 :
4233 : /* x is an n x k matrix, rank(x) = k <= n. Return an invertible n x n matrix
4234 : * whose first k columns are given by x. If rank(x) < k, undefined result. */
4235 : GEN
4236 175 : suppl(GEN x)
4237 : {
4238 175 : pari_sp av = avma;
4239 : GEN d, M;
4240 : long r;
4241 175 : if (typ(x)!=t_MAT) pari_err_TYPE("suppl",x);
4242 175 : M = RgM_suppl_fast(x);
4243 175 : if (M) return M;
4244 70 : init_suppl(x);
4245 70 : d = gauss_pivot(x,&r);
4246 70 : set_avma(av); return get_suppl(x,d,nbrows(x),r,&col_ei);
4247 : }
4248 :
4249 : GEN
4250 7 : image2(GEN x)
4251 : {
4252 7 : pari_sp av = avma;
4253 : long k, n, i;
4254 : GEN A, B;
4255 :
4256 7 : if (typ(x)!=t_MAT) pari_err_TYPE("image2",x);
4257 7 : if (lg(x) == 1) return cgetg(1,t_MAT);
4258 7 : A = ker(x); k = lg(A)-1;
4259 7 : if (!k) { set_avma(av); return gcopy(x); }
4260 7 : A = suppl(A); n = lg(A)-1;
4261 7 : B = cgetg(n-k+1, t_MAT);
4262 21 : for (i = k+1; i <= n; i++) gel(B,i-k) = RgM_RgC_mul(x, gel(A,i));
4263 7 : return gerepileupto(av, B);
4264 : }
4265 :
4266 : GEN
4267 210 : matimage0(GEN x,long flag)
4268 : {
4269 210 : switch(flag)
4270 : {
4271 203 : case 0: return image(x);
4272 7 : case 1: return image2(x);
4273 0 : default: pari_err_FLAG("matimage");
4274 : }
4275 : return NULL; /* LCOV_EXCL_LINE */
4276 : }
4277 :
4278 : static long
4279 126 : RgM_rank_FpM(GEN x, GEN p)
4280 : {
4281 126 : pari_sp av = avma;
4282 : ulong pp;
4283 : long r;
4284 126 : x = RgM_Fp_init(x,p,&pp);
4285 126 : switch(pp)
4286 : {
4287 28 : case 0: r = FpM_rank(x,p); break;
4288 63 : case 2: r = F2m_rank(x); break;
4289 35 : default:r = Flm_rank(x,pp); break;
4290 : }
4291 126 : return gc_long(av, r);
4292 : }
4293 :
4294 : static long
4295 49 : RgM_rank_FqM(GEN x, GEN pol, GEN p)
4296 : {
4297 49 : pari_sp av = avma;
4298 : long r;
4299 49 : GEN T = RgX_to_FpX(pol, p);
4300 49 : if (signe(T) == 0) pari_err_OP("rank",x,pol);
4301 42 : r = FqM_rank(RgM_to_FqM(x, T, p), T, p);
4302 42 : return gc_long(av,r);
4303 : }
4304 :
4305 : #define code(t1,t2) ((t1 << 6) | t2)
4306 : static long
4307 315 : RgM_rank_fast(GEN x)
4308 : {
4309 : GEN p, pol;
4310 : long pa;
4311 315 : long t = RgM_type(x,&p,&pol,&pa);
4312 315 : switch(t)
4313 : {
4314 42 : case t_INT: return ZM_rank(x);
4315 21 : case t_FRAC: return QM_rank(x);
4316 126 : case t_INTMOD: return RgM_rank_FpM(x, p);
4317 70 : case t_FFELT: return FFM_rank(x, pol);
4318 49 : case code(t_POLMOD, t_INTMOD):
4319 49 : return RgM_rank_FqM(x, pol, p);
4320 7 : default: return -1;
4321 : }
4322 : }
4323 : #undef code
4324 :
4325 : long
4326 315 : rank(GEN x)
4327 : {
4328 315 : pari_sp av = avma;
4329 : long r;
4330 :
4331 315 : if (typ(x)!=t_MAT) pari_err_TYPE("rank",x);
4332 315 : r = RgM_rank_fast(x);
4333 308 : if (r >= 0) return r;
4334 7 : (void)gauss_pivot(x, &r);
4335 7 : return gc_long(av, lg(x)-1 - r);
4336 : }
4337 :
4338 : /* d a t_VECSMALL of integers in 1..n. Return the vector of the d[i]
4339 : * followed by the missing indices */
4340 : static GEN
4341 51014 : perm_complete(GEN d, long n)
4342 : {
4343 51014 : GEN y = cgetg(n+1, t_VECSMALL);
4344 51014 : long i, j = 1, k = n, l = lg(d);
4345 51014 : pari_sp av = avma;
4346 51014 : char *T = stack_calloc(n+1);
4347 242262 : for (i = 1; i < l; i++) T[d[i]] = 1;
4348 518213 : for (i = 1; i <= n; i++)
4349 467199 : if (T[i]) y[j++] = i; else y[k--] = i;
4350 51014 : return gc_const(av, y);
4351 : }
4352 :
4353 : /* n = dim x, r = dim Ker(x), d from gauss_pivot */
4354 : static GEN
4355 5971 : indeximage0(long n, long r, GEN d)
4356 : {
4357 : long i, j;
4358 : GEN v;
4359 :
4360 5971 : r = n - r; /* now r = dim Im(x) */
4361 5971 : v = cgetg(r+1,t_VECSMALL);
4362 32837 : if (d) for (i=j=1; j<=n; j++)
4363 26866 : if (d[j]) v[i++] = j;
4364 5971 : return v;
4365 : }
4366 : /* x an m x n t_MAT, n > 0, r = dim Ker(x), d from gauss_pivot */
4367 : static void
4368 25507 : indexrank_all(long m, long n, long r, GEN d, GEN *prow, GEN *pcol)
4369 : {
4370 25507 : GEN IR = indexrank0(n, r, d);
4371 25507 : *prow = perm_complete(gel(IR,1), m);
4372 25507 : *pcol = perm_complete(gel(IR,2), n);
4373 25507 : }
4374 :
4375 : static GEN
4376 28 : RgM_indexrank_FpM(GEN x, GEN p)
4377 : {
4378 28 : pari_sp av = avma;
4379 : ulong pp;
4380 : GEN r;
4381 28 : x = RgM_Fp_init(x,p,&pp);
4382 28 : switch(pp)
4383 : {
4384 7 : case 0: r = FpM_indexrank(x,p); break;
4385 7 : case 2: r = F2m_indexrank(x); break;
4386 14 : default: r = Flm_indexrank(x,pp); break;
4387 : }
4388 28 : return gerepileupto(av, r);
4389 : }
4390 :
4391 : static GEN
4392 0 : RgM_indexrank_FqM(GEN x, GEN pol, GEN p)
4393 : {
4394 0 : pari_sp av = avma;
4395 0 : GEN r, T = RgX_to_FpX(pol, p);
4396 0 : if (signe(T) == 0) pari_err_OP("indexrank",x,pol);
4397 0 : r = FqM_indexrank(RgM_to_FqM(x, T, p), T, p);
4398 0 : return gerepileupto(av, r);
4399 : }
4400 :
4401 : #define code(t1,t2) ((t1 << 6) | t2)
4402 : static GEN
4403 60485 : RgM_indexrank_fast(GEN x)
4404 : {
4405 : GEN p, pol;
4406 : long pa;
4407 60485 : long t = RgM_type(x,&p,&pol,&pa);
4408 60485 : switch(t)
4409 : {
4410 392 : case t_INT: return ZM_indexrank(x);
4411 1148 : case t_FRAC: return QM_indexrank(x);
4412 28 : case t_INTMOD: return RgM_indexrank_FpM(x, p);
