isl_equalities.c

   1 /*
   2  * Copyright 2008-2009 Katholieke Universiteit Leuven
   3  *
   4  * Use of this software is governed by the MIT license
   5  *
   6  * Written by Sven Verdoolaege, K.U.Leuven, Departement
   7  * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
   8  */
   9
  10 #include <isl_mat_private.h>
  11 #include <isl/seq.h>
  12 #include "isl_map_private.h"
  13 #include "isl_equalities.h"
  14
  15 /* Given a set of modulo constraints
  16  *
  17  *              c + A y = 0 mod d
  18  *
  19  * this function computes a particular solution y_0
  20  *
  21  * The input is given as a matrix B = [ c A ] and a vector d.
  22  *
  23  * The output is matrix containing the solution y_0 or
  24  * a zero-column matrix if the constraints admit no integer solution.
  25  *
  26  * The given set of constrains is equivalent to
  27  *
  28  *              c + A y = -D x
  29  *
  30  * with D = diag d and x a fresh set of variables.
  31  * Reducing both c and A modulo d does not change the
  32  * value of y in the solution and may lead to smaller coefficients.
  33  * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
  34  * Then
  35  *                [ x ]
  36  *              M [ y ] = - c
  37  * and so
  38  *                             [ x ]
  39  *              [ H 0 ] U^{-1} [ y ] = - c
  40  * Let
  41  *              [ A ]          [ x ]
  42  *              [ B ] = U^{-1} [ y ]
  43  * then
  44  *              H A + 0 B = -c
  45  *
  46  * so B may be chosen arbitrarily, e.g., B = 0, and then
  47  *
  48  *                     [ x ] = [ -c ]
  49  *              U^{-1} [ y ] = [  0 ]
  50  * or
  51  *              [ x ]     [ -c ]
  52  *              [ y ] = U [  0 ]
  53  * specifically,
  54  *
  55  *              y = U_{2,1} (-c)
  56  *
  57  * If any of the coordinates of this y are non-integer
  58  * then the constraints admit no integer solution and
  59  * a zero-column matrix is returned.
  60  */
  61 static struct isl_mat *particular_solution(struct isl_mat *B, struct isl_vec *d)
  62 {
  63         int i, j;
  64         struct isl_mat *M = NULL;
  65         struct isl_mat *C = NULL;
  66         struct isl_mat *U = NULL;
  67         struct isl_mat *H = NULL;
  68         struct isl_mat *cst = NULL;
  69         struct isl_mat *T = NULL;
  70
  71         M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
  72         C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
  73         if (!M || !C)
  74                 goto error;
  75         isl_int_set_si(C->row[0][0], 1);
  76         for (i = 0; i < B->n_row; ++i) {
  77                 isl_seq_clr(M->row[i], B->n_row);
  78                 isl_int_set(M->row[i][i], d->block.data[i]);
  79                 isl_int_neg(C->row[1 + i][0], B->row[i][0]);
  80                 isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
  81                 for (j = 0; j < B->n_col - 1; ++j)
  82                         isl_int_fdiv_r(M->row[i][B->n_row + j],
  83                                         B->row[i][1 + j], M->row[i][i]);
  84         }
  85         M = isl_mat_left_hermite(M, 0, &U, NULL);
  86         if (!M || !U)
  87                 goto error;
  88         H = isl_mat_sub_alloc(M, 0, B->n_row, 0, B->n_row);
  89         H = isl_mat_lin_to_aff(H);
  90         C = isl_mat_inverse_product(H, C);
  91         if (!C)
  92                 goto error;
  93         for (i = 0; i < B->n_row; ++i) {
  94                 if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
  95                         break;
  96                 isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
  97         }
  98         if (i < B->n_row)
  99                 cst = isl_mat_alloc(B->ctx, B->n_row, 0);
 100         else
 101                 cst = isl_mat_sub_alloc(C, 1, B->n_row, 0, 1);
 102         T = isl_mat_sub_alloc(U, B->n_row, B->n_col - 1, 0, B->n_row);
 103         cst = isl_mat_product(T, cst);
 104         isl_mat_free(M);
 105         isl_mat_free(C);
 106         isl_mat_free(U);
 107         return cst;
 108 error:
 109         isl_mat_free(M);
 110         isl_mat_free(C);
 111         isl_mat_free(U);
 112         return NULL;
 113 }
 114
 115 /* Compute and return the matrix
 116  *
 117  *              U_1^{-1} diag(d_1, 1, ..., 1)
 118  *
 119  * with U_1 the unimodular completion of the first (and only) row of B.
