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 static struct isl_mat *unimodular_complete(struct isl_ctx *ctx,
 108                                                 struct isl_mat *M, int row)
 109 {
 110         int r;
 111         struct isl_mat *H = NULL, *Q = NULL;
 112
 113         isl_assert(ctx, M->n_row == M->n_col, goto error);
 114         M->n_row = row;
 115         H = isl_mat_left_hermite(ctx, isl_mat_copy(ctx, M), 0, NULL, &Q);
 116         M->n_row = M->n_col;
 117         if (!H)
 118                 goto error;
 119         for (r = 0; r < row; ++r)
 120                 isl_assert(ctx, isl_int_is_one(H->row[r][r]), goto error);
 121         for (r = row; r < M->n_row; ++r)
 122                 isl_seq_cpy(M->row[r], Q->row[r], M->n_col);
 123         isl_mat_free(ctx, H);
 124         isl_mat_free(ctx, Q);
 125         return M;
 126 error:
 127         isl_mat_free(ctx, H);
 128         isl_mat_free(ctx, Q);
 129         isl_mat_free(ctx, M);
 130         return NULL;
 131 }
 132
 133 /* Compute and return the matrix
 134  *
 135  *              U_1^{-1} diag(d_1, 1, ..., 1)
 136  *
 137  * with U_1 the unimodular completion of the first (and only) row of B.
 138  * The columns of this matrix generate the lattice that satisfies
 139  * the single (linear) modulo constraint.
 140  */
 141 static struct isl_mat *parameter_compression_1(struct isl_ctx *ctx,
 142                         struct isl_mat *B, struct isl_vec *d)
 143 {
 144         struct isl_mat *U;
 145
 146         U = isl_mat_alloc(ctx, B->n_col - 1, B->n_col - 1);
 147         if (!U)
 148                 return NULL;
 149         isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
 150         U = unimodular_complete(ctx, U, 1);
 151         U = isl_mat_right_inverse(ctx, U);
 152         if (!U)
 153                 return NULL;
 154         isl_mat_col_mul(U, 0, d->block.data[0], 0);
 155         U = isl_mat_lin_to_aff(ctx, U);
 156         return U;
 157 error:
 158         isl_mat_free(ctx, U);
 159         return NULL;
 160 }
 161
 162 /* Compute a common lattice of solutions to the linear modulo
 163  * constraints specified by B and d.
 164  * See also the documentation of isl_mat_parameter_compression.
 165  * We put the matrix
 166  *
 167  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 168  *
 169  * on a common denominator.  This denominator D is the lcm of modulos d.
 170  * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
 171  * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
 172  * Putting this on the common denominator, we have
 173  * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
 174  */
 175 static struct isl_mat *parameter_compression_multi(struct isl_ctx *ctx,
 176                         struct isl_mat *B, struct isl_vec *d)
 177 {
 178         int i, j, k;
 179         int ok;
 180         isl_int D;
 181         struct isl_mat *A = NULL, *U = NULL;
 182         struct isl_mat *T;
 183         unsigned size;
 184
 185         isl_int_init(D);
 186
 187         isl_vec_lcm(ctx, d, &D);
 188
 189         size = B->n_col - 1;
 190         A = isl_mat_alloc(ctx, size, B->n_row * size);
 191         U = isl_mat_alloc(ctx, size, size);
 192         if (!U || !A)
 193                 goto error;
 194         for (i = 0; i < B->n_row; ++i) {
 195                 isl_seq_cpy(U->row[0], B->row[i] + 1, size);
 196                 U = unimodular_complete(ctx, U, 1);
 197                 if (!U)
 198                         goto error;
 199                 isl_int_divexact(D, D, d->block.data[i]);
 200                 for (k = 0; k < U->n_col; ++k)
 201                         isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
 202                 isl_int_mul(D, D, d->block.data[i]);
 203                 for (j = 1; j < U->n_row; ++j)
 204                         for (k = 0; k < U->n_col; ++k)
 205                                 isl_int_mul(A->row[k][i*size+j],
 206                                                 D, U->row[j][k]);
 207         }
 208         A = isl_mat_left_hermite(ctx, A, 0, NULL, NULL);
 209         T = isl_mat_sub_alloc(ctx, A->row, 0, A->n_row, 0, A->n_row);
 210         T = isl_mat_lin_to_aff(ctx, T);
 211         isl_int_set(T->row[0][0], D);
 212         T = isl_mat_right_inverse(ctx, T);
 213         isl_assert(ctx, isl_int_is_one(T->row[0][0]), goto error);
 214         T = isl_mat_transpose(ctx, T);
 215         isl_mat_free(ctx, A);
 216         isl_mat_free(ctx, U);
 217
 218         isl_int_clear(D);
 219         return T;
 220 error:
 221         isl_mat_free(ctx, A);
 222         isl_mat_free(ctx, U);
 223         isl_int_clear(D);
 224         return NULL;
 225 }
 226
 227 /* Given a set of modulo constraints
 228  *
 229  *              c + A y = 0 mod d
 230  *
 231  * this function returns an affine transformation T,
 232  *
 233  *              y = T y'
 234  *
 235  * that bijectively maps the integer vectors y' to integer
 236  * vectors y that satisfy the modulo constraints.