4413 21 : case t_FFELT: return FFM_indexrank(x, pol);
4414 0 : case code(t_POLMOD, t_INTMOD):
4415 0 : return RgM_indexrank_FqM(x, pol, p);
4416 58896 : default: return NULL;
4417 : }
4418 : }
4419 : #undef code
4420 :
4421 : GEN
4422 60485 : indexrank(GEN x)
4423 : {
4424 : pari_sp av;
4425 : long r;
4426 : GEN d;
4427 60485 : if (typ(x)!=t_MAT) pari_err_TYPE("indexrank",x);
4428 60485 : d = RgM_indexrank_fast(x);
4429 60485 : if (d) return d;
4430 58896 : av = avma;
4431 58896 : init_indexrank(x);
4432 58896 : d = gauss_pivot(x, &r);
4433 58894 : set_avma(av); return indexrank0(lg(x)-1, r, d);
4434 : }
4435 :
4436 : GEN
4437 5971 : ZM_indeximage(GEN x) {
4438 5971 : pari_sp av = avma;
4439 : long r;
4440 : GEN d;
4441 5971 : init_indexrank(x);
4442 5971 : d = ZM_pivots(x,&r);
4443 5971 : set_avma(av); return indeximage0(lg(x)-1, r, d);
4444 : }
4445 : long
4446 2241732 : ZM_rank(GEN x) {
4447 2241732 : pari_sp av = avma;
4448 : long r;
4449 2241732 : (void)ZM_pivots(x,&r);
4450 2241728 : return gc_long(av, lg(x)-1-r);
4451 : }
4452 : GEN
4453 1762354 : ZM_indexrank(GEN x) {
4454 1762354 : pari_sp av = avma;
4455 : long r;
4456 : GEN d;
4457 1762354 : init_indexrank(x);
4458 1762353 : d = ZM_pivots(x,&r);
4459 1762362 : set_avma(av); return indexrank0(lg(x)-1, r, d);
4460 : }
4461 :
4462 : long
4463 21 : QM_rank(GEN x)
4464 : {
4465 21 : pari_sp av = avma;
4466 21 : long r = ZM_rank(Q_primpart(x));
4467 21 : set_avma(av);
4468 21 : return r;
4469 : }
4470 :
4471 : GEN
4472 1148 : QM_indexrank(GEN x)
4473 : {
4474 1148 : pari_sp av = avma;
4475 1148 : GEN r = ZM_indexrank(Q_primpart(x));
4476 1148 : return gerepileupto(av, r);
4477 : }
4478 :
4479 : /*******************************************************************/
4480 : /* */
4481 : /* ZabM */
4482 : /* */
4483 : /*******************************************************************/
4484 :
4485 : static GEN
4486 1276 : FpXM_ratlift(GEN a, GEN q)
4487 : {
4488 : GEN B, y;
4489 1276 : long i, j, l = lg(a), n;
4490 1276 : B = sqrti(shifti(q,-1));
4491 1276 : y = cgetg(l, t_MAT);
4492 1276 : if (l==1) return y;
4493 1276 : n = lgcols(a);
4494 3059 : for (i=1; i<l; i++)
4495 : {
4496 2404 : GEN yi = cgetg(n, t_COL);
4497 32311 : for (j=1; j<n; j++)
4498 : {
4499 30528 : GEN v = FpX_ratlift(gmael(a,i,j), q, B, B, NULL);
4500 30528 : if (!v) return NULL;
4501 29907 : gel(yi, j) = RgX_renormalize(v);
4502 : }
4503 1783 : gel(y,i) = yi;
4504 : }
4505 655 : return y;
4506 : }
4507 :
4508 : static GEN
4509 4477 : FlmV_recover_pre(GEN a, GEN M, ulong p, ulong pi, long sv)
4510 : {
4511 4477 : GEN a1 = gel(a,1);
4512 4477 : long i, j, k, l = lg(a1), n, lM = lg(M);
4513 4477 : GEN v = cgetg(lM, t_VECSMALL);
4514 4477 : GEN y = cgetg(l, t_MAT);
4515 4477 : if (l==1) return y;
4516 4477 : n = lgcols(a1);
4517 22466 : for (i=1; i<l; i++)
4518 : {
4519 17988 : GEN yi = cgetg(n, t_COL);
4520 347264 : for (j=1; j<n; j++)
4521 : {
4522 4673726 : for (k=1; k<lM; k++) uel(v,k) = umael(gel(a,k),i,j);
4523 329275 : gel(yi, j) = Flm_Flc_mul_pre_Flx(M, v, p, pi, sv);
4524 : }
4525 17989 : gel(y,i) = yi;
4526 : }
4527 4478 : return y;
4528 : }
4529 :
4530 : static GEN
4531 0 : FlkM_inv(GEN M, GEN P, ulong p)
4532 : {
4533 0 : ulong PI = get_Fl_red(p), pi = SMALL_ULONG(p)? 0: PI;
4534 0 : GEN R = Flx_roots_pre(P, p, pi);
4535 0 : long l = lg(R), i;
4536 0 : GEN W = Flv_invVandermonde(R, 1UL, p);
4537 0 : GEN V = cgetg(l, t_VEC);
4538 0 : for(i=1; i<l; i++)
4539 : {
4540 0 : GEN pows = Fl_powers_pre(uel(R,i), degpol(P), p, PI);
4541 0 : GEN H = Flm_inv_sp(FlxM_eval_powers_pre(M, pows, p, pi), NULL, p);
4542 0 : if (!H) return NULL;
4543 0 : gel(V, i) = H;
4544 : }
4545 0 : return FlmV_recover_pre(V, W, p, pi, P[1]);
4546 : }
4547 :
4548 : static GEN
4549 3201 : FlkM_adjoint(GEN M, GEN P, ulong p)
4550 : {
4551 3201 : ulong PI = get_Fl_red(p), pi = SMALL_ULONG(p)? 0: PI;
4552 3201 : GEN R = Flx_roots_pre(P, p, pi);
4553 3201 : long l = lg(R), i;
4554 3201 : GEN W = Flv_invVandermonde(R, 1UL, p);
4555 3201 : GEN V = cgetg(l, t_VEC);
4556 15521 : for(i=1; i<l; i++)
4557 : {
4558 12320 : GEN pows = Fl_powers_pre(uel(R,i), degpol(P), p, PI);
4559 12320 : gel(V, i) = Flm_adjoint(FlxM_eval_powers_pre(M, pows, p, pi), p);
4560 : }
4561 3201 : return FlmV_recover_pre(V, W, p, pi, P[1]);
4562 : }
4563 :
4564 : static GEN
4565 1978 : ZabM_inv_slice(GEN A, GEN Q, GEN P, GEN *mod)
4566 : {
4567 1978 : pari_sp av = avma;
4568 1978 : long i, n = lg(P)-1, w = varn(Q);
4569 : GEN H, T;
4570 1978 : if (n == 1)
4571 : {
4572 1548 : ulong p = uel(P,1);
4573 1548 : GEN Qp = ZX_to_Flx(Q, p);
4574 1548 : GEN Ap = ZXM_to_FlxM(A, p, get_Flx_var(Qp));
4575 1548 : GEN Hp = FlkM_adjoint(Ap, Qp, p);
4576 1548 : Hp = gerepileupto(av, FlxM_to_ZXM(Hp));
4577 1548 : *mod = utoipos(p); return Hp;
4578 : }
4579 430 : T = ZV_producttree(P);
4580 430 : A = ZXM_nv_mod_tree(A, P, T, w);
4581 430 : Q = ZX_nv_mod_tree(Q, P, T);
4582 430 : H = cgetg(n+1, t_VEC);
4583 2083 : for(i=1; i <= n; i++)
4584 : {
4585 1653 : ulong p = P[i];
4586 1653 : GEN a = gel(A,i), q = gel(Q, i);
4587 1653 : gel(H,i) = FlkM_adjoint(a, q, p);
4588 : }
4589 430 : H = nxMV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P,T));