 120  * The columns of this matrix generate the lattice that satisfies
 121  * the single (linear) modulo constraint.
 122  */
 123 static struct isl_mat *parameter_compression_1(
 124                         struct isl_mat *B, struct isl_vec *d)
 125 {
 126         struct isl_mat *U;
 127
 128         U = isl_mat_alloc(B->ctx, B->n_col - 1, B->n_col - 1);
 129         if (!U)
 130                 return NULL;
 131         isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
 132         U = isl_mat_unimodular_complete(U, 1);
 133         U = isl_mat_right_inverse(U);
 134         if (!U)
 135                 return NULL;
 136         isl_mat_col_mul(U, 0, d->block.data[0], 0);
 137         U = isl_mat_lin_to_aff(U);
 138         return U;
 139 }
 140
 141 /* Compute a common lattice of solutions to the linear modulo
 142  * constraints specified by B and d.
 143  * See also the documentation of isl_mat_parameter_compression.
 144  * We put the matrix
 145  *
 146  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 147  *
 148  * on a common denominator.  This denominator D is the lcm of modulos d.
 149  * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
 150  * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
 151  * Putting this on the common denominator, we have
 152  * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
 153  */
 154 static struct isl_mat *parameter_compression_multi(
 155                         struct isl_mat *B, struct isl_vec *d)
 156 {
 157         int i, j, k;
 158         isl_int D;
 159         struct isl_mat *A = NULL, *U = NULL;
 160         struct isl_mat *T;
 161         unsigned size;
 162
 163         isl_int_init(D);
 164
 165         isl_vec_lcm(d, &D);
 166
 167         size = B->n_col - 1;
 168         A = isl_mat_alloc(B->ctx, size, B->n_row * size);
 169         U = isl_mat_alloc(B->ctx, size, size);
 170         if (!U || !A)
 171                 goto error;
 172         for (i = 0; i < B->n_row; ++i) {
 173                 isl_seq_cpy(U->row[0], B->row[i] + 1, size);
 174                 U = isl_mat_unimodular_complete(U, 1);
 175                 if (!U)
 176                         goto error;
 177                 isl_int_divexact(D, D, d->block.data[i]);
 178                 for (k = 0; k < U->n_col; ++k)
 179                         isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
 180                 isl_int_mul(D, D, d->block.data[i]);
 181                 for (j = 1; j < U->n_row; ++j)
 182                         for (k = 0; k < U->n_col; ++k)
 183                                 isl_int_mul(A->row[k][i*size+j],
 184                                                 D, U->row[j][k]);
 185         }
 186         A = isl_mat_left_hermite(A, 0, NULL, NULL);
 187         T = isl_mat_sub_alloc(A, 0, A->n_row, 0, A->n_row);
 188         T = isl_mat_lin_to_aff(T);
 189         if (!T)
 190                 goto error;
 191         isl_int_set(T->row[0][0], D);
 192         T = isl_mat_right_inverse(T);
 193         if (!T)
 194                 goto error;
 195         isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
 196         T = isl_mat_transpose(T);
 197         isl_mat_free(A);
 198         isl_mat_free(U);
 199
 200         isl_int_clear(D);
 201         return T;
 202 error:
 203         isl_mat_free(A);
 204         isl_mat_free(U);
 205         isl_int_clear(D);
 206         return NULL;
 207 }
 208
 209 /* Given a set of modulo constraints
 210  *
 211  *              c + A y = 0 mod d
 212  *
 213  * this function returns an affine transformation T,
 214  *
 215  *              y = T y'
 216  *
 217  * that bijectively maps the integer vectors y' to integer
 218  * vectors y that satisfy the modulo constraints.
 219  *
 220  * This function is inspired by Section 2.5.3
 221  * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
 222  * Model.  Applications to Program Analysis and Optimization".
