isl_equalities.c

   1 /*
   2  * Copyright 2008-2009 Katholieke Universiteit Leuven
   3  * Copyright 2010      INRIA Saclay
   4  *
   5  * Use of this software is governed by the MIT license
   6  *
   7  * Written by Sven Verdoolaege, K.U.Leuven, Departement
   8  * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
   9  * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
  10  * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
  11  */
  12
  13 #include <isl_mat_private.h>
  14 #include <isl/seq.h>
  15 #include "isl_map_private.h"
  16 #include "isl_equalities.h"
  17 #include <isl_val_private.h>
  18
  19 /* Given a set of modulo constraints
  20  *
  21  *              c + A y = 0 mod d
  22  *
  23  * this function computes a particular solution y_0
  24  *
  25  * The input is given as a matrix B = [ c A ] and a vector d.
  26  *
  27  * The output is matrix containing the solution y_0 or
  28  * a zero-column matrix if the constraints admit no integer solution.
  29  *
  30  * The given set of constrains is equivalent to
  31  *
  32  *              c + A y = -D x
  33  *
  34  * with D = diag d and x a fresh set of variables.
  35  * Reducing both c and A modulo d does not change the
  36  * value of y in the solution and may lead to smaller coefficients.
  37  * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
  38  * Then
  39  *                [ x ]
  40  *              M [ y ] = - c
  41  * and so
  42  *                             [ x ]
  43  *              [ H 0 ] U^{-1} [ y ] = - c
  44  * Let
  45  *              [ A ]          [ x ]
  46  *              [ B ] = U^{-1} [ y ]
  47  * then
  48  *              H A + 0 B = -c
  49  *
  50  * so B may be chosen arbitrarily, e.g., B = 0, and then
  51  *
  52  *                     [ x ] = [ -c ]
  53  *              U^{-1} [ y ] = [  0 ]
  54  * or
  55  *              [ x ]     [ -c ]
  56  *              [ y ] = U [  0 ]
  57  * specifically,
  58  *
  59  *              y = U_{2,1} (-c)
  60  *
  61  * If any of the coordinates of this y are non-integer
  62  * then the constraints admit no integer solution and
  63  * a zero-column matrix is returned.
  64  */
  65 static struct isl_mat *particular_solution(struct isl_mat *B, struct isl_vec *d)
  66 {
  67         int i, j;
  68         struct isl_mat *M = NULL;
  69         struct isl_mat *C = NULL;
  70         struct isl_mat *U = NULL;
  71         struct isl_mat *H = NULL;
  72         struct isl_mat *cst = NULL;
  73         struct isl_mat *T = NULL;
  74
  75         M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
  76         C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
  77         if (!M || !C)
  78                 goto error;
  79         isl_int_set_si(C->row[0][0], 1);
  80         for (i = 0; i < B->n_row; ++i) {
  81                 isl_seq_clr(M->row[i], B->n_row);
  82                 isl_int_set(M->row[i][i], d->block.data[i]);
  83                 isl_int_neg(C->row[1 + i][0], B->row[i][0]);
  84                 isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
  85                 for (j = 0; j < B->n_col - 1; ++j)
  86                         isl_int_fdiv_r(M->row[i][B->n_row + j],
  87                                         B->row[i][1 + j], M->row[i][i]);
  88         }
  89         M = isl_mat_left_hermite(M, 0, &U, NULL);
  90         if (!M || !U)
  91                 goto error;
  92         H = isl_mat_sub_alloc(M, 0, B->n_row, 0, B->n_row);
  93         H = isl_mat_lin_to_aff(H);
  94         C = isl_mat_inverse_product(H, C);
  95         if (!C)
  96                 goto error;
  97         for (i = 0; i < B->n_row; ++i) {
  98                 if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
  99                         break;
 100                 isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
 101         }
 102         if (i < B->n_row)
 103                 cst = isl_mat_alloc(B->ctx, B->n_row, 0);
 104         else
 105                 cst = isl_mat_sub_alloc(C, 1, B->n_row, 0, 1);
 106         T = isl_mat_sub_alloc(U, B->n_row, B->n_col - 1, 0, B->n_row);
 107         cst = isl_mat_product(T, cst);
 108         isl_mat_free(M);
 109         isl_mat_free(C);
 110         isl_mat_free(U);
 111         return cst;
 112 error:
 113         isl_mat_free(M);
 114         isl_mat_free(C);
 115         isl_mat_free(U);
 116         return NULL;
 117 }
 118
 119 /* Compute and return the matrix
 120  *
 121  *              U_1^{-1} diag(d_1, 1, ..., 1)
 122  *
 123  * with U_1 the unimodular completion of the first (and only) row of B.