 237  *
 238  * This function is inspired by Section 2.5.3
 239  * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
 240  * Model.  Applications to Program Analysis and Optimization".
 241  * However, the implementation only follows the algorithm of that
 242  * section for computing a particular solution and not for computing
 243  * a general homogeneous solution.  The latter is incomplete and
 244  * may remove some valid solutions.
 245  * Instead, we use an adaptation of the algorithm in Section 7 of
 246  * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
 247  * Model: Bringing the Power of Quasi-Polynomials to the Masses".
 248  *
 249  * The input is given as a matrix B = [ c A ] and a vector d.
 250  * Each element of the vector d corresponds to a row in B.
 251  * The output is a lower triangular matrix.
 252  * If no integer vector y satisfies the given constraints then
 253  * a matrix with zero columns is returned.
 254  *
 255  * We first compute a particular solution y_0 to the given set of
 256  * modulo constraints in particular_solution.  If no such solution
 257  * exists, then we return a zero-columned transformation matrix.
 258  * Otherwise, we compute the generic solution to
 259  *
 260  *              A y = 0 mod d
 261  *
 262  * That is we want to compute G such that
 263  *
 264  *              y = G y''
 265  *
 266  * with y'' integer, describes the set of solutions.
 267  *
 268  * We first remove the common factors of each row.
 269  * In particular if gcd(A_i,d_i) != 1, then we divide the whole
 270  * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
 271  * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
 272  * In the later case, we simply drop the row (in both A and d).
 273  *
 274  * If there are no rows left in A, the G is the identity matrix. Otherwise,
 275  * for each row i, we now determine the lattice of integer vectors
 276  * that satisfies this row.  Let U_i be the unimodular extension of the
 277  * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
 278  * The first component of
 279  *
 280  *              y' = U_i y
 281  *
 282  * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
 283  * Then,
 284  *
 285  *              y = U_i^{-1} diag(d_i, 1, ..., 1) y''
 286  *
 287  * for arbitrary integer vectors y''.  That is, y belongs to the lattice
 288  * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
 289  * If there is only one row, then G = L_1.
 290  *
 291  * If there is more than one row left, we need to compute the intersection
 292  * of the lattices.  That is, we need to compute an L such that
 293  *
 294  *              L = L_i L_i'    for all i
 295  *
 296  * with L_i' some integer matrices.  Let A be constructed as follows
 297  *
 298  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 299  *
 300  * and computed the Hermite Normal Form of A = [ H 0 ] U
 301  * Then,
 302  *
 303  *              L_i^{-T} = H U_{1,i}
 304  *
 305  * or
 306  *
 307  *              H^{-T} = L_i U_{1,i}^T
 308  *
 309  * In other words G = L = H^{-T}.
 310  * To ensure that G is lower triangular, we compute and use its Hermite
 311  * normal form.
 312  *
 313  * The affine transformation matrix returned is then
 314  *
 315  *              [  1   0  ]
 316  *              [ y_0  G  ]
 317  *
 318  * as any y = y_0 + G y' with y' integer is a solution to the original
 319  * modulo constraints.