 223  * However, the implementation only follows the algorithm of that
 224  * section for computing a particular solution and not for computing
 225  * a general homogeneous solution.  The latter is incomplete and
 226  * may remove some valid solutions.
 227  * Instead, we use an adaptation of the algorithm in Section 7 of
 228  * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
 229  * Model: Bringing the Power of Quasi-Polynomials to the Masses".
 230  *
 231  * The input is given as a matrix B = [ c A ] and a vector d.
 232  * Each element of the vector d corresponds to a row in B.
 233  * The output is a lower triangular matrix.
 234  * If no integer vector y satisfies the given constraints then
 235  * a matrix with zero columns is returned.
 236  *
 237  * We first compute a particular solution y_0 to the given set of
 238  * modulo constraints in particular_solution.  If no such solution
 239  * exists, then we return a zero-columned transformation matrix.
 240  * Otherwise, we compute the generic solution to
 241  *
 242  *              A y = 0 mod d
 243  *
 244  * That is we want to compute G such that
 245  *
 246  *              y = G y''
 247  *
 248  * with y'' integer, describes the set of solutions.
 249  *
 250  * We first remove the common factors of each row.
 251  * In particular if gcd(A_i,d_i) != 1, then we divide the whole
 252  * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
 253  * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
 254  * In the later case, we simply drop the row (in both A and d).
 255  *
 256  * If there are no rows left in A, then G is the identity matrix. Otherwise,
 257  * for each row i, we now determine the lattice of integer vectors
 258  * that satisfies this row.  Let U_i be the unimodular extension of the
 259  * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
 260  * The first component of
 261  *
 262  *              y' = U_i y
 263  *
 264  * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
 265  * Then,
 266  *
 267  *              y = U_i^{-1} diag(d_i, 1, ..., 1) y''
 268  *
 269  * for arbitrary integer vectors y''.  That is, y belongs to the lattice
 270  * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
 271  * If there is only one row, then G = L_1.
 272  *
 273  * If there is more than one row left, we need to compute the intersection
 274  * of the lattices.  That is, we need to compute an L such that
 275  *
 276  *              L = L_i L_i'    for all i
 277  *
 278  * with L_i' some integer matrices.  Let A be constructed as follows
 279  *
 280  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 281  *
 282  * and computed the Hermite Normal Form of A = [ H 0 ] U
 283  * Then,
 284  *
 285  *              L_i^{-T} = H U_{1,i}
 286  *
 287  * or
 288  *
 289  *              H^{-T} = L_i U_{1,i}^T
 290  *
 291  * In other words G = L = H^{-T}.
 292  * To ensure that G is lower triangular, we compute and use its Hermite
 293  * normal form.
 294  *
 295  * The affine transformation matrix returned is then
 296  *
 297  *              [  1   0  ]
 298  *              [ y_0  G  ]
 299  *
 300  * as any y = y_0 + G y' with y' integer is a solution to the original
 301  * modulo constraints.
 302  */
 303 struct isl_mat *isl_mat_parameter_compression(
 304                         struct isl_mat *B, struct isl_vec *d)
 305 {
 306         int i;
 307         struct isl_mat *cst = NULL;
 308         struct isl_mat *T = NULL;
 309         isl_int D;
 310
 311         if (!B || !d)
 312                 goto error;
 313         isl_assert(B->ctx, B->n_row == d->size, goto error);
 314         cst = particular_solution(B, d);
 315         if (!cst)
 316                 goto error;
 317         if (cst->n_col == 0) {
 318                 T = isl_mat_alloc(B->ctx, B->n_col, 0);
 319                 isl_mat_free(cst);
 320                 isl_mat_free(B);
 321                 isl_vec_free(d);
 322                 return T;
 323         }
 324         isl_int_init(D);
 325         /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
 326         for (i = 0; i < B->n_row; ++i) {
 327                 isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
 328                 if (isl_int_is_one(D))
 329                         continue;
 330                 if (isl_int_is_zero(D)) {
 331                         B = isl_mat_drop_rows(B, i, 1);
 332                         d = isl_vec_cow(d);
 333                         if (!B || !d)