 124  * The columns of this matrix generate the lattice that satisfies
 125  * the single (linear) modulo constraint.
 126  */
 127 static struct isl_mat *parameter_compression_1(
 128                         struct isl_mat *B, struct isl_vec *d)
 129 {
 130         struct isl_mat *U;
 131
 132         U = isl_mat_alloc(B->ctx, B->n_col - 1, B->n_col - 1);
 133         if (!U)
 134                 return NULL;
 135         isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
 136         U = isl_mat_unimodular_complete(U, 1);
 137         U = isl_mat_right_inverse(U);
 138         if (!U)
 139                 return NULL;
 140         isl_mat_col_mul(U, 0, d->block.data[0], 0);
 141         U = isl_mat_lin_to_aff(U);
 142         return U;
 143 }
 144
 145 /* Compute a common lattice of solutions to the linear modulo
 146  * constraints specified by B and d.
 147  * See also the documentation of isl_mat_parameter_compression.
 148  * We put the matrix
 149  *
 150  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 151  *
 152  * on a common denominator.  This denominator D is the lcm of modulos d.
 153  * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
 154  * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
 155  * Putting this on the common denominator, we have
 156  * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
 157  */
 158 static struct isl_mat *parameter_compression_multi(
 159                         struct isl_mat *B, struct isl_vec *d)
 160 {
 161         int i, j, k;
 162         isl_int D;
 163         struct isl_mat *A = NULL, *U = NULL;
 164         struct isl_mat *T;
 165         unsigned size;
 166
 167         isl_int_init(D);
 168
 169         isl_vec_lcm(d, &D);
 170
 171         size = B->n_col - 1;
 172         A = isl_mat_alloc(B->ctx, size, B->n_row * size);
 173         U = isl_mat_alloc(B->ctx, size, size);
 174         if (!U || !A)
 175                 goto error;
 176         for (i = 0; i < B->n_row; ++i) {
 177                 isl_seq_cpy(U->row[0], B->row[i] + 1, size);
 178                 U = isl_mat_unimodular_complete(U, 1);
 179                 if (!U)
 180                         goto error;
 181                 isl_int_divexact(D, D, d->block.data[i]);
 182                 for (k = 0; k < U->n_col; ++k)
 183                         isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
 184                 isl_int_mul(D, D, d->block.data[i]);
 185                 for (j = 1; j < U->n_row; ++j)
 186                         for (k = 0; k < U->n_col; ++k)
 187                                 isl_int_mul(A->row[k][i*size+j],
 188                                                 D, U->row[j][k]);
 189         }
 190         A = isl_mat_left_hermite(A, 0, NULL, NULL);
 191         T = isl_mat_sub_alloc(A, 0, A->n_row, 0, A->n_row);
 192         T = isl_mat_lin_to_aff(T);
 193         if (!T)
 194                 goto error;
 195         isl_int_set(T->row[0][0], D);
 196         T = isl_mat_right_inverse(T);
 197         if (!T)
 198                 goto error;
 199         isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
 200         T = isl_mat_transpose(T);
 201         isl_mat_free(A);
 202         isl_mat_free(U);
 203
 204         isl_int_clear(D);
 205         return T;
 206 error:
 207         isl_mat_free(A);
 208         isl_mat_free(U);
 209         isl_int_clear(D);
 210         return NULL;
 211 }
 212
 213 /* Given a set of modulo constraints
 214  *
 215  *              c + A y = 0 mod d
 216  *
 217  * this function returns an affine transformation T,
 218  *
 219  *              y = T y'
 220  *
 221  * that bijectively maps the integer vectors y' to integer
 222  * vectors y that satisfy the modulo constraints.