 320  */
 321 struct isl_mat *isl_mat_parameter_compression(struct isl_ctx *ctx,
 322                         struct isl_mat *B, struct isl_vec *d)
 323 {
 324         int i;
 325         struct isl_mat *cst = NULL;
 326         struct isl_mat *T = NULL;
 327         isl_int D;
 328
 329         if (!B || !d)
 330                 goto error;
 331         isl_assert(ctx, B->n_row == d->size, goto error);
 332         cst = particular_solution(ctx, B, d);
 333         if (!cst)
 334                 goto error;
 335         if (cst->n_col == 0) {
 336                 T = isl_mat_alloc(ctx, B->n_col, 0);
 337                 isl_mat_free(ctx, cst);
 338                 isl_mat_free(ctx, B);
 339                 isl_vec_free(ctx, d);
 340                 return T;
 341         }
 342         isl_int_init(D);
 343         /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
 344         for (i = 0; i < B->n_row; ++i) {
 345                 isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
 346                 if (isl_int_is_one(D))
 347                         continue;
 348                 if (isl_int_is_zero(D)) {
 349                         B = isl_mat_drop_rows(ctx, B, i, 1);
 350                         d = isl_vec_cow(ctx, d);
 351                         if (!B || !d)
 352                                 goto error2;
 353                         isl_seq_cpy(d->block.data+i, d->block.data+i+1,
 354                                                         d->size - (i+1));
 355                         d->size--;
 356                         i--;
 357                         continue;
 358                 }
 359                 B = isl_mat_cow(ctx, B);
 360                 if (!B)
 361                         goto error2;
 362                 isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
 363                 isl_int_gcd(D, D, d->block.data[i]);
 364                 d = isl_vec_cow(ctx, d);
 365                 if (!d)
 366                         goto error2;
 367                 isl_int_divexact(d->block.data[i], d->block.data[i], D);
 368         }
 369         isl_int_clear(D);
 370         if (B->n_row == 0)
 371                 T = isl_mat_identity(ctx, B->n_col);
 372         else if (B->n_row == 1)
 373                 T = parameter_compression_1(ctx, B, d);
 374         else
 375                 T = parameter_compression_multi(ctx, B, d);
 376         T = isl_mat_left_hermite(ctx, T, 0, NULL, NULL);
 377         if (!T)
 378                 goto error;
 379         isl_mat_sub_copy(ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
 380         isl_mat_free(ctx, cst);
 381         isl_mat_free(ctx, B);
 382         isl_vec_free(ctx, d);
 383         return T;
 384 error2:
 385         isl_int_clear(D);
 386 error:
 387         isl_mat_free(ctx, cst);
 388         isl_mat_free(ctx, B);
 389         isl_vec_free(ctx, d);
 390         return NULL;
 391 }
 392
 393 /* Given a set of equalities
 394  *
 395  *              M x - c = 0
 396  *
 397  * this function computes unimodular transformation from a lower-dimensional
 398  * space to the original space that bijectively maps the integer points x'
 399  * in the lower-dimensional space to the integer points x in the original
 400  * space that satisfy the equalities.
 401  *
 402  * The input is given as a matrix B = [ -c M ] and the out is a
 403  * matrix that maps [1 x'] to [1 x].
 404  * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 405  *
 406  * First compute the (left) Hermite normal form of M,
 407  *
 408  *              M [U1 U2] = M U = H = [H1 0]
 409  * or
 410  *                            M = H Q = [H1 0] [Q1]
 411  *                                             [Q2]
 412  *
 413  * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
 414  * Define the transformed variables as
 415  *
 416  *              x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
 417  *                          [ x2' ]           [Q2]
 418  *
 419  * The equalities then become
 420  *
 421  *              H1 x1' - c = 0   or   x1' = H1^{-1} c = c'
 422  *
 423  * If any of the c' is non-integer, then the original set has no
 424  * integer solutions (since the x' are a unimodular transformation
 425  * of the x).
