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_mat *B, struct isl_vec *d)
  53 {
  54         int i, j;
  55         struct isl_mat *M = NULL;
  56         struct isl_mat *C = NULL;
  57         struct isl_mat *U = NULL;
  58         struct isl_mat *H = NULL;
  59         struct isl_mat *cst = NULL;
  60         struct isl_mat *T = NULL;
  61
  62         M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
  63         C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
  64         if (!M || !C)
  65                 goto error;
  66         isl_int_set_si(C->row[0][0], 1);
  67         for (i = 0; i < B->n_row; ++i) {
  68                 isl_seq_clr(M->row[i], B->n_row);
  69                 isl_int_set(M->row[i][i], d->block.data[i]);
  70                 isl_int_neg(C->row[1 + i][0], B->row[i][0]);
  71                 isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
  72                 for (j = 0; j < B->n_col - 1; ++j)
  73                         isl_int_fdiv_r(M->row[i][B->n_row + j],
  74                                         B->row[i][1 + j], M->row[i][i]);
  75         }
  76         M = isl_mat_left_hermite(M, 0, &U, NULL);
  77         if (!M || !U)
  78                 goto error;
  79         H = isl_mat_sub_alloc(B->ctx, M->row, 0, B->n_row, 0, B->n_row);
  80         H = isl_mat_lin_to_aff(H);
  81         C = isl_mat_inverse_product(H, C);
  82         if (!C)
  83                 goto error;
  84         for (i = 0; i < B->n_row; ++i) {
  85                 if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
  86                         break;
  87                 isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
  88         }
  89         if (i < B->n_row)
  90                 cst = isl_mat_alloc(B->ctx, B->n_row, 0);
  91         else
  92                 cst = isl_mat_sub_alloc(C->ctx, C->row, 1, B->n_row, 0, 1);
  93         T = isl_mat_sub_alloc(U->ctx, U->row, B->n_row, B->n_col - 1, 0, B->n_row);
  94         cst = isl_mat_product(T, cst);
  95         isl_mat_free(M);
  96         isl_mat_free(C);
  97         isl_mat_free(U);
  98         return cst;
  99 error:
 100         isl_mat_free(M);
 101         isl_mat_free(C);
 102         isl_mat_free(U);
 103         return NULL;
 104 }
 105
 106 /* Compute and return the matrix
 107  *
 108  *              U_1^{-1} diag(d_1, 1, ..., 1)
 109  *
 110  * with U_1 the unimodular completion of the first (and only) row of B.
 111  * The columns of this matrix generate the lattice that satisfies
 112  * the single (linear) modulo constraint.
 113  */
 114 static struct isl_mat *parameter_compression_1(
 115                         struct isl_mat *B, struct isl_vec *d)
 116 {
 117         struct isl_mat *U;
 118
 119         U = isl_mat_alloc(B->ctx, B->n_col - 1, B->n_col - 1);
 120         if (!U)
 121                 return NULL;
 122         isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
 123         U = isl_mat_unimodular_complete(U, 1);
 124         U = isl_mat_right_inverse(U);
 125         if (!U)
 126                 return NULL;
 127         isl_mat_col_mul(U, 0, d->block.data[0], 0);
 128         U = isl_mat_lin_to_aff(U);
 129         return U;
 130 error:
 131         isl_mat_free(U);
 132         return NULL;
 133 }
 134
 135 /* Compute a common lattice of solutions to the linear modulo
 136  * constraints specified by B and d.
 137  * See also the documentation of isl_mat_parameter_compression.
 138  * We put the matrix
 139  *
 140  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 141  *
 142  * on a common denominator.  This denominator D is the lcm of modulos d.