 223  *
 224  * This function is inspired by Section 2.5.3
 225  * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
 226  * Model.  Applications to Program Analysis and Optimization".
 227  * However, the implementation only follows the algorithm of that
 228  * section for computing a particular solution and not for computing
 229  * a general homogeneous solution.  The latter is incomplete and
 230  * may remove some valid solutions.
 231  * Instead, we use an adaptation of the algorithm in Section 7 of
 232  * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
 233  * Model: Bringing the Power of Quasi-Polynomials to the Masses".
 234  *
 235  * The input is given as a matrix B = [ c A ] and a vector d.
 236  * Each element of the vector d corresponds to a row in B.
 237  * The output is a lower triangular matrix.
 238  * If no integer vector y satisfies the given constraints then
 239  * a matrix with zero columns is returned.
 240  *
 241  * We first compute a particular solution y_0 to the given set of
 242  * modulo constraints in particular_solution.  If no such solution
 243  * exists, then we return a zero-columned transformation matrix.
 244  * Otherwise, we compute the generic solution to
 245  *
 246  *              A y = 0 mod d
 247  *
 248  * That is we want to compute G such that
 249  *
 250  *              y = G y''
 251  *
 252  * with y'' integer, describes the set of solutions.
 253  *
 254  * We first remove the common factors of each row.
 255  * In particular if gcd(A_i,d_i) != 1, then we divide the whole
 256  * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
 257  * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
 258  * In the later case, we simply drop the row (in both A and d).
 259  *
 260  * If there are no rows left in A, then G is the identity matrix. Otherwise,
 261  * for each row i, we now determine the lattice of integer vectors
 262  * that satisfies this row.  Let U_i be the unimodular extension of the
 263  * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
 264  * The first component of
 265  *
 266  *              y' = U_i y
 267  *
 268  * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
 269  * Then,
 270  *
 271  *              y = U_i^{-1} diag(d_i, 1, ..., 1) y''
 272  *
 273  * for arbitrary integer vectors y''.  That is, y belongs to the lattice
 274  * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
 275  * If there is only one row, then G = L_1.
 276  *
 277  * If there is more than one row left, we need to compute the intersection
 278  * of the lattices.  That is, we need to compute an L such that
 279  *
 280  *              L = L_i L_i'    for all i
 281  *
 282  * with L_i' some integer matrices.  Let A be constructed as follows
 283  *
 284  *              A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 285  *
 286  * and computed the Hermite Normal Form of A = [ H 0 ] U
 287  * Then,
 288  *
 289  *              L_i^{-T} = H U_{1,i}
 290  *
 291  * or
 292  *
 293  *              H^{-T} = L_i U_{1,i}^T
 294  *
 295  * In other words G = L = H^{-T}.
 296  * To ensure that G is lower triangular, we compute and use its Hermite
 297  * normal form.
 298  *
 299  * The affine transformation matrix returned is then
 300  *
 301  *              [  1   0  ]
 302  *              [ y_0  G  ]
 303  *
 304  * as any y = y_0 + G y' with y' integer is a solution to the original
 305  * modulo constraints.