4590 430 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
4591 : }
4592 :
4593 : GEN
4594 1978 : ZabM_inv_worker(GEN P, GEN A, GEN Q)
4595 : {
4596 1978 : GEN V = cgetg(3, t_VEC);
4597 1978 : gel(V,1) = ZabM_inv_slice(A, Q, P, &gel(V,2));
4598 1978 : return V;
4599 : }
4600 :
4601 : static GEN
4602 5467 : vecnorml1(GEN a)
4603 : {
4604 : long i, l;
4605 5467 : GEN g = cgetg_copy(a, &l);
4606 60214 : for (i=1; i<l; i++)
4607 54747 : gel(g, i) = gnorml1_fake(gel(a,i));
4608 5467 : return g;
4609 : }
4610 :
4611 : static GEN
4612 1820 : ZabM_true_Hadamard(GEN a)
4613 : {
4614 1820 : pari_sp av = avma;
4615 1820 : long n = lg(a)-1, i;
4616 : GEN B;
4617 1820 : if (n == 0) return gen_1;
4618 1820 : if (n == 1) return gnorml1_fake(gcoeff(a,1,1));
4619 1176 : B = gen_1;
4620 6643 : for (i = 1; i <= n; i++)
4621 5467 : B = gmul(B, gnorml2(RgC_gtofp(vecnorml1(gel(a,i)),DEFAULTPREC)));
4622 1176 : return gerepileuptoint(av, ceil_safe(sqrtr_abs(B)));
4623 : }
4624 :
4625 : GEN
4626 1820 : ZabM_inv(GEN A, GEN Q, long n, GEN *pt_den)
4627 : {
4628 1820 : pari_sp av = avma;
4629 : forprime_t S;
4630 : GEN bnd, H, D, d, mod, worker;
4631 1820 : if (lg(A) == 1)
4632 : {
4633 0 : if (pt_den) *pt_den = gen_1;
4634 0 : return cgetg(1, t_MAT);
4635 : }
4636 1820 : bnd = ZabM_true_Hadamard(A);
4637 1820 : worker = snm_closure(is_entry("_ZabM_inv_worker"), mkvec2(A, Q));
4638 1820 : u_forprime_arith_init(&S, HIGHBIT+1, ULONG_MAX, 1, n);
4639 1820 : H = gen_crt("ZabM_inv", worker, &S, NULL, expi(bnd), 0, &mod,
4640 : nxMV_chinese_center, FpXM_center);
4641 1820 : D = RgMrow_RgC_mul(H, gel(A,1), 1);
4642 1820 : D = ZX_rem(D, Q);
4643 1820 : d = Z_content(mkvec2(H, D));
4644 1820 : if (d)
4645 : {
4646 511 : D = ZX_Z_divexact(D, d);
4647 511 : H = Q_div_to_int(H, d);
4648 : }
4649 1820 : if (!pt_den) return gerepileupto(av, H);
4650 1820 : *pt_den = D; return gc_all(av, 2, &H, pt_den);
4651 : }
4652 :
4653 : GEN
4654 0 : ZabM_inv_ratlift(GEN M, GEN P, long n, GEN *pden)
4655 : {
4656 0 : pari_sp av2, av = avma;
4657 : GEN q, H;
4658 0 : ulong m = LONG_MAX>>1;
4659 0 : ulong p= 1 + m - (m % n);
4660 0 : long lM = lg(M);
4661 0 : if (lM == 1) { *pden = gen_1; return cgetg(1,t_MAT); }
4662 :
4663 0 : av2 = avma;
4664 0 : H = NULL;
4665 : for(;;)
4666 0 : {
4667 : GEN Hp, Pp, Mp, Hr;
4668 0 : do p += n; while(!uisprime(p));
4669 0 : Pp = ZX_to_Flx(P, p);
4670 0 : Mp = ZXM_to_FlxM(M, p, get_Flx_var(Pp));
4671 0 : Hp = FlkM_inv(Mp, Pp, p);
4672 0 : if (!Hp) continue;
4673 0 : if (!H)
4674 : {
4675 0 : H = ZXM_init_CRT(Hp, degpol(P)-1, p);
4676 0 : q = utoipos(p);
4677 : }
4678 : else
4679 0 : ZXM_incremental_CRT(&H, Hp, &q, p);
4680 0 : Hr = FpXM_ratlift(H, q);
4681 0 : if (DEBUGLEVEL>5) err_printf("ZabM_inv mod %ld (ratlift=%ld)\n", p,!!Hr);
4682 0 : if (Hr) {/* DONE ? */
4683 0 : GEN Hl = Q_remove_denom(Hr, pden);
4684 0 : GEN MH = ZXQM_mul(Hl, M, P);
4685 0 : if (*pden)
4686 0 : { if (RgM_isscalar(MH, *pden)) { H = Hl; break; }}
4687 : else
4688 0 : { if (RgM_isidentity(MH)) { H = Hl; *pden = gen_1; break; } }
4689 : }
4690 :
4691 0 : if (gc_needed(av,2))
4692 : {
4693 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZabM_inv");
4694 0 : gerepileall(av2, 2, &H, &q);
4695 : }
4696 : }
4697 0 : return gc_all(av, 2, &H, pden);
4698 : }
4699 :
4700 : static GEN
4701 1276 : FlkM_ker(GEN M, GEN P, ulong p)
4702 : {
4703 1276 : ulong PI = get_Fl_red(p), pi = SMALL_ULONG(p)? 0: PI;
4704 1276 : GEN R = Flx_roots_pre(P, p, pi);
4705 1276 : long l = lg(R), i, dP = degpol(P), r;
4706 : GEN M1, K, D;
4707 1276 : GEN W = Flv_invVandermonde(R, 1UL, p);
4708 1276 : GEN V = cgetg(l, t_VEC);
4709 1276 : M1 = FlxM_eval_powers_pre(M, Fl_powers_pre(uel(R,1), dP, p, PI), p, pi);
4710 1276 : K = Flm_ker_sp(M1, p, 2);
4711 1276 : r = lg(gel(K,1)); D = gel(K,2);
4712 1276 : gel(V, 1) = gel(K,1);
4713 2652 : for(i=2; i<l; i++)
4714 : {
4715 1376 : GEN Mi = FlxM_eval_powers_pre(M, Fl_powers_pre(uel(R,i), dP, p, PI), p, pi);
4716 1376 : GEN K = Flm_ker_sp(Mi, p, 2);
4717 1376 : if (lg(gel(K,1)) != r || !zv_equal(D, gel(K,2))) return NULL;
4718 1376 : gel(V, i) = gel(K,1);
4719 : }
4720 1276 : return mkvec2(FlmV_recover_pre(V, W, p, pi, P[1]), D);
4721 : }
4722 :
4723 : static int
4724 655 : ZabM_ker_check(GEN M, GEN H, ulong p, GEN P, long n)
4725 : {
4726 : GEN pow;
4727 655 : long j, l = lg(H);
4728 : ulong pi, r;
4729 3899 : do p += n; while(!uisprime(p));
4730 655 : pi = get_Fl_red(p);
4731 655 : P = ZX_to_Flx(P, p);
4732 655 : r = Flx_oneroot_pre(P, p, pi);
4733 655 : pow = Fl_powers_pre(r, degpol(P),p, (p & HIGHMASK)? pi: 0);
4734 655 : M = ZXM_to_FlxM(M, p, P[1]); M = FlxM_eval_powers_pre(M, pow, p, pi);
4735 655 : H = ZXM_to_FlxM(H, p, P[1]); H = FlxM_eval_powers_pre(H, pow, p, pi);
4736 2178 : for (j = 1; j < l; j++)
4737 1555 : if (!zv_equal0(Flm_Flc_mul_pre(M, gel(H,j), p, pi))) return 0;
4738 623 : return 1;
4739 : }
4740 :
4741 : GEN
4742 623 : ZabM_ker(GEN M, GEN P, long n)
4743 : {
4744 623 : pari_sp av = avma;
4745 : pari_timer ti;
4746 623 : GEN q, H = NULL, D = NULL;
4747 623 : ulong m = LONG_MAX>>1;
4748 623 : ulong p = 1 + m - (m % n);
4749 :
4750 623 : if (DEBUGLEVEL>5) timer_start(&ti);
4751 : for(;;)
4752 653 : {
4753 : GEN Kp, Hp, Dp, Pp, Mp, Hr;