 334                                 goto error2;
 335                         isl_seq_cpy(d->block.data+i, d->block.data+i+1,
 336                                                         d->size - (i+1));
 337                         d->size--;
 338                         i--;
 339                         continue;
 340                 }
 341                 B = isl_mat_cow(B);
 342                 if (!B)
 343                         goto error2;
 344                 isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
 345                 isl_int_gcd(D, D, d->block.data[i]);
 346                 d = isl_vec_cow(d);
 347                 if (!d)
 348                         goto error2;
 349                 isl_int_divexact(d->block.data[i], d->block.data[i], D);
 350         }
 351         isl_int_clear(D);
 352         if (B->n_row == 0)
 353                 T = isl_mat_identity(B->ctx, B->n_col);
 354         else if (B->n_row == 1)
 355                 T = parameter_compression_1(B, d);
 356         else
 357                 T = parameter_compression_multi(B, d);
 358         T = isl_mat_left_hermite(T, 0, NULL, NULL);
 359         if (!T)
 360                 goto error;
 361         isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
 362         isl_mat_free(cst);
 363         isl_mat_free(B);
 364         isl_vec_free(d);
 365         return T;
 366 error2:
 367         isl_int_clear(D);
 368 error:
 369         isl_mat_free(cst);
 370         isl_mat_free(B);
 371         isl_vec_free(d);
 372         return NULL;
 373 }
 374
 375 /* Given a set of equalities
 376  *
 377  *              M x - c = 0
 378  *
 379  * this function computes a unimodular transformation from a lower-dimensional
 380  * space to the original space that bijectively maps the integer points x'
 381  * in the lower-dimensional space to the integer points x in the original
 382  * space that satisfy the equalities.
 383  *
 384  * The input is given as a matrix B = [ -c M ] and the output is a
 385  * matrix that maps [1 x'] to [1 x].
 386  * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 387  *
 388  * First compute the (left) Hermite normal form of M,
 389  *
 390  *              M [U1 U2] = M U = H = [H1 0]
 391  * or
 392  *                            M = H Q = [H1 0] [Q1]
 393  *                                             [Q2]
 394  *
 395  * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
 396  * Define the transformed variables as
 397  *
 398  *              x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
 399  *                          [ x2' ]           [Q2]
 400  *
 401  * The equalities then become
 402  *
 403  *              H1 x1' - c = 0   or   x1' = H1^{-1} c = c'
 404  *
 405  * If any of the c' is non-integer, then the original set has no
 406  * integer solutions (since the x' are a unimodular transformation
 407  * of the x) and a zero-column matrix is returned.
 408  * Otherwise, the transformation is given by
 409  *
 410  *              x = U1 H1^{-1} c + U2 x2'
 411  *
 412  * The inverse transformation is simply
 413  *
 414  *              x2' = Q2 x
 415  */
 416 __isl_give isl_mat *isl_mat_variable_compression(__isl_take isl_mat *B,
 417         __isl_give isl_mat **T2)
 418 {
 419         int i;
 420         struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
 421         unsigned dim;
 422
 423         if (T2)
 424                 *T2 = NULL;
 425         if (!B)
 426                 goto error;
 427
 428         dim = B->n_col - 1;
 429         H = isl_mat_sub_alloc(B, 0, B->n_row, 1, dim);
 430         H = isl_mat_left_hermite(H, 0, &U, T2);
 431         if (!H || !U || (T2 && !*T2))
 432                 goto error;
 433         if (T2) {
 434                 *T2 = isl_mat_drop_rows(*T2, 0, B->n_row);
 435                 *T2 = isl_mat_lin_to_aff(*T2);
 436                 if (!*T2)
 437                         goto error;
 438         }
 439         C = isl_mat_alloc(B->ctx, 1+B->n_row, 1);
 440         if (!C)
 441                 goto error;
 442         isl_int_set_si(C->row[0][0], 1);
 443         isl_mat_sub_neg(C->ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
 444         H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
 445         H1 = isl_mat_lin_to_aff(H1);
 446         TC = isl_mat_inverse_product(H1, C);
 447         if (!TC)
 448                 goto error;
 449         isl_mat_free(H);
 450         if (!isl_int_is_one(TC->row[0][0])) {