 306  */
 307 struct isl_mat *isl_mat_parameter_compression(
 308                         struct isl_mat *B, struct isl_vec *d)
 309 {
 310         int i;
 311         struct isl_mat *cst = NULL;
 312         struct isl_mat *T = NULL;
 313         isl_int D;
 314
 315         if (!B || !d)
 316                 goto error;
 317         isl_assert(B->ctx, B->n_row == d->size, goto error);
 318         cst = particular_solution(B, d);
 319         if (!cst)
 320                 goto error;
 321         if (cst->n_col == 0) {
 322                 T = isl_mat_alloc(B->ctx, B->n_col, 0);
 323                 isl_mat_free(cst);
 324                 isl_mat_free(B);
 325                 isl_vec_free(d);
 326                 return T;
 327         }
 328         isl_int_init(D);
 329         /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
 330         for (i = 0; i < B->n_row; ++i) {
 331                 isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
 332                 if (isl_int_is_one(D))
 333                         continue;
 334                 if (isl_int_is_zero(D)) {
 335                         B = isl_mat_drop_rows(B, i, 1);
 336                         d = isl_vec_cow(d);
 337                         if (!B || !d)
 338                                 goto error2;
 339                         isl_seq_cpy(d->block.data+i, d->block.data+i+1,
 340                                                         d->size - (i+1));
 341                         d->size--;
 342                         i--;
 343                         continue;
 344                 }
 345                 B = isl_mat_cow(B);
 346                 if (!B)
 347                         goto error2;
 348                 isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
 349                 isl_int_gcd(D, D, d->block.data[i]);
 350                 d = isl_vec_cow(d);
 351                 if (!d)
 352                         goto error2;
 353                 isl_int_divexact(d->block.data[i], d->block.data[i], D);
 354         }
 355         isl_int_clear(D);
 356         if (B->n_row == 0)
 357                 T = isl_mat_identity(B->ctx, B->n_col);
 358         else if (B->n_row == 1)
 359                 T = parameter_compression_1(B, d);
 360         else
 361                 T = parameter_compression_multi(B, d);
 362         T = isl_mat_left_hermite(T, 0, NULL, NULL);
 363         if (!T)
 364                 goto error;
 365         isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
 366         isl_mat_free(cst);
 367         isl_mat_free(B);
 368         isl_vec_free(d);
 369         return T;
 370 error2:
 371         isl_int_clear(D);
 372 error:
 373         isl_mat_free(cst);
 374         isl_mat_free(B);
 375         isl_vec_free(d);
 376         return NULL;
 377 }
 378
 379 /* Given a set of equalities
 380  *
 381  *              B(y) + A x = 0                                          (*)
 382  *
 383  * compute and return an affine transformation T,
 384  *
 385  *              y = T y'
 386  *
 387  * that bijectively maps the integer vectors y' to integer
 388  * vectors y that satisfy the modulo constraints for some value of x.
 389  *
 390  * Let [H 0] be the Hermite Normal Form of A, i.e.,
 391  *
 392  *              A = [H 0] Q
 393  *
 394  * Then y is a solution of (*) iff
 395  *
 396  *              H^-1 B(y) (= - [I 0] Q x)
 397  *
 398  * is an integer vector.  Let d be the common denominator of H^-1.
 399  * We impose
 400  *
 401  *              d H^-1 B(y) = 0 mod d
 402  *
 403  * and compute the solution using isl_mat_parameter_compression.
 404  */
 405 __isl_give isl_mat *isl_mat_parameter_compression_ext(__isl_take isl_mat *B,
 406         __isl_take isl_mat *A)
 407 {
 408         isl_ctx *ctx;
 409         isl_vec *d;
 410         int n_row, n_col;
 411
 412         if (!A)
 413                 return isl_mat_free(B);
 414
 415         ctx = isl_mat_get_ctx(A);
 416         n_row = A->n_row;
 417         n_col = A->n_col;
 418         A = isl_mat_left_hermite(A, 0, NULL, NULL);
 419         A = isl_mat_drop_cols(A, n_row, n_col - n_row);
 420         A = isl_mat_lin_to_aff(A);
 421         A = isl_mat_right_inverse(A);
 422         d = isl_vec_alloc(ctx, n_row);
 423         if (A)
 424                 d = isl_vec_set(d, A->row[0][0]);
 425         A = isl_mat_drop_rows(A, 0, 1);
 426         A = isl_mat_drop_cols(A, 0, 1);
 427         B = isl_mat_product(A, B);
 428
 429         return isl_mat_parameter_compression(B, d);
 430 }
 431
 432 /* Given a set of equalities
 433  *
 434  *              M x - c = 0
 435  *
 436  * this function computes a unimodular transformation from a lower-dimensional
 437  * space to the original space that bijectively maps the integer points x'
 438  * in the lower-dimensional space to the integer points x in the original
 439  * space that satisfy the equalities.