4754 22341 : do p += n; while(!uisprime(p));
4755 1276 : Pp = ZX_to_Flx(P, p);
4756 1276 : Mp = ZXM_to_FlxM(M, p, get_Flx_var(Pp));
4757 1276 : Kp = FlkM_ker(Mp, Pp, p);
4758 1276 : if (!Kp) continue;
4759 1276 : Hp = gel(Kp,1); Dp = gel(Kp,2);
4760 1276 : if (H && (lg(Hp)>lg(H) || (lg(Hp)==lg(H) && vecsmall_lexcmp(Dp,D)>0))) continue;
4761 1276 : if (!H || (lg(Hp)<lg(H) || vecsmall_lexcmp(Dp,D)<0))
4762 : {
4763 623 : H = ZXM_init_CRT(Hp, degpol(P)-1, p); D = Dp;
4764 623 : q = utoipos(p);
4765 : }
4766 : else
4767 653 : ZXM_incremental_CRT(&H, Hp, &q, p);
4768 1276 : Hr = FpXM_ratlift(H, q);
4769 1276 : if (DEBUGLEVEL>5) timer_printf(&ti,"ZabM_ker mod %ld (ratlift=%ld)", p,!!Hr);
4770 1276 : if (Hr) {/* DONE ? */
4771 655 : GEN Hl = vec_Q_primpart(Hr);
4772 655 : if (ZabM_ker_check(M, Hl, p, P, n)) { H = Hl; break; }
4773 : }
4774 :
4775 653 : if (gc_needed(av,2))
4776 : {
4777 4 : if (DEBUGMEM>1) pari_warn(warnmem,"ZabM_ker");
4778 4 : gerepileall(av, 3, &H, &D, &q);
4779 : }
4780 : }
4781 623 : return gerepilecopy(av, H);
4782 : }
4783 :
4784 : GEN
4785 2387 : ZabM_indexrank(GEN M, GEN P, long n)
4786 : {
4787 2387 : pari_sp av = avma;
4788 2387 : ulong m = LONG_MAX>>1;
4789 2387 : ulong p = 1+m-(m%n), D = degpol(P);
4790 2387 : long lM = lg(M), lmax = 0, c = 0;
4791 : GEN v;
4792 : for(;;)
4793 735 : {
4794 : GEN R, Pp, Mp, K;
4795 : ulong pi;
4796 : long l;
4797 61415 : do p += n; while (!uisprime(p));
4798 3122 : pi = (p & HIGHMASK)? get_Fl_red(p): 0;
4799 3122 : Pp = ZX_to_Flx(P, p);
4800 3122 : R = Flx_roots_pre(Pp, p, pi);
4801 3122 : Mp = ZXM_to_FlxM(M, p, get_Flx_var(Pp));
4802 3122 : K = FlxM_eval_powers_pre(Mp, Fl_powers_pre(uel(R,1), D,p,pi), p,pi);
4803 3122 : v = Flm_indexrank(K, p);
4804 3122 : l = lg(gel(v,2));
4805 3122 : if (l == lM) break;
4806 980 : if (lmax >= 0 && l > lmax) { lmax = l; c = 0; } else c++;
4807 980 : if (c > 2)
4808 : { /* probably not maximal rank, expensive check */
4809 245 : lM -= lg(ZabM_ker(M, P, n))-1; /* actual rank (+1) */
4810 245 : if (lmax == lM) break;
4811 0 : lmax = -1; /* disable check */
4812 : }
4813 : }
4814 2387 : return gerepileupto(av, v);
4815 : }
4816 :
4817 : #if 0
4818 : GEN
4819 : ZabM_gauss(GEN M, GEN P, long n, GEN *den)
4820 : {
4821 : pari_sp av = avma;
4822 : GEN v, S, W;
4823 : v = ZabM_indexrank(M, P, n);
4824 : S = shallowmatextract(M,gel(v,1),gel(v,2));
4825 : W = ZabM_inv(S, P, n, den);
4826 : return gc_all(av,2,&W,den);
4827 : }
4828 : #endif
4829 :
4830 : GEN
4831 140 : ZabM_pseudoinv(GEN M, GEN P, long n, GEN *pv, GEN *den)
4832 : {
4833 140 : GEN v = ZabM_indexrank(M, P, n);
4834 140 : if (pv) *pv = v;
4835 140 : M = shallowmatextract(M,gel(v,1),gel(v,2));
4836 140 : return ZabM_inv(M, P, n, den);
4837 : }
4838 : GEN
4839 5012 : ZM_pseudoinv(GEN M, GEN *pv, GEN *den)
4840 : {
4841 5012 : GEN v = ZM_indexrank(M);
4842 5012 : if (pv) *pv = v;
4843 5012 : M = shallowmatextract(M,gel(v,1),gel(v,2));
4844 5012 : return ZM_inv(M, den);
4845 : }
4846 :
4847 : /*******************************************************************/
4848 : /* */
4849 : /* Structured Elimination */
4850 : /* */
4851 : /*******************************************************************/
4852 :
4853 : static void
4854 111665 : rem_col(GEN c, long i, GEN iscol, GEN Wrow, long *rcol, long *rrow)
4855 : {
4856 111665 : long lc = lg(c), k;
4857 111665 : iscol[i] = 0; (*rcol)--;
4858 1078730 : for (k = 1; k < lc; ++k)
4859 : {
4860 967065 : Wrow[c[k]]--;
4861 967065 : if (Wrow[c[k]]==0) (*rrow)--;
4862 : }
4863 111665 : }
4864 :
4865 : static void
4866 6275 : rem_singleton(GEN M, GEN iscol, GEN Wrow, long idx, long *rcol, long *rrow)
4867 : {
4868 : long i, j;
4869 6275 : long nbcol = lg(iscol)-1, last;
4870 : do
4871 : {
4872 8349 : last = 0;
4873 19436913 : for (i = 1; i <= nbcol; ++i)
4874 19428564 : if (iscol[i])
4875 : {
4876 10128552 : GEN c = idx ? gmael(M, i, idx): gel(M,i);
4877 10128552 : long lc = lg(c);
4878 97282262 : for (j = 1; j < lc; ++j)
4879 87176020 : if (Wrow[c[j]] == 1)
4880 : {
4881 22310 : rem_col(c, i, iscol, Wrow, rcol, rrow);
4882 22310 : last=1; break;
4883 : }
4884 : }
4885 8349 : } while (last);
4886 6275 : }
4887 :
4888 : static GEN
4889 6103 : fill_wcol(GEN M, GEN iscol, GEN Wrow, long *w, GEN wcol)
4890 : {
4891 6103 : long nbcol = lg(iscol)-1;
4892 : long i, j, m, last;
4893 : GEN per;
4894 15293 : for (m = 2, last=0; !last ; m++)
4895 : {
4896 22446556 : for (i = 1; i <= nbcol; ++i)
4897 : {
4898 22437366 : wcol[i] = 0;
4899 22437366 : if (iscol[i])
4900 : {
4901 11520537 : GEN c = gmael(M, i, 1);
4902 11520537 : long lc = lg(c);
4903 107249414 : for (j = 1; j < lc; ++j)
4904 95728877 : if (Wrow[c[j]] == m) { wcol[i]++; last = 1; }
4905 : }
4906 : }
4907 : }
4908 6103 : per = vecsmall_indexsort(wcol);
4909 6103 : *w = wcol[per[nbcol]];
4910 6103 : return per;
4911 : }
4912 :
4913 : /* M is a RgMs with nbrow rows, A a list of row indices.
4914 : Eliminate rows of M with a single entry that do not belong to A,
4915 : and the corresponding columns. Also eliminate columns until #colums=#rows.
4916 : Return pcol and prow:
4917 : pcol is a map from the new columns indices to the old one.
4918 : prow is a map from the old rows indices to the new one (0 if removed).
