isl_equalities.c

   1 #include "isl_mat.h"
   2 #include "isl_seq.h"
   3 #include "isl_map_private.h"
   4 #include "isl_equalities.h"
   5
   6 /* Given a set of modulo constraints
   7  *
   8  *              c + A y = 0 mod d
   9  *
  10  * this function computes a particular solution y_0
  11  *
  12  * The input is given as a matrix B = [ c A ] and a vector d.
  13  *
  14  * The output is matrix containing the solution y_0 or
  15  * a zero-column matrix if the constraints admit no integer solution.
  16  *
  17  * The given set of constrains is equivalent to
  18  *
  19  *              c + A y = -D x
  20  *
  21  * with D = diag d and x a fresh set of variables.
  22  * Reducing both c and A modulo d does not change the
  23  * value of y in the solution and may lead to smaller coefficients.
  24  * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
  25  * Then
  26  *                [ x ]
  27  *              M [ y ] = - c
  28  * and so
  29  *                             [ x ]
  30  *              [ H 0 ] U^{-1} [ y ] = - c
  31  * Let
  32  *              [ A ]          [ x ]
  33  *              [ B ] = U^{-1} [ y ]
  34  * then
  35  *              H A + 0 B = -c
  36  *
  37  * so B may be chosen arbitrarily, e.g., B = 0, and then
  38  *
  39  *                     [ x ] = [ -c ]
  40  *              U^{-1} [ y ] = [  0 ]
  41  * or
  42  *              [ x ]     [ -c ]
  43  *              [ y ] = U [  0 ]
  44  * specifically,
  45  *
  46  *              y = U_{2,1} (-c)
  47  *
  48  * If any of the coordinates of this y are non-integer
  49  * then the constraints admit no integer solution and
  50  * a zero-column matrix is returned.
  51  */
  52 static struct isl_mat *particular_solution(struct isl_ctx *ctx,
  53                         struct isl_mat *B, struct isl_vec *d)
  54 {
  55         int i, j;
  56         struct isl_mat *M = NULL;
  57         struct isl_mat *C = NULL;
  58         struct isl_mat *U = NULL;
  59         struct isl_mat *H = NULL;
  60         struct isl_mat *cst = NULL;
  61         struct isl_mat *T = NULL;
  62
  63         M = isl_mat_alloc(ctx, B->n_row, B->n_row + B->n_col - 1);
  64         C = isl_mat_alloc(ctx, 1 + B->n_row, 1);
  65         if (!M || !C)
  66                 goto error;
  67         isl_int_set_si(C->row[0][0], 1);
  68         for (i = 0; i < B->n_row; ++i) {
  69                 isl_seq_clr(M->row[i], B->n_row);
  70                 isl_int_set(M->row[i][i], d->block.data[i]);
  71                 isl_int_neg(C->row[1 + i][0], B->row[i][0]);
  72                 isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
  73                 for (j = 0; j < B->n_col - 1; ++j)
  74                         isl_int_fdiv_r(M->row[i][B->n_row + j],
  75                                         B->row[i][1 + j], M->row[i][i]);
  76         }
  77         M = isl_mat_left_hermite(ctx, M, 0, &U, NULL);
  78         if (!M || !U)
  79                 goto error;
  80         H = isl_mat_sub_alloc(ctx, M->row, 0, B->n_row, 0, B->n_row);
  81         H = isl_mat_lin_to_aff(ctx, H);
  82         C = isl_mat_inverse_product(ctx, H, C);
  83         if (!C)
  84                 goto error;
  85         for (i = 0; i < B->n_row; ++i) {
  86                 if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
  87                         break;
  88                 isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
  89         }
  90         if (i < B->n_row)
  91                 cst = isl_mat_alloc(ctx, B->n_row, 0);
  92         else
  93                 cst = isl_mat_sub_alloc(ctx, C->row, 1, B->n_row, 0, 1);
  94         T = isl_mat_sub_alloc(ctx, U->row, B->n_row, B->n_col - 1, 0, B->n_row);
  95         cst = isl_mat_product(ctx, T, cst);
  96         isl_mat_free(ctx, M);
  97         isl_mat_free(ctx, C);
  98         isl_mat_free(ctx, U);
  99         return cst;
 100 error:
 101         isl_mat_free(ctx, M);
 102         isl_mat_free(ctx, C);
 103         isl_mat_free(ctx, U);
 104         return NULL;
 105 }
 106
 107 /* Compute and return the matrix
 108  *
 109  *              U_1^{-1} diag(d_1, 1, ..., 1)
 110  *
 111  * with U_1 the unimodular completion of the first (and only) row of B.