 440  *
 441  * The input is given as a matrix B = [ -c M ] and the output is a
 442  * matrix that maps [1 x'] to [1 x].
 443  * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 444  *
 445  * First compute the (left) Hermite normal form of M,
 446  *
 447  *              M [U1 U2] = M U = H = [H1 0]
 448  * or
 449  *                            M = H Q = [H1 0] [Q1]
 450  *                                             [Q2]
 451  *
 452  * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
 453  * Define the transformed variables as
 454  *
 455  *              x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
 456  *                          [ x2' ]           [Q2]
 457  *
 458  * The equalities then become
 459  *
 460  *              H1 x1' - c = 0   or   x1' = H1^{-1} c = c'
 461  *
 462  * If any of the c' is non-integer, then the original set has no
 463  * integer solutions (since the x' are a unimodular transformation
 464  * of the x) and a zero-column matrix is returned.
 465  * Otherwise, the transformation is given by
 466  *
 467  *              x = U1 H1^{-1} c + U2 x2'
 468  *
 469  * The inverse transformation is simply
 470  *
 471  *              x2' = Q2 x
 472  */
 473 __isl_give isl_mat *isl_mat_variable_compression(__isl_take isl_mat *B,
 474         __isl_give isl_mat **T2)
 475 {
 476         int i;
 477         struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
 478         unsigned dim;
 479
 480         if (T2)
 481                 *T2 = NULL;
 482         if (!B)
 483                 goto error;
 484
 485         dim = B->n_col - 1;
 486         H = isl_mat_sub_alloc(B, 0, B->n_row, 1, dim);
 487         H = isl_mat_left_hermite(H, 0, &U, T2);
 488         if (!H || !U || (T2 && !*T2))
 489                 goto error;
 490         if (T2) {
 491                 *T2 = isl_mat_drop_rows(*T2, 0, B->n_row);
 492                 *T2 = isl_mat_lin_to_aff(*T2);
 493                 if (!*T2)
 494                         goto error;
 495         }
 496         C = isl_mat_alloc(B->ctx, 1+B->n_row, 1);
 497         if (!C)
 498                 goto error;
 499         isl_int_set_si(C->row[0][0], 1);
 500         isl_mat_sub_neg(C->ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
 501         H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
 502         H1 = isl_mat_lin_to_aff(H1);
 503         TC = isl_mat_inverse_product(H1, C);
 504         if (!TC)
 505                 goto error;
 506         isl_mat_free(H);
 507         if (!isl_int_is_one(TC->row[0][0])) {
 508                 for (i = 0; i < B->n_row; ++i) {
 509                         if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
 510                                 struct isl_ctx *ctx = B->ctx;
 511                                 isl_mat_free(B);
 512                                 isl_mat_free(TC);
 513                                 isl_mat_free(U);
 514                                 if (T2) {
 515                                         isl_mat_free(*T2);
 516                                         *T2 = NULL;
 517                                 }
 518                                 return isl_mat_alloc(ctx, 1 + dim, 0);
 519                         }
 520                         isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
 521                 }
 522                 isl_int_set_si(TC->row[0][0], 1);
 523         }
 524         U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, B->n_row);
 525         U1 = isl_mat_lin_to_aff(U1);
 526         U2 = isl_mat_sub_alloc(U, 0, U->n_row, B->n_row, U->n_row - B->n_row);
 527         U2 = isl_mat_lin_to_aff(U2);
 528         isl_mat_free(U);
 529         TC = isl_mat_product(U1, TC);
 530         TC = isl_mat_aff_direct_sum(TC, U2);
 531
 532         isl_mat_free(B);
 533
 534         return TC;
 535 error:
 536         isl_mat_free(B);
 537         isl_mat_free(H);
 538         isl_mat_free(U);
 539         if (T2) {
 540                 isl_mat_free(*T2);
 541                 *T2 = NULL;
 542         }
 543         return NULL;
 544 }
 545
 546 /* Use the n equalities of bset to unimodularly transform the
 547  * variables x such that n transformed variables x1' have a constant value
 548  * and rewrite the constraints of bset in terms of the remaining
 549  * transformed variables x2'.  The matrix pointed to by T maps
 550  * the new variables x2' back to the original variables x, while T2
 551  * maps the original variables to the new variables.