 426  * Otherwise, the transformation is given by
 427  *
 428  *              x = U1 H1^{-1} c + U2 x2'
 429  *
 430  * The inverse transformation is simply
 431  *
 432  *              x2' = Q2 x
 433  */
 434 struct isl_mat *isl_mat_variable_compression(struct isl_ctx *ctx,
 435                         struct isl_mat *B, struct isl_mat **T2)
 436 {
 437         int i;
 438         struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
 439         unsigned dim;
 440
 441         if (T2)
 442                 *T2 = NULL;
 443         if (!B)
 444                 goto error;
 445
 446         dim = B->n_col - 1;
 447         H = isl_mat_sub_alloc(ctx, B->row, 0, B->n_row, 1, dim);
 448         H = isl_mat_left_hermite(ctx, H, 0, &U, T2);
 449         if (!H || !U || (T2 && !*T2))
 450                 goto error;
 451         if (T2) {
 452                 *T2 = isl_mat_drop_rows(ctx, *T2, 0, B->n_row);
 453                 *T2 = isl_mat_lin_to_aff(ctx, *T2);
 454                 if (!*T2)
 455                         goto error;
 456         }
 457         C = isl_mat_alloc(ctx, 1+B->n_row, 1);
 458         if (!C)
 459                 goto error;
 460         isl_int_set_si(C->row[0][0], 1);
 461         isl_mat_sub_neg(ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
 462         H1 = isl_mat_sub_alloc(ctx, H->row, 0, H->n_row, 0, H->n_row);
 463         H1 = isl_mat_lin_to_aff(ctx, H1);
 464         TC = isl_mat_inverse_product(ctx, H1, C);
 465         if (!TC)
 466                 goto error;
 467         isl_mat_free(ctx, H);
 468         if (!isl_int_is_one(TC->row[0][0])) {
 469                 for (i = 0; i < B->n_row; ++i) {
 470                         if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
 471                                 isl_mat_free(ctx, B);
 472                                 isl_mat_free(ctx, TC);
 473                                 isl_mat_free(ctx, U);
 474                                 if (T2) {
 475                                         isl_mat_free(ctx, *T2);
 476                                         *T2 = NULL;
 477                                 }
 478                                 return isl_mat_alloc(ctx, 1 + dim, 0);
 479                         }
 480                         isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
 481                 }
 482                 isl_int_set_si(TC->row[0][0], 1);
 483         }
 484         U1 = isl_mat_sub_alloc(ctx, U->row, 0, U->n_row, 0, B->n_row);
 485         U1 = isl_mat_lin_to_aff(ctx, U1);
 486         U2 = isl_mat_sub_alloc(ctx, U->row, 0, U->n_row,
 487                                 B->n_row, U->n_row - B->n_row);
 488         U2 = isl_mat_lin_to_aff(ctx, U2);
 489         isl_mat_free(ctx, U);
 490         TC = isl_mat_product(ctx, U1, TC);
 491         TC = isl_mat_aff_direct_sum(ctx, TC, U2);
 492
 493         isl_mat_free(ctx, B);
 494
 495         return TC;
 496 error:
 497         isl_mat_free(ctx, B);
 498         isl_mat_free(ctx, H);
 499         isl_mat_free(ctx, U);
 500         if (T2) {
 501                 isl_mat_free(ctx, *T2);
 502                 *T2 = NULL;
 503         }
 504         return NULL;
 505 }
 506
 507 /* Use the n equalities of bset to unimodularly transform the
 508  * variables x such that n transformed variables x1' have a constant value
 509  * and rewrite the constraints of bset in terms of the remaining
 510  * transformed variables x2'.  The matrix pointed to by T maps
 511  * the new variables x2' back to the original variables x, while T2
 512  * maps the original variables to the new variables.