 143  * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
 144  * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
 145  * Putting this on the common denominator, we have
 146  * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
 147  */
 148 static struct isl_mat *parameter_compression_multi(
 149                         struct isl_mat *B, struct isl_vec *d)
 150 {
 151         int i, j, k;
 152         isl_int D;
 153         struct isl_mat *A = NULL, *U = NULL;
 154         struct isl_mat *T;
 155         unsigned size;
 156
 157         isl_int_init(D);
 158
 159         isl_vec_lcm(d, &D);
 160
 161         size = B->n_col - 1;
 162         A = isl_mat_alloc(B->ctx, size, B->n_row * size);
 163         U = isl_mat_alloc(B->ctx, size, size);
 164         if (!U || !A)
 165                 goto error;
 166         for (i = 0; i < B->n_row; ++i) {
 167                 isl_seq_cpy(U->row[0], B->row[i] + 1, size);
 168                 U = isl_mat_unimodular_complete(U, 1);
 169                 if (!U)
 170                         goto error;
 171                 isl_int_divexact(D, D, d->block.data[i]);
 172                 for (k = 0; k < U->n_col; ++k)
 173                         isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
 174                 isl_int_mul(D, D, d->block.data[i]);
 175                 for (j = 1; j < U->n_row; ++j)
 176                         for (k = 0; k < U->n_col; ++k)
 177                                 isl_int_mul(A->row[k][i*size+j],
 178                                                 D, U->row[j][k]);
 179         }
 180         A = isl_mat_left_hermite(A, 0, NULL, NULL);
 181         T = isl_mat_sub_alloc(A->ctx, A->row, 0, A->n_row, 0, A->n_row);
 182         T = isl_mat_lin_to_aff(T);
 183         isl_int_set(T->row[0][0], D);
 184         T = isl_mat_right_inverse(T);
 185         isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
 186         T = isl_mat_transpose(T);
 187         isl_mat_free(A);
 188         isl_mat_free(U);
 189
 190         isl_int_clear(D);
 191         return T;
 192 error:
 193         isl_mat_free(A);
 194         isl_mat_free(U);
 195         isl_int_clear(D);
 196         return NULL;
 197 }
 198
 199 /* Given a set of modulo constraints
 200  *
 201  *              c + A y = 0 mod d
 202  *
 203  * this function returns an affine transformation T,
 204  *
 205  *              y = T y'
 206  *
 207  * that bijectively maps the integer vectors y' to integer
 208  * vectors y that satisfy the modulo constraints.
 209  *
 210  * This function is inspired by Section 2.5.3
 211  * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
 212  * Model.  Applications to Program Analysis and Optimization".
 213  * However, the implementation only follows the algorithm of that
 214  * section for computing a particular solution and not for computing
 215  * a general homogeneous solution.  The latter is incomplete and
 216  * may remove some valid solutions.
 217  * Instead, we use an adaptation of the algorithm in Section 7 of
 218  * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
 219  * Model: Bringing the Power of Quasi-Polynomials to the Masses".
 220  *
 221  * The input is given as a matrix B = [ c A ] and a vector d.
 222  * Each element of the vector d corresponds to a row in B.
 223  * The output is a lower triangular matrix.
 224  * If no integer vector y satisfies the given constraints then
 225  * a matrix with zero columns is returned.
 226  *
 227  * We first compute a particular solution y_0 to the given set of
 228  * modulo constraints in particular_solution.  If no such solution
 229  * exists, then we return a zero-columned transformation matrix.
 230  * Otherwise, we compute the generic solution to
 231  *
 232  *              A y = 0 mod d
 233  *
 234  * That is we want to compute G such that
 235  *
 236  *              y = G y''
 237  *
 238  * with y'' integer, describes the set of solutions.
 239  *
 240  * We first remove the common factors of each row.
 241  * In particular if gcd(A_i,d_i) != 1, then we divide the whole
 242  * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
 243  * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
 244  * In the later case, we simply drop the row (in both A and d).
 245  *
 246  * If there are no rows left in A, the G is the identity matrix. Otherwise,
 247  * for each row i, we now determine the lattice of integer vectors
 248  * that satisfies this row.  Let U_i be the unimodular extension of the
 249  * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
 250  * The first component of
 251  *
 252  *              y' = U_i y
 253  *
 254  * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
 255  * Then,
 256  *
 257  *              y = U_i^{-1} diag(d_i, 1, ..., 1) y''
 258  *
 259  * for arbitrary integer vectors y''.  That is, y belongs to the lattice
 260  * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
 261  * If there is only one row, then G = L_1.
 262  *
 263  * If there is more than one row left, we need to compute the intersection
 264  * of the lattices.  That is, we need to compute an L such that
 265  *
 266  *              L = L_i L_i'    for all i
 267  *
 268  * with L_i' some integer matrices.  Let A be constructed as follows
 269  *
 270  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 271  *
 272  * and computed the Hermite Normal Form of A = [ H 0 ] U
 273  * Then,
 274  *
 275  *              L_i^{-T} = H U_{1,i}
 276  *
 277  * or
 278  *
 279  *              H^{-T} = L_i U_{1,i}^T
 280  *
 281  * In other words G = L = H^{-T}.