 451                 for (i = 0; i < B->n_row; ++i) {
 452                         if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
 453                                 struct isl_ctx *ctx = B->ctx;
 454                                 isl_mat_free(B);
 455                                 isl_mat_free(TC);
 456                                 isl_mat_free(U);
 457                                 if (T2) {
 458                                         isl_mat_free(*T2);
 459                                         *T2 = NULL;
 460                                 }
 461                                 return isl_mat_alloc(ctx, 1 + dim, 0);
 462                         }
 463                         isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
 464                 }
 465                 isl_int_set_si(TC->row[0][0], 1);
 466         }
 467         U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, B->n_row);
 468         U1 = isl_mat_lin_to_aff(U1);
 469         U2 = isl_mat_sub_alloc(U, 0, U->n_row, B->n_row, U->n_row - B->n_row);
 470         U2 = isl_mat_lin_to_aff(U2);
 471         isl_mat_free(U);
 472         TC = isl_mat_product(U1, TC);
 473         TC = isl_mat_aff_direct_sum(TC, U2);
 474
 475         isl_mat_free(B);
 476
 477         return TC;
 478 error:
 479         isl_mat_free(B);
 480         isl_mat_free(H);
 481         isl_mat_free(U);
 482         if (T2) {
 483                 isl_mat_free(*T2);
 484                 *T2 = NULL;
 485         }
 486         return NULL;
 487 }
 488
 489 /* Use the n equalities of bset to unimodularly transform the
 490  * variables x such that n transformed variables x1' have a constant value
 491  * and rewrite the constraints of bset in terms of the remaining
 492  * transformed variables x2'.  The matrix pointed to by T maps
 493  * the new variables x2' back to the original variables x, while T2
 494  * maps the original variables to the new variables.
 495  */
 496 static struct isl_basic_set *compress_variables(
 497         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 498 {
 499         struct isl_mat *B, *TC;
 500         unsigned dim;
 501
 502         if (T)
 503                 *T = NULL;
 504         if (T2)
 505                 *T2 = NULL;
 506         if (!bset)
 507                 goto error;
 508         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 509         isl_assert(bset->ctx, bset->n_div == 0, goto error);
 510         dim = isl_basic_set_n_dim(bset);
 511         isl_assert(bset->ctx, bset->n_eq <= dim, goto error);
 512         if (bset->n_eq == 0)
 513                 return bset;
 514
 515         B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
 516         TC = isl_mat_variable_compression(B, T2);
 517         if (!TC)
 518                 goto error;
 519         if (TC->n_col == 0) {
 520                 isl_mat_free(TC);
 521                 if (T2) {
 522                         isl_mat_free(*T2);
 523                         *T2 = NULL;
 524                 }
 525                 return isl_basic_set_set_to_empty(bset);
 526         }
 527
 528         bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(TC) : TC);
 529         if (T)
 530                 *T = TC;
 531         return bset;
 532 error:
 533         isl_basic_set_free(bset);
 534         return NULL;
 535 }
 536
 537 struct isl_basic_set *isl_basic_set_remove_equalities(
 538         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 539 {
 540         if (T)
 541                 *T = NULL;
 542         if (T2)
 543                 *T2 = NULL;
 544         if (!bset)
 545                 return NULL;
 546         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 547         bset = isl_basic_set_gauss(bset, NULL);
 548         if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
 549                 return bset;
 550         bset = compress_variables(bset, T, T2);
 551         return bset;
 552 error:
 553         isl_basic_set_free(bset);
 554         *T = NULL;
 555         return NULL;
 556 }
 557
 558 /* Check if dimension dim belongs to a residue class
 559  *              i_dim \equiv r mod m
 560  * with m != 1 and if so return m in *modulo and r in *residue.
 561  * As a special case, when i_dim has a fixed value v, then
 562  * *modulo is set to 0 and *residue to v.
 563  *
 564  * If i_dim does not belong to such a residue class, then *modulo
 565  * is set to 1 and *residue is set to 0.