 112  * The columns of this matrix generate the lattice that satisfies
 113  * the single (linear) modulo constraint.
 114  */
 115 static struct isl_mat *parameter_compression_1(struct isl_ctx *ctx,
 116                         struct isl_mat *B, struct isl_vec *d)
 117 {
 118         struct isl_mat *U;
 119
 120         U = isl_mat_alloc(ctx, B->n_col - 1, B->n_col - 1);
 121         if (!U)
 122                 return NULL;
 123         isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
 124         U = isl_mat_unimodular_complete(ctx, U, 1);
 125         U = isl_mat_right_inverse(ctx, U);
 126         if (!U)
 127                 return NULL;
 128         isl_mat_col_mul(U, 0, d->block.data[0], 0);
 129         U = isl_mat_lin_to_aff(ctx, U);
 130         return U;
 131 error:
 132         isl_mat_free(ctx, U);
 133         return NULL;
 134 }
 135
 136 /* Compute a common lattice of solutions to the linear modulo
 137  * constraints specified by B and d.
 138  * See also the documentation of isl_mat_parameter_compression.
 139  * We put the matrix
 140  *
 141  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 142  *
 143  * on a common denominator.  This denominator D is the lcm of modulos d.
 144  * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
 145  * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
 146  * Putting this on the common denominator, we have
 147  * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
 148  */
 149 static struct isl_mat *parameter_compression_multi(struct isl_ctx *ctx,
 150                         struct isl_mat *B, struct isl_vec *d)
 151 {
 152         int i, j, k;
 153         int ok;
 154         isl_int D;
 155         struct isl_mat *A = NULL, *U = NULL;
 156         struct isl_mat *T;
 157         unsigned size;
 158
 159         isl_int_init(D);
 160
 161         isl_vec_lcm(ctx, d, &D);
 162
 163         size = B->n_col - 1;
 164         A = isl_mat_alloc(ctx, size, B->n_row * size);
 165         U = isl_mat_alloc(ctx, size, size);
 166         if (!U || !A)
 167                 goto error;
 168         for (i = 0; i < B->n_row; ++i) {
 169                 isl_seq_cpy(U->row[0], B->row[i] + 1, size);
 170                 U = isl_mat_unimodular_complete(ctx, U, 1);
 171                 if (!U)
 172                         goto error;
 173                 isl_int_divexact(D, D, d->block.data[i]);
 174                 for (k = 0; k < U->n_col; ++k)
 175                         isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
 176                 isl_int_mul(D, D, d->block.data[i]);
 177                 for (j = 1; j < U->n_row; ++j)
 178                         for (k = 0; k < U->n_col; ++k)
 179                                 isl_int_mul(A->row[k][i*size+j],
 180                                                 D, U->row[j][k]);
 181         }
 182         A = isl_mat_left_hermite(ctx, A, 0, NULL, NULL);
 183         T = isl_mat_sub_alloc(ctx, A->row, 0, A->n_row, 0, A->n_row);
 184         T = isl_mat_lin_to_aff(ctx, T);
 185         isl_int_set(T->row[0][0], D);
 186         T = isl_mat_right_inverse(ctx, T);
 187         isl_assert(ctx, isl_int_is_one(T->row[0][0]), goto error);
 188         T = isl_mat_transpose(ctx, T);
 189         isl_mat_free(ctx, A);
 190         isl_mat_free(ctx, U);
 191
 192         isl_int_clear(D);
 193         return T;
 194 error:
 195         isl_mat_free(ctx, A);
 196         isl_mat_free(ctx, U);
 197         isl_int_clear(D);
 198         return NULL;
 199 }
 200
 201 /* Given a set of modulo constraints
 202  *
 203  *              c + A y = 0 mod d
 204  *
 205  * this function returns an affine transformation T,
 206  *
 207  *              y = T y'
 208  *
 209  * that bijectively maps the integer vectors y' to integer
 210  * vectors y that satisfy the modulo constraints.