 282  * To ensure that G is lower triangular, we compute and use its Hermite
 283  * normal form.
 284  *
 285  * The affine transformation matrix returned is then
 286  *
 287  *              [  1   0  ]
 288  *              [ y_0  G  ]
 289  *
 290  * as any y = y_0 + G y' with y' integer is a solution to the original
 291  * modulo constraints.
 292  */
 293 struct isl_mat *isl_mat_parameter_compression(
 294                         struct isl_mat *B, struct isl_vec *d)
 295 {
 296         int i;
 297         struct isl_mat *cst = NULL;
 298         struct isl_mat *T = NULL;
 299         isl_int D;
 300
 301         if (!B || !d)
 302                 goto error;
 303         isl_assert(B->ctx, B->n_row == d->size, goto error);
 304         cst = particular_solution(B, d);
 305         if (!cst)
 306                 goto error;
 307         if (cst->n_col == 0) {
 308                 T = isl_mat_alloc(B->ctx, B->n_col, 0);
 309                 isl_mat_free(cst);
 310                 isl_mat_free(B);
 311                 isl_vec_free(d);
 312                 return T;
 313         }
 314         isl_int_init(D);
 315         /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
 316         for (i = 0; i < B->n_row; ++i) {
 317                 isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
 318                 if (isl_int_is_one(D))
 319                         continue;
 320                 if (isl_int_is_zero(D)) {
 321                         B = isl_mat_drop_rows(B, i, 1);
 322                         d = isl_vec_cow(d);
 323                         if (!B || !d)
 324                                 goto error2;
 325                         isl_seq_cpy(d->block.data+i, d->block.data+i+1,
 326                                                         d->size - (i+1));
 327                         d->size--;
 328                         i--;
 329                         continue;
 330                 }
 331                 B = isl_mat_cow(B);
 332                 if (!B)
 333                         goto error2;
 334                 isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
 335                 isl_int_gcd(D, D, d->block.data[i]);
 336                 d = isl_vec_cow(d);
 337                 if (!d)
 338                         goto error2;
 339                 isl_int_divexact(d->block.data[i], d->block.data[i], D);
 340         }
 341         isl_int_clear(D);
 342         if (B->n_row == 0)
 343                 T = isl_mat_identity(B->ctx, B->n_col);
 344         else if (B->n_row == 1)
 345                 T = parameter_compression_1(B, d);
 346         else
 347                 T = parameter_compression_multi(B, d);
 348         T = isl_mat_left_hermite(T, 0, NULL, NULL);
 349         if (!T)
 350                 goto error;
 351         isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
 352         isl_mat_free(cst);
 353         isl_mat_free(B);
 354         isl_vec_free(d);
 355         return T;
 356 error2:
 357         isl_int_clear(D);
 358 error:
 359         isl_mat_free(cst);
 360         isl_mat_free(B);
 361         isl_vec_free(d);
 362         return NULL;
 363 }
 364
 365 /* Given a set of equalities
 366  *
 367  *              M x - c = 0
 368  *
 369  * this function computes unimodular transformation from a lower-dimensional
 370  * space to the original space that bijectively maps the integer points x'
 371  * in the lower-dimensional space to the integer points x in the original
 372  * space that satisfy the equalities.
 373  *
 374  * The input is given as a matrix B = [ -c M ] and the out is a
 375  * matrix that maps [1 x'] to [1 x].
 376  * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 377  *
 378  * First compute the (left) Hermite normal form of M,
 379  *
 380  *              M [U1 U2] = M U = H = [H1 0]
 381  * or
 382  *                            M = H Q = [H1 0] [Q1]
 383  *                                             [Q2]
 384  *
 385  * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
 386  * Define the transformed variables as
 387  *
 388  *              x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
 389  *                          [ x2' ]           [Q2]
 390  *
 391  * The equalities then become
 392  *
 393  *              H1 x1' - c = 0   or   x1' = H1^{-1} c = c'
 394  *
 395  * If any of the c' is non-integer, then the original set has no
 396  * integer solutions (since the x' are a unimodular transformation
 397  * of the x).