 552  */
 553 static struct isl_basic_set *compress_variables(
 554         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 555 {
 556         struct isl_mat *B, *TC;
 557         unsigned dim;
 558
 559         if (T)
 560                 *T = NULL;
 561         if (T2)
 562                 *T2 = NULL;
 563         if (!bset)
 564                 goto error;
 565         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 566         isl_assert(bset->ctx, bset->n_div == 0, goto error);
 567         dim = isl_basic_set_n_dim(bset);
 568         isl_assert(bset->ctx, bset->n_eq <= dim, goto error);
 569         if (bset->n_eq == 0)
 570                 return bset;
 571
 572         B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
 573         TC = isl_mat_variable_compression(B, T2);
 574         if (!TC)
 575                 goto error;
 576         if (TC->n_col == 0) {
 577                 isl_mat_free(TC);
 578                 if (T2) {
 579                         isl_mat_free(*T2);
 580                         *T2 = NULL;
 581                 }
 582                 return isl_basic_set_set_to_empty(bset);
 583         }
 584
 585         bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(TC) : TC);
 586         if (T)
 587                 *T = TC;
 588         return bset;
 589 error:
 590         isl_basic_set_free(bset);
 591         return NULL;
 592 }
 593
 594 struct isl_basic_set *isl_basic_set_remove_equalities(
 595         struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
 596 {
 597         if (T)
 598                 *T = NULL;
 599         if (T2)
 600                 *T2 = NULL;
 601         if (!bset)
 602                 return NULL;
 603         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 604         bset = isl_basic_set_gauss(bset, NULL);
 605         if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
 606                 return bset;
 607         bset = compress_variables(bset, T, T2);
 608         return bset;
 609 error:
 610         isl_basic_set_free(bset);
 611         *T = NULL;
 612         return NULL;
 613 }
 614
 615 /* Check if dimension dim belongs to a residue class
 616  *              i_dim \equiv r mod m
 617  * with m != 1 and if so return m in *modulo and r in *residue.
 618  * As a special case, when i_dim has a fixed value v, then
 619  * *modulo is set to 0 and *residue to v.
 620  *
 621  * If i_dim does not belong to such a residue class, then *modulo
 622  * is set to 1 and *residue is set to 0.