 211  *
 212  * This function is inspired by Section 2.5.3
 213  * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
 214  * Model.  Applications to Program Analysis and Optimization".
 215  * However, the implementation only follows the algorithm of that
 216  * section for computing a particular solution and not for computing
 217  * a general homogeneous solution.  The latter is incomplete and
 218  * may remove some valid solutions.
 219  * Instead, we use an adaptation of the algorithm in Section 7 of
 220  * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
 221  * Model: Bringing the Power of Quasi-Polynomials to the Masses".
 222  *
 223  * The input is given as a matrix B = [ c A ] and a vector d.
 224  * Each element of the vector d corresponds to a row in B.
 225  * The output is a lower triangular matrix.
 226  * If no integer vector y satisfies the given constraints then
 227  * a matrix with zero columns is returned.
 228  *
 229  * We first compute a particular solution y_0 to the given set of
 230  * modulo constraints in particular_solution.  If no such solution
 231  * exists, then we return a zero-columned transformation matrix.
 232  * Otherwise, we compute the generic solution to
 233  *
 234  *              A y = 0 mod d
 235  *
 236  * That is we want to compute G such that
 237  *
 238  *              y = G y''
 239  *
 240  * with y'' integer, describes the set of solutions.
 241  *
 242  * We first remove the common factors of each row.
 243  * In particular if gcd(A_i,d_i) != 1, then we divide the whole
 244  * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
 245  * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
 246  * In the later case, we simply drop the row (in both A and d).
 247  *
 248  * If there are no rows left in A, the G is the identity matrix. Otherwise,
 249  * for each row i, we now determine the lattice of integer vectors
 250  * that satisfies this row.  Let U_i be the unimodular extension of the
 251  * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
 252  * The first component of
 253  *
 254  *              y' = U_i y
 255  *
 256  * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
 257  * Then,
 258  *
 259  *              y = U_i^{-1} diag(d_i, 1, ..., 1) y''
 260  *
 261  * for arbitrary integer vectors y''.  That is, y belongs to the lattice
 262  * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
 263  * If there is only one row, then G = L_1.
 264  *
 265  * If there is more than one row left, we need to compute the intersection
 266  * of the lattices.  That is, we need to compute an L such that
 267  *
 268  *              L = L_i L_i'    for all i
 269  *
 270  * with L_i' some integer matrices.  Let A be constructed as follows
 271  *
 272  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 273  *
 274  * and computed the Hermite Normal Form of A = [ H 0 ] U
 275  * Then,
 276  *
 277  *              L_i^{-T} = H U_{1,i}
 278  *
 279  * or
 280  *
 281  *              H^{-T} = L_i U_{1,i}^T
 282  *
 283  * In other words G = L = H^{-T}.
 284  * To ensure that G is lower triangular, we compute and use its Hermite
 285  * normal form.
 286  *
 287  * The affine transformation matrix returned is then
 288  *
 289  *              [  1   0  ]
 290  *              [ y_0  G  ]
 291  *
 292  * as any y = y_0 + G y' with y' integer is a solution to the original
 293  * modulo constraints.