 398  * Otherwise, the transformation is given by
 399  *
 400  *              x = U1 H1^{-1} c + U2 x2'
 401  *
 402  * The inverse transformation is simply
 403  *
 404  *              x2' = Q2 x
 405  */
 406 struct isl_mat *isl_mat_variable_compression(struct isl_mat *B,
 407         struct isl_mat **T2)
 408 {
 409         int i;
 410         struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
 411         unsigned dim;
 412
 413         if (T2)
 414                 *T2 = NULL;
 415         if (!B)
 416                 goto error;
 417
 418         dim = B->n_col - 1;
 419         H = isl_mat_sub_alloc(B->ctx, B->row, 0, B->n_row, 1, dim);
 420         H = isl_mat_left_hermite(H, 0, &U, T2);
 421         if (!H || !U || (T2 && !*T2))
 422                 goto error;
 423         if (T2) {
 424                 *T2 = isl_mat_drop_rows(*T2, 0, B->n_row);
 425                 *T2 = isl_mat_lin_to_aff(*T2);
 426                 if (!*T2)
 427                         goto error;
 428         }
 429         C = isl_mat_alloc(B->ctx, 1+B->n_row, 1);
 430         if (!C)
 431                 goto error;
 432         isl_int_set_si(C->row[0][0], 1);
 433         isl_mat_sub_neg(C->ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
 434         H1 = isl_mat_sub_alloc(H->ctx, H->row, 0, H->n_row, 0, H->n_row);
 435         H1 = isl_mat_lin_to_aff(H1);
 436         TC = isl_mat_inverse_product(H1, C);
 437         if (!TC)
 438                 goto error;
 439         isl_mat_free(H);
 440         if (!isl_int_is_one(TC->row[0][0])) {
 441                 for (i = 0; i < B->n_row; ++i) {
 442                         if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
 443                                 struct isl_ctx *ctx = B->ctx;
 444                                 isl_mat_free(B);
 445                                 isl_mat_free(TC);
 446                                 isl_mat_free(U);
 447                                 if (T2) {
 448                                         isl_mat_free(*T2);
 449                                         *T2 = NULL;
 450                                 }
 451                                 return isl_mat_alloc(ctx, 1 + dim, 0);
 452                         }
 453                         isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
 454                 }
 455                 isl_int_set_si(TC->row[0][0], 1);
 456         }
 457         U1 = isl_mat_sub_alloc(U->ctx, U->row, 0, U->n_row, 0, B->n_row);
 458         U1 = isl_mat_lin_to_aff(U1);
 459         U2 = isl_mat_sub_alloc(U->ctx, U->row, 0, U->n_row,
 460                                 B->n_row, U->n_row - B->n_row);
 461         U2 = isl_mat_lin_to_aff(U2);
 462         isl_mat_free(U);
 463         TC = isl_mat_product(U1, TC);
 464         TC = isl_mat_aff_direct_sum(TC, U2);
 465
 466         isl_mat_free(B);
 467
 468         return TC;
 469 error:
 470         isl_mat_free(B);
 471         isl_mat_free(H);
 472         isl_mat_free(U);
 473         if (T2) {
 474                 isl_mat_free(*T2);
 475                 *T2 = NULL;
 476         }
 477         return NULL;
 478 }
 479
 480 /* Use the n equalities of bset to unimodularly transform the
 481  * variables x such that n transformed variables x1' have a constant value
 482  * and rewrite the constraints of bset in terms of the remaining
 483  * transformed variables x2'.  The matrix pointed to by T maps
 484  * the new variables x2' back to the original variables x, while T2
 485  * maps the original variables to the new variables.