4919 : */
4920 :
4921 : void
4922 119 : RgMs_structelim_col(GEN M, long nbcol, long nbrow, GEN A, GEN *p_col, GEN *p_row)
4923 : {
4924 119 : long i, j, k, lA = lg(A);
4925 119 : GEN prow = cgetg(nbrow+1, t_VECSMALL);
4926 119 : GEN pcol = zero_zv(nbcol);
4927 119 : pari_sp av = avma;
4928 119 : long rcol = nbcol, rrow = 0, imin = nbcol - usqrt(nbcol);
4929 119 : GEN iscol = const_vecsmall(nbcol, 1);
4930 119 : GEN Wrow = zero_zv(nbrow);
4931 119 : GEN wcol = cgetg(nbcol+1, t_VECSMALL);
4932 119 : pari_sp av2 = avma;
4933 126763 : for (i = 1; i <= nbcol; ++i)
4934 : {
4935 126644 : GEN F = gmael(M, i, 1);
4936 126644 : long l = lg(F)-1;
4937 1115879 : for (j = 1; j <= l; ++j) Wrow[F[j]]++;
4938 : }
4939 119 : for (j = 1; j < lA; ++j)
4940 : {
4941 0 : if (Wrow[A[j]] == 0) { *p_col=NULL; return; }
4942 0 : Wrow[A[j]] = -1;
4943 : }
4944 235298 : for (i = 1; i <= nbrow; ++i)
4945 235179 : if (Wrow[i]) rrow++;
4946 119 : rem_singleton(M, iscol, Wrow, 1, &rcol, &rrow);
4947 119 : if (rcol < rrow) pari_err_BUG("RgMs_structelim, rcol<rrow");
4948 6222 : while (rcol > rrow)
4949 : {
4950 : long w;
4951 6103 : GEN per = fill_wcol(M, iscol, Wrow, &w, wcol);
4952 95458 : for (i = nbcol; i>=imin && wcol[per[i]]>=w && rcol>rrow; i--)
4953 89355 : rem_col(gmael(M, per[i], 1), per[i], iscol, Wrow, &rcol, &rrow);
4954 6103 : rem_singleton(M, iscol, Wrow, 1, &rcol, &rrow); set_avma(av2);
4955 : }
4956 126763 : for (j = 1, i = 1; i <= nbcol; ++i)
4957 126644 : if (iscol[i]) pcol[j++] = i;
4958 119 : setlg(pcol,j);
4959 235298 : for (k = 1, i = 1; i <= nbrow; ++i) prow[i] = Wrow[i]? k++: 0;
4960 119 : *p_col = pcol; *p_row = prow; set_avma(av);
4961 : }
4962 :
4963 : void
4964 0 : RgMs_structelim(GEN M, long nbrow, GEN A, GEN *p_col, GEN *p_row)
4965 0 : { RgMs_structelim_col(M, lg(M)-1, nbrow, A, p_col, p_row); }
4966 :
4967 : GEN
4968 53 : F2Ms_colelim(GEN M, long nbrow)
4969 : {
4970 53 : long i,j, nbcol = lg(M)-1, rcol = nbcol, rrow = 0;
4971 53 : GEN pcol = zero_zv(nbcol);
4972 53 : pari_sp av = avma;
4973 53 : GEN iscol = const_vecsmall(nbcol, 1), Wrow = zero_zv(nbrow);
4974 85968 : for (i = 1; i <= nbcol; ++i)
4975 : {
4976 85915 : GEN F = gel(M, i);
4977 85915 : long l = lg(F)-1;
4978 1587632 : for (j = 1; j <= l; ++j) Wrow[F[j]]++;
4979 : }
4980 53 : rem_singleton(M, iscol, Wrow, 0, &rcol, &rrow);
4981 85968 : for (j = 1, i = 1; i <= nbcol; ++i)
4982 85915 : if (iscol[i]) pcol[j++] = i;
4983 53 : fixlg(pcol,j); return gc_const(av, pcol);
4984 : }
4985 :
4986 : /*******************************************************************/
4987 : /* */
4988 : /* EIGENVECTORS */
4989 : /* (independent eigenvectors, sorted by increasing eigenvalue) */
4990 : /* */
4991 : /*******************************************************************/
4992 : /* assume x is square of dimension > 0 */
4993 : static int
4994 53 : RgM_is_symmetric_cx(GEN x, long bit)
4995 : {
4996 53 : pari_sp av = avma;
4997 53 : long i, j, l = lg(x);
4998 239 : for (i = 1; i < l; i++)
4999 708 : for (j = 1; j < i; j++)
5000 : {
5001 522 : GEN a = gcoeff(x,i,j), b = gcoeff(x,j,i), c = gsub(a,b);
5002 522 : if (!gequal0(c) && gexpo(c) - gexpo(a) > -bit) return gc_long(av,0);
5003 : }
5004 21 : return gc_long(av,1);
5005 : }
5006 : static GEN
5007 53 : eigen_err(int exact, GEN x, long flag, long prec)
5008 : {
5009 53 : pari_sp av = avma;
5010 : GEN y;
5011 53 : if (RgM_is_symmetric_cx(x, prec - 10))
5012 : { /* approximately symmetric: recover */
5013 21 : x = jacobi(x, prec); if (flag) return x;
5014 14 : return gerepilecopy(av, gel(x,2));
5015 : }
5016 32 : if (!exact) x = bestappr(x, NULL);
5017 32 : y = mateigen(x, flag, precdbl(prec));
5018 32 : if (exact)
5019 18 : y = gprec_wtrunc(y, prec);
5020 14 : else if (flag)
5021 7 : y = mkvec2(RgV_gtofp(gel(y,1), prec), RgM_gtofp(gel(y,2), prec));
5022 : else
5023 7 : y = RgM_gtofp(y, prec);
5024 32 : return gerepilecopy(av, y);
5025 : }
5026 : GEN
5027 144 : mateigen(GEN x, long flag, long prec)
5028 : {
5029 : GEN y, R, T;
5030 144 : long k, l, ex, n = lg(x);
5031 : int exact;
5032 144 : pari_sp av = avma;
5033 :
5034 144 : if (typ(x)!=t_MAT) pari_err_TYPE("eigen",x);
5035 144 : if (n != 1 && n != lgcols(x)) pari_err_DIM("eigen");
5036 144 : if (flag < 0 || flag > 1) pari_err_FLAG("mateigen");
5037 144 : if (n == 1)
5038 : {
5039 14 : if (flag) retmkvec2(cgetg(1,t_COL), cgetg(1,t_MAT));
5040 7 : return cgetg(1,t_MAT);
5041 : }
5042 130 : if (n == 2)
5043 : {
5044 14 : if (flag) retmkvec2(mkcolcopy(gcoeff(x,1,1)), matid(1));
5045 7 : return matid(1);
5046 : }
5047 :
5048 116 : ex = 16 - prec;
5049 116 : T = charpoly(x,0);
5050 116 : exact = RgX_is_QX(T);
5051 116 : if (exact)
5052 : {
5053 74 : T = ZX_radical( Q_primpart(T) );
5054 74 : R = nfrootsQ(T); settyp(R, t_COL);
5055 74 : if (lg(R)-1 < degpol(T))
5056 : { /* add missing complex roots */
5057 60 : GEN r = cleanroots(RgX_div(T, roots_to_pol(R, 0)), prec);
5058 60 : R = shallowconcat(R, r);
5059 : }
5060 : }
5061 : else
5062 : {
5063 42 : GEN r1, v = vectrunc_init(lg(T));
5064 : long e;
5065 42 : R = cleanroots(T,prec);
5066 42 : r1 = NULL;
5067 266 : for (k = 1; k < lg(R); k++)
5068 : {
5069 224 : GEN r2 = gel(R,k), r = grndtoi(r2, &e);
5070 224 : if (e < ex) r2 = r;
5071 224 : if (r1)
5072 : {
5073 182 : r = gsub(r1,r2);
5074 182 : if (gequal0(r) || gexpo(r) < ex) continue;
5075 : }
5076 182 : vectrunc_append(v, r2);
5077 182 : r1 = r2;
5078 : }
5079 42 : R = v;
5080 : }
5081 : /* R = distinct complex roots of charpoly(x) */
5082 116 : l = lg(R); y = cgetg(l, t_VEC);
5083 452 : for (k = 1; k < l; k++)
5084 : {
5085 389 : GEN F = ker_aux(RgM_Rg_sub_shallow(x, gel(R,k)), x);
5086 389 : long d = lg(F)-1;
5087 389 : if (!d) { set_avma(av); return eigen_err(exact, x, flag, prec); }
5088 336 : gel(y,k) = F;
5089 336 : if (flag) gel(R,k) = const_col(d, gel(R,k));