 294  */
 295 struct isl_mat *isl_mat_parameter_compression(struct isl_ctx *ctx,
 296                         struct isl_mat *B, struct isl_vec *d)
 297 {
 298         int i;
 299         struct isl_mat *cst = NULL;
 300         struct isl_mat *T = NULL;
 301         isl_int D;
 302
 303         if (!B || !d)
 304                 goto error;
 305         isl_assert(ctx, B->n_row == d->size, goto error);
 306         cst = particular_solution(ctx, B, d);
 307         if (!cst)
 308                 goto error;
 309         if (cst->n_col == 0) {
 310                 T = isl_mat_alloc(ctx, B->n_col, 0);
 311                 isl_mat_free(ctx, cst);
 312                 isl_mat_free(ctx, B);
 313                 isl_vec_free(ctx, d);
 314                 return T;
 315         }
 316         isl_int_init(D);
 317         /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
 318         for (i = 0; i < B->n_row; ++i) {
 319                 isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
 320                 if (isl_int_is_one(D))
 321                         continue;
 322                 if (isl_int_is_zero(D)) {
 323                         B = isl_mat_drop_rows(ctx, B, i, 1);
 324                         d = isl_vec_cow(ctx, d);
 325                         if (!B || !d)
 326                                 goto error2;
 327                         isl_seq_cpy(d->block.data+i, d->block.data+i+1,
 328                                                         d->size - (i+1));
 329                         d->size--;
 330                         i--;
 331                         continue;
 332                 }
 333                 B = isl_mat_cow(ctx, B);
 334                 if (!B)
 335                         goto error2;
 336                 isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
 337                 isl_int_gcd(D, D, d->block.data[i]);
 338                 d = isl_vec_cow(ctx, d);
 339                 if (!d)
 340                         goto error2;
 341                 isl_int_divexact(d->block.data[i], d->block.data[i], D);
 342         }
 343         isl_int_clear(D);
 344         if (B->n_row == 0)
 345                 T = isl_mat_identity(ctx, B->n_col);
 346         else if (B->n_row == 1)
 347                 T = parameter_compression_1(ctx, B, d);
 348         else
 349                 T = parameter_compression_multi(ctx, B, d);
 350         T = isl_mat_left_hermite(ctx, T, 0, NULL, NULL);
 351         if (!T)
 352                 goto error;
 353         isl_mat_sub_copy(ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
 354         isl_mat_free(ctx, cst);
 355         isl_mat_free(ctx, B);
 356         isl_vec_free(ctx, d);
 357         return T;
 358 error2:
 359         isl_int_clear(D);
 360 error:
 361         isl_mat_free(ctx, cst);
 362         isl_mat_free(ctx, B);
 363         isl_vec_free(ctx, d);
 364         return NULL;
 365 }
 366
 367 /* Given a set of equalities
 368  *
 369  *              M x - c = 0
 370  *
 371  * this function computes unimodular transformation from a lower-dimensional
 372  * space to the original space that bijectively maps the integer points x'
 373  * in the lower-dimensional space to the integer points x in the original
 374  * space that satisfy the equalities.
 375  *
 376  * The input is given as a matrix B = [ -c M ] and the out is a
 377  * matrix that maps [1 x'] to [1 x].
 378  * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 379  *
 380  * First compute the (left) Hermite normal form of M,
 381  *
 382  *              M [U1 U2] = M U = H = [H1 0]
 383  * or
 384  *                            M = H Q = [H1 0] [Q1]
 385  *                                             [Q2]
 386  *
 387  * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
 388  * Define the transformed variables as
 389  *
 390  *              x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
 391  *                          [ x2' ]           [Q2]
 392  *
 393  * The equalities then become
 394  *
 395  *              H1 x1' - c = 0   or   x1' = H1^{-1} c = c'
 396  *
 397  * If any of the c' is non-integer, then the original set has no
 398  * integer solutions (since the x' are a unimodular transformation
 399  * of the x).