 513  */
 514 static struct isl_basic_set *compress_variables(struct isl_ctx *ctx,
 515         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 516 {
 517         struct isl_mat *B, *TC;
 518         unsigned dim;
 519
 520         if (T)
 521                 *T = NULL;
 522         if (T2)
 523                 *T2 = NULL;
 524         if (!bset)
 525                 goto error;
 526         isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
 527         isl_assert(ctx, bset->n_div == 0, goto error);
 528         dim = isl_basic_set_n_dim(bset);
 529         isl_assert(ctx, bset->n_eq <= dim, goto error);
 530         if (bset->n_eq == 0)
 531                 return bset;
 532
 533         B = isl_mat_sub_alloc(ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
 534         TC = isl_mat_variable_compression(ctx, B, T2);
 535         if (!TC)
 536                 goto error;
 537         if (TC->n_col == 0) {
 538                 isl_mat_free(ctx, TC);
 539                 if (T2) {
 540                         isl_mat_free(ctx, *T2);
 541                         *T2 = NULL;
 542                 }
 543                 return isl_basic_set_set_to_empty(bset);
 544         }
 545
 546         bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(ctx, TC) : TC);
 547         if (T)
 548                 *T = TC;
 549         return bset;
 550 error:
 551         isl_basic_set_free(bset);
 552         return NULL;
 553 }
 554
 555 struct isl_basic_set *isl_basic_set_remove_equalities(
 556         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 557 {
 558         if (T)
 559                 *T = NULL;
 560         if (T2)
 561                 *T2 = NULL;
 562         if (!bset)
 563                 return NULL;
 564         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 565         bset = isl_basic_set_gauss(bset, NULL);
 566         if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
 567                 return bset;
 568         bset = compress_variables(bset->ctx, bset, T, T2);
 569         return bset;
 570 error:
 571         isl_basic_set_free(bset);
 572         *T = NULL;
 573         return NULL;
 574 }
 575
 576 /* Check if dimension dim belongs to a residue class
 577  *              i_dim \equiv r mod m
 578  * with m != 1 and if so return m in *modulo and r in *residue.
 579  */
 580 int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
 581         int pos, isl_int *modulo, isl_int *residue)
 582 {
 583         struct isl_ctx *ctx;
 584         struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
 585         unsigned total;
 586         unsigned nparam;
 587
 588         if (!bset || !modulo || !residue)
 589                 return -1;
 590
 591         ctx = bset->ctx;
 592         total = isl_basic_set_total_dim(bset);
 593         nparam = isl_basic_set_n_param(bset);
 594         H = isl_mat_sub_alloc(ctx, bset->eq, 0, bset->n_eq, 1, total);
 595         H = isl_mat_left_hermite(ctx, H, 0, &U, NULL);
 596         if (!H)
 597                 return -1;
 598
 599         isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
 600                         total-bset->n_eq, modulo);
 601         if (isl_int_is_zero(*modulo) || isl_int_is_one(*modulo)) {
 602                 isl_int_set_si(*residue, 0);
 603                 isl_mat_free(ctx, H);
 604                 isl_mat_free(ctx, U);
 605                 return 0;
 606         }
 607
 608         C = isl_mat_alloc(ctx, 1+bset->n_eq, 1);
 609         if (!C)
 610                 goto error;
 611         isl_int_set_si(C->row[0][0], 1);
 612         isl_mat_sub_neg(ctx, C->row+1, bset->eq, bset->n_eq, 0, 0, 1);
 613         H1 = isl_mat_sub_alloc(ctx, H->row, 0, H->n_row, 0, H->n_row);
 614         H1 = isl_mat_lin_to_aff(ctx, H1);
 615         C = isl_mat_inverse_product(ctx, H1, C);
 616         isl_mat_free(ctx, H);
 617         U1 = isl_mat_sub_alloc(ctx, U->row, nparam+pos, 1, 0, bset->n_eq);
 618         U1 = isl_mat_lin_to_aff(ctx, U1);
 619         isl_mat_free(ctx, U);
 620         C = isl_mat_product(ctx, U1, C);
 621         if (!C)
 622                 goto error;
 623         if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
 624                 bset = isl_basic_set_copy(bset);
 625                 bset = isl_basic_set_set_to_empty(bset);
 626                 isl_basic_set_free(bset);
 627                 isl_int_set_si(*modulo, 0);
 628                 isl_int_set_si(*residue, 0);
 629                 return 0;
 630         }
 631         isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
 632         isl_int_fdiv_r(*residue, *residue, *modulo);
 633         isl_mat_free(ctx, C);
 634         return 0;
 635 error:
 636         isl_mat_free(ctx, H);
 637         isl_mat_free(ctx, U);
 638         return -1;
 639 }