5090 : }
5091 63 : y = shallowconcat1(y);
5092 63 : if (lg(y) > n) { set_avma(av); return eigen_err(exact, x, flag, prec); }
5093 : /* lg(y) < n if x is not diagonalizable */
5094 63 : if (flag) y = mkvec2(shallowconcat1(R), y);
5095 63 : return gerepilecopy(av,y);
5096 : }
5097 : GEN
5098 0 : eigen(GEN x, long prec) { return mateigen(x, 0, prec); }
5099 :
5100 : /*******************************************************************/
5101 : /* */
5102 : /* DETERMINANT */
5103 : /* */
5104 : /*******************************************************************/
5105 :
5106 : GEN
5107 26474 : det0(GEN a,long flag)
5108 : {
5109 26474 : switch(flag)
5110 : {
5111 26460 : case 0: return det(a);
5112 14 : case 1: return det2(a);
5113 0 : default: pari_err_FLAG("matdet");
5114 : }
5115 : return NULL; /* LCOV_EXCL_LINE */
5116 : }
5117 :
5118 : /* M a 2x2 matrix, returns det(M) */
5119 : static GEN
5120 84646 : RgM_det2(GEN M)
5121 : {
5122 84646 : pari_sp av = avma;
5123 84646 : GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
5124 84646 : GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
5125 84646 : return gerepileupto(av, gsub(gmul(a,d), gmul(b,c)));
5126 : }
5127 : /* M a 2x2 ZM, returns det(M) */
5128 : static GEN
5129 6377 : ZM_det2(GEN M)
5130 : {
5131 6377 : pari_sp av = avma;
5132 6377 : GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
5133 6377 : GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
5134 6377 : return gerepileuptoint(av, subii(mulii(a,d), mulii(b, c)));
5135 : }
5136 : /* M a 3x3 ZM, return det(M) */
5137 : static GEN
5138 99954 : ZM_det3(GEN M)
5139 : {
5140 99954 : pari_sp av = avma;
5141 99954 : GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2), c = gcoeff(M,1,3);
5142 99954 : GEN d = gcoeff(M,2,1), e = gcoeff(M,2,2), f = gcoeff(M,2,3);
5143 99954 : GEN g = gcoeff(M,3,1), h = gcoeff(M,3,2), i = gcoeff(M,3,3);
5144 99954 : GEN t, D = signe(i)? mulii(subii(mulii(a,e), mulii(b,d)), i): gen_0;
5145 99954 : if (signe(g))
5146 : {
5147 66097 : t = mulii(subii(mulii(b,f), mulii(c,e)), g);
5148 66097 : D = addii(D, t);
5149 : }
5150 99954 : if (signe(h))
5151 : {
5152 77366 : t = mulii(subii(mulii(c,d), mulii(a,f)), h);
5153 77366 : D = addii(D, t);
5154 : }
5155 99954 : return gerepileuptoint(av, D);
5156 : }
5157 :
5158 : static GEN
5159 58395 : det_simple_gauss(GEN a, GEN data, pivot_fun pivot)
5160 : {
5161 58395 : pari_sp av = avma;
5162 58395 : long i,j,k, s = 1, nbco = lg(a)-1;
5163 58395 : GEN p, x = gen_1;
5164 :
5165 58395 : a = RgM_shallowcopy(a);
5166 342099 : for (i=1; i<nbco; i++)
5167 : {
5168 283711 : k = pivot(a, data, i, NULL);
5169 283712 : if (k > nbco) return gerepilecopy(av, gcoeff(a,i,i));
5170 283705 : if (k != i)
5171 : { /* exchange the lines s.t. k = i */
5172 1153698 : for (j=i; j<=nbco; j++) swap(gcoeff(a,i,j), gcoeff(a,k,j));
5173 118538 : s = -s;
5174 : }
5175 283705 : p = gcoeff(a,i,i);
5176 :
5177 283705 : x = gmul(x,p);
5178 1785012 : for (k=i+1; k<=nbco; k++)
5179 : {
5180 1501307 : GEN m = gcoeff(a,i,k);
5181 1501307 : if (gequal0(m)) continue;
5182 :
5183 1066943 : m = gdiv(m,p);
5184 9938002 : for (j=i+1; j<=nbco; j++)
5185 8871059 : gcoeff(a,j,k) = gsub(gcoeff(a,j,k), gmul(m,gcoeff(a,j,i)));
5186 : }
5187 283705 : if (gc_needed(av,2))
5188 : {
5189 0 : if(DEBUGMEM>1) pari_warn(warnmem,"det. col = %ld",i);
5190 0 : gerepileall(av,2, &a,&x);
5191 : }
5192 : }
5193 58388 : if (s < 0) x = gneg_i(x);
5194 58388 : return gerepileupto(av, gmul(x, gcoeff(a,nbco,nbco)));
5195 : }
5196 :
5197 : GEN
5198 115662 : det2(GEN a)
5199 : {
5200 : GEN data;
5201 : pivot_fun pivot;
5202 115662 : long n = lg(a)-1;
5203 115662 : if (typ(a)!=t_MAT) pari_err_TYPE("det2",a);
5204 115662 : if (!n) return gen_1;
5205 115662 : if (n != nbrows(a)) pari_err_DIM("det2");
5206 115662 : if (n == 1) return gcopy(gcoeff(a,1,1));
5207 76592 : if (n == 2) return RgM_det2(a);
5208 27078 : pivot = get_pivot_fun(a, a, &data);
5209 27078 : return det_simple_gauss(a, data, pivot);
5210 : }
5211 :
5212 : /* Assumes a a square t_MAT of dimension n > 0. Returns det(a) using
5213 : * Gauss-Bareiss. */
5214 : static GEN
5215 462 : det_bareiss(GEN a)
5216 : {
5217 462 : pari_sp av = avma;
5218 462 : long nbco = lg(a)-1,i,j,k,s = 1;
5219 : GEN p, pprec;
5220 :
5221 462 : a = RgM_shallowcopy(a);
5222 1337 : for (pprec=gen_1,i=1; i<nbco; i++,pprec=p)
5223 : {
5224 882 : int diveuc = (gequal1(pprec)==0);
5225 : GEN ci;
5226 :
5227 882 : p = gcoeff(a,i,i);
5228 882 : if (gequal0(p))
5229 : {
5230 14 : k=i+1; while (k<=nbco && gequal0(gcoeff(a,i,k))) k++;
5231 7 : if (k>nbco) return gerepilecopy(av, p);
5232 0 : swap(gel(a,k), gel(a,i)); s = -s;
5233 0 : p = gcoeff(a,i,i);
5234 : }
5235 875 : ci = gel(a,i);
5236 2373 : for (k=i+1; k<=nbco; k++)
5237 : {
5238 1498 : GEN ck = gel(a,k), m = gel(ck,i);
5239 1498 : if (gequal0(m))
5240 : {
5241 7 : if (gequal1(p))
5242 : {
5243 0 : if (diveuc)
5244 0 : gel(a,k) = gdiv(gel(a,k), pprec);
5245 : }
5246 : else
5247 42 : for (j=i+1; j<=nbco; j++)
5248 : {
5249 35 : GEN p1 = gmul(p, gel(ck,j));
5250 35 : if (diveuc) p1 = gdiv(p1,pprec);
5251 35 : gel(ck,j) = p1;
5252 : }
5253 : }
5254 : else
5255 4662 : for (j=i+1; j<=nbco; j++)
5256 : {
5257 3171 : pari_sp av2 = avma;
5258 3171 : GEN p1 = gsub(gmul(p,gel(ck,j)), gmul(m,gel(ci,j)));
5259 3171 : if (diveuc) p1 = gdiv(p1,pprec);
5260 3171 : gel(ck,j) = gerepileupto(av2, p1);
5261 : }
5262 1498 : if (gc_needed(av,2))
5263 : {
5264 0 : if(DEBUGMEM>1) pari_warn(warnmem,"det. col = %ld",i);
5265 0 : gerepileall(av,2, &a,&pprec);
5266 0 : ci = gel(a,i);
5267 0 : p = gcoeff(a,i,i);
5268 : }
5269 : }
5270 : }
5271 455 : p = gcoeff(a,nbco,nbco);
5272 455 : p = (s < 0)? gneg(p): gcopy(p);
5273 455 : return gerepileupto(av, p);
5274 : }
5275 :
5276 : /* count nonzero entries in col j, at most 'max' of them.