 400  * Otherwise, the transformation is given by
 401  *
 402  *              x = U1 H1^{-1} c + U2 x2'
 403  *
 404  * The inverse transformation is simply
 405  *
 406  *              x2' = Q2 x
 407  */
 408 struct isl_mat *isl_mat_variable_compression(struct isl_ctx *ctx,
 409                         struct isl_mat *B, struct isl_mat **T2)
 410 {
 411         int i;
 412         struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
 413         unsigned dim;
 414
 415         if (T2)
 416                 *T2 = NULL;
 417         if (!B)
 418                 goto error;
 419
 420         dim = B->n_col - 1;
 421         H = isl_mat_sub_alloc(ctx, B->row, 0, B->n_row, 1, dim);
 422         H = isl_mat_left_hermite(ctx, H, 0, &U, T2);
 423         if (!H || !U || (T2 && !*T2))
 424                 goto error;
 425         if (T2) {
 426                 *T2 = isl_mat_drop_rows(ctx, *T2, 0, B->n_row);
 427                 *T2 = isl_mat_lin_to_aff(ctx, *T2);
 428                 if (!*T2)
 429                         goto error;
 430         }
 431         C = isl_mat_alloc(ctx, 1+B->n_row, 1);
 432         if (!C)
 433                 goto error;
 434         isl_int_set_si(C->row[0][0], 1);
 435         isl_mat_sub_neg(ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
 436         H1 = isl_mat_sub_alloc(ctx, H->row, 0, H->n_row, 0, H->n_row);
 437         H1 = isl_mat_lin_to_aff(ctx, H1);
 438         TC = isl_mat_inverse_product(ctx, H1, C);
 439         if (!TC)
 440                 goto error;
 441         isl_mat_free(ctx, H);
 442         if (!isl_int_is_one(TC->row[0][0])) {
 443                 for (i = 0; i < B->n_row; ++i) {
 444                         if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
 445                                 isl_mat_free(ctx, B);
 446                                 isl_mat_free(ctx, TC);
 447                                 isl_mat_free(ctx, U);
 448                                 if (T2) {
 449                                         isl_mat_free(ctx, *T2);
 450                                         *T2 = NULL;
 451                                 }
 452                                 return isl_mat_alloc(ctx, 1 + dim, 0);
 453                         }
 454                         isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
 455                 }
 456                 isl_int_set_si(TC->row[0][0], 1);
 457         }
 458         U1 = isl_mat_sub_alloc(ctx, U->row, 0, U->n_row, 0, B->n_row);
 459         U1 = isl_mat_lin_to_aff(ctx, U1);
 460         U2 = isl_mat_sub_alloc(ctx, U->row, 0, U->n_row,
 461                                 B->n_row, U->n_row - B->n_row);
 462         U2 = isl_mat_lin_to_aff(ctx, U2);
 463         isl_mat_free(ctx, U);
 464         TC = isl_mat_product(ctx, U1, TC);
 465         TC = isl_mat_aff_direct_sum(ctx, TC, U2);
 466
 467         isl_mat_free(ctx, B);
 468
 469         return TC;
 470 error:
 471         isl_mat_free(ctx, B);
 472         isl_mat_free(ctx, H);
 473         isl_mat_free(ctx, U);
 474         if (T2) {
 475                 isl_mat_free(ctx, *T2);
 476                 *T2 = NULL;
 477         }
 478         return NULL;
 479 }
 480
 481 /* Use the n equalities of bset to unimodularly transform the
 482  * variables x such that n transformed variables x1' have a constant value
 483  * and rewrite the constraints of bset in terms of the remaining
 484  * transformed variables x2'.  The matrix pointed to by T maps
 485  * the new variables x2' back to the original variables x, while T2
 486  * maps the original variables to the new variables.