 486  */
 487 static struct isl_basic_set *compress_variables(
 488         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 489 {
 490         struct isl_mat *B, *TC;
 491         unsigned dim;
 492
 493         if (T)
 494                 *T = NULL;
 495         if (T2)
 496                 *T2 = NULL;
 497         if (!bset)
 498                 goto error;
 499         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 500         isl_assert(bset->ctx, bset->n_div == 0, goto error);
 501         dim = isl_basic_set_n_dim(bset);
 502         isl_assert(bset->ctx, bset->n_eq <= dim, goto error);
 503         if (bset->n_eq == 0)
 504                 return bset;
 505
 506         B = isl_mat_sub_alloc(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
 507         TC = isl_mat_variable_compression(B, T2);
 508         if (!TC)
 509                 goto error;
 510         if (TC->n_col == 0) {
 511                 isl_mat_free(TC);
 512                 if (T2) {
 513                         isl_mat_free(*T2);
 514                         *T2 = NULL;
 515                 }
 516                 return isl_basic_set_set_to_empty(bset);
 517         }
 518
 519         bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(TC) : TC);
 520         if (T)
 521                 *T = TC;
 522         return bset;
 523 error:
 524         isl_basic_set_free(bset);
 525         return NULL;
 526 }
 527
 528 struct isl_basic_set *isl_basic_set_remove_equalities(
 529         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 530 {
 531         if (T)
 532                 *T = NULL;
 533         if (T2)
 534                 *T2 = NULL;
 535         if (!bset)
 536                 return NULL;
 537         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 538         bset = isl_basic_set_gauss(bset, NULL);
 539         if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
 540                 return bset;
 541         bset = compress_variables(bset, T, T2);
 542         return bset;
 543 error:
 544         isl_basic_set_free(bset);
 545         *T = NULL;
 546         return NULL;
 547 }
 548
 549 /* Check if dimension dim belongs to a residue class
 550  *              i_dim \equiv r mod m
 551  * with m != 1 and if so return m in *modulo and r in *residue.
 552  * As a special case, when i_dim has a fixed value v, then
 553  * *modulo is set to 0 and *residue to v.
 554  *
 555  * If i_dim does not belong to such a residue class, then *modulo
 556  * is set to 1 and *residue is set to 0.
 557  */
 558 int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
 559         int pos, isl_int *modulo, isl_int *residue)
 560 {
 561         struct isl_ctx *ctx;
 562         struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
 563         unsigned total;
 564         unsigned nparam;
 565
 566         if (!bset || !modulo || !residue)
 567                 return -1;
 568
 569         if (isl_basic_set_fast_dim_is_fixed(bset, pos, residue)) {
 570                 isl_int_set_si(*modulo, 0);
 571                 return 0;
 572         }
 573
 574         ctx = bset->ctx;
 575         total = isl_basic_set_total_dim(bset);
 576         nparam = isl_basic_set_n_param(bset);
 577         H = isl_mat_sub_alloc(bset->ctx, bset->eq, 0, bset->n_eq, 1, total);
 578         H = isl_mat_left_hermite(H, 0, &U, NULL);
 579         if (!H)
 580                 return -1;
 581
 582         isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
 583                         total-bset->n_eq, modulo);
 584         if (isl_int_is_zero(*modulo))
 585                 isl_int_set_si(*modulo, 1);
 586         if (isl_int_is_one(*modulo)) {
 587                 isl_int_set_si(*residue, 0);
 588                 isl_mat_free(H);
 589                 isl_mat_free(U);
 590                 return 0;
 591         }
 592
 593         C = isl_mat_alloc(bset->ctx, 1+bset->n_eq, 1);
 594         if (!C)
 595                 goto error;
 596         isl_int_set_si(C->row[0][0], 1);
 597         isl_mat_sub_neg(C->ctx, C->row+1, bset->eq, bset->n_eq, 0, 0, 1);
 598         H1 = isl_mat_sub_alloc(H->ctx, H->row, 0, H->n_row, 0, H->n_row);
 599         H1 = isl_mat_lin_to_aff(H1);
 600         C = isl_mat_inverse_product(H1, C);
 601         isl_mat_free(H);
 602         U1 = isl_mat_sub_alloc(U->ctx, U->row, nparam+pos, 1, 0, bset->n_eq);
 603         U1 = isl_mat_lin_to_aff(U1);
 604         isl_mat_free(U);
 605         C = isl_mat_product(U1, C);
 606         if (!C)
 607                 goto error;
 608         if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
 609                 bset = isl_basic_set_copy(bset);
 610                 bset = isl_basic_set_set_to_empty(bset);
 611                 isl_basic_set_free(bset);
 612                 isl_int_set_si(*modulo, 1);
 613                 isl_int_set_si(*residue, 0);
 614                 return 0;
 615         }
 616         isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
 617         isl_int_fdiv_r(*residue, *residue, *modulo);
 618         isl_mat_free(C);
 619         return 0;
 620 error:
 621         isl_mat_free(H);
 622         isl_mat_free(U);
 623         return -1;
 624 }