5277 : * Return their indices */
5278 : static GEN
5279 1470 : col_count_non_zero(GEN a, long j, long max)
5280 : {
5281 1470 : GEN v = cgetg(max+1, t_VECSMALL);
5282 1470 : GEN c = gel(a,j);
5283 1470 : long i, l = lg(a), k = 1;
5284 5614 : for (i = 1; i < l; i++)
5285 5376 : if (!gequal0(gel(c,i)))
5286 : {
5287 5110 : if (k > max) return NULL; /* fail */
5288 3878 : v[k++] = i;
5289 : }
5290 238 : setlg(v, k); return v;
5291 : }
5292 : /* count nonzero entries in row i, at most 'max' of them.
5293 : * Return their indices */
5294 : static GEN
5295 1456 : row_count_non_zero(GEN a, long i, long max)
5296 : {
5297 1456 : GEN v = cgetg(max+1, t_VECSMALL);
5298 1456 : long j, l = lg(a), k = 1;
5299 5558 : for (j = 1; j < l; j++)
5300 5334 : if (!gequal0(gcoeff(a,i,j)))
5301 : {
5302 5096 : if (k > max) return NULL; /* fail */
5303 3864 : v[k++] = j;
5304 : }
5305 224 : setlg(v, k); return v;
5306 : }
5307 :
5308 : static GEN det_develop(GEN a, long max, double bound);
5309 : /* (-1)^(i+j) a[i,j] * det RgM_minor(a,i,j) */
5310 : static GEN
5311 406 : coeff_det(GEN a, long i, long j, long max, double bound)
5312 : {
5313 406 : GEN c = gcoeff(a, i, j);
5314 406 : c = gmul(c, det_develop(RgM_minor(a, i,j), max, bound));
5315 406 : if (odd(i+j)) c = gneg(c);
5316 406 : return c;
5317 : }
5318 : /* a square t_MAT, 'bound' a rough upper bound for the number of
5319 : * multiplications we are willing to pay while developing rows/columns before
5320 : * switching to Gaussian elimination */
5321 : static GEN
5322 658 : det_develop(GEN M, long max, double bound)
5323 : {
5324 658 : pari_sp av = avma;
5325 658 : long i,j, n = lg(M)-1, lbest = max+2, best_col = 0, best_row = 0;
5326 658 : GEN best = NULL;
5327 :
5328 658 : if (bound < 1.) return det_bareiss(M); /* too costly now */
5329 :
5330 434 : switch(n)
5331 : {
5332 0 : case 0: return gen_1;
5333 0 : case 1: return gcopy(gcoeff(M,1,1));
5334 14 : case 2: return RgM_det2(M);
5335 : }
5336 420 : if (max > ((n+2)>>1)) max = (n+2)>>1;
5337 1876 : for (j = 1; j <= n; j++)
5338 : {
5339 1470 : pari_sp av2 = avma;
5340 1470 : GEN v = col_count_non_zero(M, j, max);
5341 : long lv;
5342 1470 : if (!v || (lv = lg(v)) >= lbest) { set_avma(av2); continue; }
5343 182 : if (lv == 1) { set_avma(av); return gen_0; }
5344 182 : if (lv == 2) {
5345 14 : set_avma(av);
5346 14 : return gerepileupto(av, coeff_det(M,v[1],j,max,bound));
5347 : }
5348 168 : best = v; lbest = lv; best_col = j;
5349 : }
5350 1862 : for (i = 1; i <= n; i++)
5351 : {
5352 1456 : pari_sp av2 = avma;
5353 1456 : GEN v = row_count_non_zero(M, i, max);
5354 : long lv;
5355 1456 : if (!v || (lv = lg(v)) >= lbest) { set_avma(av2); continue; }
5356 0 : if (lv == 1) { set_avma(av); return gen_0; }
5357 0 : if (lv == 2) {
5358 0 : set_avma(av);
5359 0 : return gerepileupto(av, coeff_det(M,i,v[1],max,bound));
5360 : }
5361 0 : best = v; lbest = lv; best_row = i;
5362 : }
5363 406 : if (best_row)
5364 : {
5365 0 : double d = lbest-1;
5366 0 : GEN s = NULL;
5367 : long k;
5368 0 : bound /= d*d*d;
5369 0 : for (k = 1; k < lbest; k++)
5370 : {
5371 0 : GEN c = coeff_det(M, best_row, best[k], max, bound);
5372 0 : s = s? gadd(s, c): c;
5373 : }
5374 0 : return gerepileupto(av, s);
5375 : }
5376 406 : if (best_col)
5377 : {
5378 168 : double d = lbest-1;
5379 168 : GEN s = NULL;
5380 : long k;
5381 168 : bound /= d*d*d;
5382 560 : for (k = 1; k < lbest; k++)
5383 : {
5384 392 : GEN c = coeff_det(M, best[k], best_col, max, bound);
5385 392 : s = s? gadd(s, c): c;
5386 : }
5387 168 : return gerepileupto(av, s);
5388 : }
5389 238 : return det_bareiss(M);
5390 : }
5391 :
5392 : /* area of parallelogram bounded by (v1,v2) */
5393 : static GEN
5394 62301 : parallelogramarea(GEN v1, GEN v2)
5395 62301 : { return gsub(gmul(gnorml2(v1), gnorml2(v2)), gsqr(RgV_dotproduct(v1, v2))); }
5396 :
5397 : /* Square of Hadamard bound for det(a), a square matrix.
5398 : * Slight improvement: instead of using the column norms, use the area of
5399 : * the parallelogram formed by pairs of consecutive vectors */
5400 : GEN
5401 19450 : RgM_Hadamard(GEN a)
5402 : {
5403 19450 : pari_sp av = avma;
5404 19450 : long n = lg(a)-1, i;
5405 : GEN B;
5406 19450 : if (n == 0) return gen_1;
5407 19450 : if (n == 1) return gsqr(gcoeff(a,1,1));
5408 19450 : a = RgM_gtofp(a, LOWDEFAULTPREC);
5409 19450 : B = gen_1;
5410 81751 : for (i = 1; i <= n/2; i++)
5411 62301 : B = gmul(B, parallelogramarea(gel(a,2*i-1), gel(a,2*i)));
5412 19450 : if (odd(n)) B = gmul(B, gnorml2(gel(a, n)));
5413 19450 : return gerepileuptoint(av, ceil_safe(B));
5414 : }
5415 :
5416 : /* If B=NULL, assume B=A' */
5417 : static GEN
5418 20692 : ZM_det_slice(GEN A, GEN P, GEN *mod)
5419 : {
5420 20692 : pari_sp av = avma;
5421 20692 : long i, n = lg(P)-1;
5422 : GEN H, T;
5423 20692 : if (n == 1)
5424 : {
5425 0 : ulong Hp, p = uel(P,1);
5426 0 : GEN a = ZM_to_Flm(A, p);
5427 0 : Hp = Flm_det_sp(a, p);
5428 0 : set_avma(av); *mod = utoipos(p); return utoi(Hp);
5429 : }
5430 20692 : T = ZV_producttree(P);
5431 20692 : A = ZM_nv_mod_tree(A, P, T);
5432 20692 : H = cgetg(n+1, t_VECSMALL);
5433 86627 : for(i=1; i <= n; i++)
5434 : {
5435 65935 : ulong p = P[i];
5436 65935 : GEN a = gel(A,i);
5437 65935 : H[i] = Flm_det_sp(a, p);
5438 : }
5439 20692 : H = ZV_chinese_tree(H, P, T, ZV_chinesetree(P,T));
5440 20692 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
5441 : }
5442 :
5443 : GEN
5444 20692 : ZM_det_worker(GEN P, GEN A)
5445 : {
5446 20692 : GEN V = cgetg(3, t_VEC);
5447 20692 : gel(V,1) = ZM_det_slice(A, P, &gel(V,2));
5448 20692 : return V;