 487  */
 488 static struct isl_basic_set *compress_variables(struct isl_ctx *ctx,
 489         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 490 {
 491         struct isl_mat *B, *TC;
 492         unsigned dim;
 493
 494         if (T)
 495                 *T = NULL;
 496         if (T2)
 497                 *T2 = NULL;
 498         if (!bset)
 499                 goto error;
 500         isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
 501         isl_assert(ctx, bset->n_div == 0, goto error);
 502         dim = isl_basic_set_n_dim(bset);
 503         isl_assert(ctx, bset->n_eq <= dim, goto error);
 504         if (bset->n_eq == 0)
 505                 return bset;
 506
 507         B = isl_mat_sub_alloc(ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
 508         TC = isl_mat_variable_compression(ctx, B, T2);
 509         if (!TC)
 510                 goto error;
 511         if (TC->n_col == 0) {
 512                 isl_mat_free(ctx, TC);
 513                 if (T2) {
 514                         isl_mat_free(ctx, *T2);
 515                         *T2 = NULL;
 516                 }
 517                 return isl_basic_set_set_to_empty(bset);
 518         }
 519
 520         bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(ctx, TC) : TC);
 521         if (T)
 522                 *T = TC;
 523         return bset;
 524 error:
 525         isl_basic_set_free(bset);
 526         return NULL;
 527 }
 528
 529 struct isl_basic_set *isl_basic_set_remove_equalities(
 530         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 531 {
 532         if (T)
 533                 *T = NULL;
 534         if (T2)
 535                 *T2 = NULL;
 536         if (!bset)
 537                 return NULL;
 538         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 539         bset = isl_basic_set_gauss(bset, NULL);
 540         if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
 541                 return bset;
 542         bset = compress_variables(bset->ctx, bset, T, T2);
 543         return bset;
 544 error:
 545         isl_basic_set_free(bset);
 546         *T = NULL;
 547         return NULL;
 548 }
 549
 550 /* Check if dimension dim belongs to a residue class
 551  *              i_dim \equiv r mod m
 552  * with m != 1 and if so return m in *modulo and r in *residue.
 553  */
 554 int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
 555         int pos, isl_int *modulo, isl_int *residue)
 556 {
 557         struct isl_ctx *ctx;
 558         struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
 559         unsigned total;
 560         unsigned nparam;
 561
 562         if (!bset || !modulo || !residue)
 563                 return -1;
 564
 565         ctx = bset->ctx;
 566         total = isl_basic_set_total_dim(bset);
 567         nparam = isl_basic_set_n_param(bset);
 568         H = isl_mat_sub_alloc(ctx, bset->eq, 0, bset->n_eq, 1, total);
 569         H = isl_mat_left_hermite(ctx, H, 0, &U, NULL);
 570         if (!H)
 571                 return -1;
 572
 573         isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
 574                         total-bset->n_eq, modulo);
 575         if (isl_int_is_zero(*modulo) || isl_int_is_one(*modulo)) {
 576                 isl_int_set_si(*residue, 0);
 577                 isl_mat_free(ctx, H);
 578                 isl_mat_free(ctx, U);
 579                 return 0;
 580         }
 581
 582         C = isl_mat_alloc(ctx, 1+bset->n_eq, 1);
 583         if (!C)
 584                 goto error;
 585         isl_int_set_si(C->row[0][0], 1);
 586         isl_mat_sub_neg(ctx, C->row+1, bset->eq, bset->n_eq, 0, 0, 1);
 587         H1 = isl_mat_sub_alloc(ctx, H->row, 0, H->n_row, 0, H->n_row);
 588         H1 = isl_mat_lin_to_aff(ctx, H1);
 589         C = isl_mat_inverse_product(ctx, H1, C);
 590         isl_mat_free(ctx, H);
 591         U1 = isl_mat_sub_alloc(ctx, U->row, nparam+pos, 1, 0, bset->n_eq);
 592         U1 = isl_mat_lin_to_aff(ctx, U1);
 593         isl_mat_free(ctx, U);
 594         C = isl_mat_product(ctx, U1, C);
 595         if (!C)
 596                 goto error;
 597         if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
 598                 bset = isl_basic_set_copy(bset);
 599                 bset = isl_basic_set_set_to_empty(bset);
 600                 isl_basic_set_free(bset);
 601                 isl_int_set_si(*modulo, 0);
 602                 isl_int_set_si(*residue, 0);
 603                 return 0;
 604         }
 605         isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
 606         isl_int_fdiv_r(*residue, *residue, *modulo);
 607         isl_mat_free(ctx, C);
 608         return 0;
 609 error:
 610         isl_mat_free(ctx, H);
 611         isl_mat_free(ctx, U);
 612         return -1;
 613 }