 566  */
 567 int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
 568         int pos, isl_int *modulo, isl_int *residue)
 569 {
 570         struct isl_ctx *ctx;
 571         struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
 572         unsigned total;
 573         unsigned nparam;
 574
 575         if (!bset || !modulo || !residue)
 576                 return -1;
 577
 578         if (isl_basic_set_plain_dim_is_fixed(bset, pos, residue)) {
 579                 isl_int_set_si(*modulo, 0);
 580                 return 0;
 581         }
 582
 583         ctx = bset->ctx;
 584         total = isl_basic_set_total_dim(bset);
 585         nparam = isl_basic_set_n_param(bset);
 586         H = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 1, total);
 587         H = isl_mat_left_hermite(H, 0, &U, NULL);
 588         if (!H)
 589                 return -1;
 590
 591         isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
 592                         total-bset->n_eq, modulo);
 593         if (isl_int_is_zero(*modulo))
 594                 isl_int_set_si(*modulo, 1);
 595         if (isl_int_is_one(*modulo)) {
 596                 isl_int_set_si(*residue, 0);
 597                 isl_mat_free(H);
 598                 isl_mat_free(U);
 599                 return 0;
 600         }
 601
 602         C = isl_mat_alloc(bset->ctx, 1+bset->n_eq, 1);
 603         if (!C)
 604                 goto error;
 605         isl_int_set_si(C->row[0][0], 1);
 606         isl_mat_sub_neg(C->ctx, C->row+1, bset->eq, bset->n_eq, 0, 0, 1);
 607         H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
 608         H1 = isl_mat_lin_to_aff(H1);
 609         C = isl_mat_inverse_product(H1, C);
 610         isl_mat_free(H);
 611         U1 = isl_mat_sub_alloc(U, nparam+pos, 1, 0, bset->n_eq);
 612         U1 = isl_mat_lin_to_aff(U1);
 613         isl_mat_free(U);
 614         C = isl_mat_product(U1, C);
 615         if (!C)
 616                 goto error;
 617         if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
 618                 bset = isl_basic_set_copy(bset);
 619                 bset = isl_basic_set_set_to_empty(bset);
 620                 isl_basic_set_free(bset);
 621                 isl_int_set_si(*modulo, 1);
 622                 isl_int_set_si(*residue, 0);
 623                 return 0;
 624         }
 625         isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
 626         isl_int_fdiv_r(*residue, *residue, *modulo);
 627         isl_mat_free(C);
 628         return 0;
 629 error:
 630         isl_mat_free(H);
 631         isl_mat_free(U);
 632         return -1;
 633 }
 634
 635 /* Check if dimension dim belongs to a residue class
 636  *              i_dim \equiv r mod m
 637  * with m != 1 and if so return m in *modulo and r in *residue.
 638  * As a special case, when i_dim has a fixed value v, then
 639  * *modulo is set to 0 and *residue to v.
 640  *
 641  * If i_dim does not belong to such a residue class, then *modulo
 642  * is set to 1 and *residue is set to 0.
 643  */
 644 int isl_set_dim_residue_class(struct isl_set *set,
 645         int pos, isl_int *modulo, isl_int *residue)
 646 {
 647         isl_int m;
 648         isl_int r;
 649         int i;
 650
 651         if (!set || !modulo || !residue)
 652                 return -1;
 653
 654         if (set->n == 0) {
 655                 isl_int_set_si(*modulo, 0);
 656                 isl_int_set_si(*residue, 0);
 657                 return 0;
 658         }
 659
 660         if (isl_basic_set_dim_residue_class(set->p[0], pos, modulo, residue)<0)
 661                 return -1;
 662
 663         if (set->n == 1)
 664                 return 0;
 665
 666         if (isl_int_is_one(*modulo))
 667                 return 0;
 668
 669         isl_int_init(m);
 670         isl_int_init(r);
 671
 672         for (i = 1; i < set->n; ++i) {
 673                 if (isl_basic_set_dim_residue_class(set->p[i], pos, &m, &r) < 0)
 674                         goto error;
 675                 isl_int_gcd(*modulo, *modulo, m);
 676                 isl_int_sub(m, *residue, r);
 677                 isl_int_gcd(*modulo, *modulo, m);
 678                 if (!isl_int_is_zero(*modulo))
 679                         isl_int_fdiv_r(*residue, *residue, *modulo);
 680                 if (isl_int_is_one(*modulo))
 681                         break;
 682         }
 683
 684         isl_int_clear(m);
 685         isl_int_clear(r);
 686
 687         return 0;
 688 error:
 689         isl_int_clear(m);
 690         isl_int_clear(r);
 691         return -1;
 692 }