5449 : }
5450 :
5451 : GEN
5452 127321 : ZM_det(GEN M)
5453 : {
5454 127321 : const long DIXON_THRESHOLD = 40;
5455 : pari_sp av, av2;
5456 127321 : long i, n = lg(M)-1;
5457 : ulong p, Dp;
5458 : forprime_t S;
5459 : pari_timer ti;
5460 : GEN H, D, mod, h, q, v, worker;
5461 : #ifdef LONG_IS_64BIT
5462 109140 : const ulong PMAX = 18446744073709551557UL;
5463 : #else
5464 18181 : const ulong PMAX = 4294967291UL;
5465 : #endif
5466 :
5467 127321 : switch(n)
5468 : {
5469 7 : case 0: return gen_1;
5470 1533 : case 1: return icopy(gcoeff(M,1,1));
5471 6377 : case 2: return ZM_det2(M);
5472 99954 : case 3: return ZM_det3(M);
5473 : }
5474 19450 : if (DEBUGLEVEL>=4) timer_start(&ti);
5475 19450 : av = avma; h = RgM_Hadamard(M); /* |D| <= sqrt(h) */
5476 19450 : if (!signe(h)) { set_avma(av); return gen_0; }
5477 19450 : h = sqrti(h);
5478 19450 : if (lgefint(h) == 3 && (ulong)h[2] <= (PMAX >> 1))
5479 : { /* h < p/2 => direct result */
5480 6766 : p = PMAX;
5481 6766 : Dp = Flm_det_sp(ZM_to_Flm(M, p), p);
5482 6766 : set_avma(av);
5483 6766 : if (!Dp) return gen_0;
5484 6766 : return (Dp <= (p>>1))? utoipos(Dp): utoineg(p - Dp);
5485 : }
5486 12684 : q = gen_1; Dp = 1;
5487 12684 : init_modular_big(&S);
5488 12684 : p = 0; /* -Wall */
5489 12684 : while (cmpii(q, h) <= 0 && (p = u_forprime_next(&S)))
5490 : {
5491 12684 : av2 = avma; Dp = Flm_det_sp(ZM_to_Flm(M, p), p);
5492 12684 : set_avma(av2);
5493 12684 : if (Dp) break;
5494 0 : q = muliu(q, p);
5495 : }
5496 12684 : if (!p) pari_err_OVERFLOW("ZM_det [ran out of primes]");
5497 12684 : if (!Dp) { set_avma(av); return gen_0; }
5498 12684 : if (mt_nbthreads() > 1 || n <= DIXON_THRESHOLD)
5499 12678 : D = q; /* never competitive when bound is sharp even with 2 threads */
5500 : else
5501 : {
5502 6 : av2 = avma;
5503 6 : v = cgetg(n+1, t_COL);
5504 6 : gel(v, 1) = gen_1; /* ensure content(v) = 1 */
5505 414 : for (i = 2; i <= n; i++) gel(v, i) = stoi(random_Fl(15) - 7);
5506 6 : D = Q_denom(ZM_gauss(M, v));
5507 6 : if (expi(D) < expi(h) >> 1)
5508 : { /* First try unlucky, try once more */
5509 414 : for (i = 2; i <= n; i++) gel(v, i) = stoi(random_Fl(15) - 7);
5510 6 : D = lcmii(D, Q_denom(ZM_gauss(M, v)));
5511 : }
5512 6 : D = gerepileuptoint(av2, D);
5513 6 : if (q != gen_1) D = lcmii(D, q);
5514 : }
5515 12684 : if (DEBUGLEVEL >=4)
5516 0 : timer_printf(&ti,"ZM_det: Dixon %ld/%ld bits",expi(D),expi(h));
5517 : /* determinant is a multiple of D */
5518 12684 : if (is_pm1(D)) D = NULL;
5519 12684 : if (D) h = divii(h, D); /* not an exact division, just a bound */
5520 12684 : worker = snm_closure(is_entry("_ZM_det_worker"), mkvec(M));
5521 12684 : H = gen_crt("ZM_det", worker, &S, D, expi(h)+1, 0, &mod,
5522 : ZV_chinese, NULL);
5523 : /* H = det(M) modulo mod, (mod,D) = 1; |det(M) / D| <= h */
5524 12684 : if (D) H = Fp_div(H, D, mod);
5525 12684 : H = Fp_center(H, mod, shifti(mod,-1));
5526 12684 : if (D) H = mulii(H, D);
5527 12684 : return gerepileuptoint(av, H);
5528 : }
5529 :
5530 : static GEN
5531 1519 : RgM_det_FpM(GEN a, GEN p)
5532 : {
5533 1519 : pari_sp av = avma;
5534 : ulong pp, d;
5535 1519 : a = RgM_Fp_init(a,p,&pp);
5536 1519 : switch(pp)
5537 : {
5538 70 : case 0: return gerepileupto(av, Fp_to_mod(FpM_det(a,p),p)); break;
5539 14 : case 2: d = F2m_det_sp(a); break;
5540 1435 : default:d = Flm_det_sp(a, pp); break;
5541 : }
5542 1449 : set_avma(av); return mkintmodu(d, pp);
5543 : }
5544 :
5545 : static GEN
5546 42 : RgM_det_FqM(GEN x, GEN pol, GEN p)
5547 : {
5548 42 : pari_sp av = avma;
5549 42 : GEN b, T = RgX_to_FpX(pol, p);
5550 42 : if (signe(T) == 0) pari_err_OP("%",x,pol);
5551 42 : b = FqM_det(RgM_to_FqM(x, T, p), T, p);
5552 42 : if (!b) return gc_NULL(av);
5553 42 : return gerepilecopy(av, mkpolmod(FpX_to_mod(b, p), FpX_to_mod(T, p)));
5554 : }
5555 :
5556 : #define code(t1,t2) ((t1 << 6) | t2)
5557 : static GEN
5558 33592 : RgM_det_fast(GEN x)
5559 : {
5560 : GEN p, pol;
5561 : long pa;
5562 33592 : long t = RgM_type(x, &p,&pol,&pa);
5563 33592 : switch(t)
5564 : {
5565 196 : case t_INT: return ZM_det(x);
5566 203 : case t_FRAC: return QM_det(x);
5567 63 : case t_FFELT: return FFM_det(x, pol);
5568 1519 : case t_INTMOD: return RgM_det_FpM(x, p);
5569 42 : case code(t_POLMOD, t_INTMOD):
5570 42 : return RgM_det_FqM(x, pol, p);
5571 31569 : default: return NULL;
5572 : }
5573 : }
5574 : #undef code
5575 :
5576 : static long
5577 252 : det_init_max(long n)
5578 : {
5579 252 : if (n > 100) return 0;
5580 252 : if (n > 50) return 1;
5581 252 : if (n > 30) return 4;
5582 252 : return 7;
5583 : }
5584 :
5585 : GEN
5586 246654 : det(GEN a)
5587 : {
5588 246654 : long n = lg(a)-1;
5589 : double B;
5590 : GEN data, b;
5591 : pivot_fun pivot;
5592 :
5593 246654 : if (typ(a)!=t_MAT) pari_err_TYPE("det",a);
5594 246654 : if (!n) return gen_1;
5595 246612 : if (n != nbrows(a)) pari_err_DIM("det");
5596 246605 : if (n == 1) return gcopy(gcoeff(a,1,1));
5597 68710 : if (n == 2) return RgM_det2(a);
5598 33592 : b = RgM_det_fast(a);
5599 33592 : if (b) return b;
5600 31569 : pivot = get_pivot_fun(a, a, &data);
5601 31569 : if (pivot != gauss_get_pivot_NZ) return det_simple_gauss(a, data, pivot);
5602 252 : B = (double)n;
5603 252 : return det_develop(a, det_init_max(n), B*B*B);
5604 : }
5605 :
5606 : GEN
5607 203 : QM_det(GEN M)
5608 : {
5609 203 : pari_sp av = avma;
5610 203 : GEN cM, pM = Q_primitive_part(M, &cM);
5611 203 : GEN b = ZM_det(pM);
5612 203 : if (cM) b = gmul(b, gpowgs(cM, lg(M)-1));
5613 203 : return gerepileupto(av, b);
5614 : }
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