 623  */
 624 int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
 625         int pos, isl_int *modulo, isl_int *residue)
 626 {
 627         struct isl_ctx *ctx;
 628         struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
 629         unsigned total;
 630         unsigned nparam;
 631
 632         if (!bset || !modulo || !residue)
 633                 return -1;
 634
 635         if (isl_basic_set_plain_dim_is_fixed(bset, pos, residue)) {
 636                 isl_int_set_si(*modulo, 0);
 637                 return 0;
 638         }
 639
 640         ctx = bset->ctx;
 641         total = isl_basic_set_total_dim(bset);
 642         nparam = isl_basic_set_n_param(bset);
 643         H = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 1, total);
 644         H = isl_mat_left_hermite(H, 0, &U, NULL);
 645         if (!H)
 646                 return -1;
 647
 648         isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
 649                         total-bset->n_eq, modulo);
 650         if (isl_int_is_zero(*modulo))
 651                 isl_int_set_si(*modulo, 1);
 652         if (isl_int_is_one(*modulo)) {
 653                 isl_int_set_si(*residue, 0);
 654                 isl_mat_free(H);
 655                 isl_mat_free(U);
 656                 return 0;
 657         }
 658
 659         C = isl_mat_alloc(bset->ctx, 1+bset->n_eq, 1);
 660         if (!C)
 661                 goto error;
 662         isl_int_set_si(C->row[0][0], 1);
 663         isl_mat_sub_neg(C->ctx, C->row+1, bset->eq, bset->n_eq, 0, 0, 1);
 664         H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
 665         H1 = isl_mat_lin_to_aff(H1);
 666         C = isl_mat_inverse_product(H1, C);
 667         isl_mat_free(H);
 668         U1 = isl_mat_sub_alloc(U, nparam+pos, 1, 0, bset->n_eq);
 669         U1 = isl_mat_lin_to_aff(U1);
 670         isl_mat_free(U);
 671         C = isl_mat_product(U1, C);
 672         if (!C)
 673                 goto error;
 674         if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
 675                 bset = isl_basic_set_copy(bset);
 676                 bset = isl_basic_set_set_to_empty(bset);
 677                 isl_basic_set_free(bset);
 678                 isl_int_set_si(*modulo, 1);
 679                 isl_int_set_si(*residue, 0);
 680                 return 0;
 681         }
 682         isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
 683         isl_int_fdiv_r(*residue, *residue, *modulo);
 684         isl_mat_free(C);
 685         return 0;
 686 error:
 687         isl_mat_free(H);
 688         isl_mat_free(U);
 689         return -1;
 690 }
 691
 692 /* Check if dimension dim belongs to a residue class
 693  *              i_dim \equiv r mod m
 694  * with m != 1 and if so return m in *modulo and r in *residue.
 695  * As a special case, when i_dim has a fixed value v, then
 696  * *modulo is set to 0 and *residue to v.
 697  *
 698  * If i_dim does not belong to such a residue class, then *modulo
 699  * is set to 1 and *residue is set to 0.
 700  */
 701 int isl_set_dim_residue_class(struct isl_set *set,
 702         int pos, isl_int *modulo, isl_int *residue)
 703 {
 704         isl_int m;
 705         isl_int r;
 706         int i;
 707
 708         if (!set || !modulo || !residue)
 709                 return -1;
 710
 711         if (set->n == 0) {
 712                 isl_int_set_si(*modulo, 0);
 713                 isl_int_set_si(*residue, 0);
 714                 return 0;
 715         }
 716
 717         if (isl_basic_set_dim_residue_class(set->p[0], pos, modulo, residue)<0)
 718                 return -1;
 719
 720         if (set->n == 1)
 721                 return 0;
 722
 723         if (isl_int_is_one(*modulo))
 724                 return 0;
 725
 726         isl_int_init(m);
 727         isl_int_init(r);
 728
 729         for (i = 1; i < set->n; ++i) {
 730                 if (isl_basic_set_dim_residue_class(set->p[i], pos, &m, &r) < 0)
 731                         goto error;
 732                 isl_int_gcd(*modulo, *modulo, m);
 733                 isl_int_sub(m, *residue, r);
 734                 isl_int_gcd(*modulo, *modulo, m);
 735                 if (!isl_int_is_zero(*modulo))
 736                         isl_int_fdiv_r(*residue, *residue, *modulo);
 737                 if (isl_int_is_one(*modulo))
 738                         break;
 739         }
 740
 741         isl_int_clear(m);
 742         isl_int_clear(r);
 743
 744         return 0;
 745 error:
 746         isl_int_clear(m);
 747         isl_int_clear(r);
 748         return -1;
 749 }
 750
 751 /* Check if dimension "dim" belongs to a residue class
 752  *              i_dim \equiv r mod m
 753  * with m != 1 and if so return m in *modulo and r in *residue.
 754  * As a special case, when i_dim has a fixed value v, then
 755  * *modulo is set to 0 and *residue to v.
 756  *
 757  * If i_dim does not belong to such a residue class, then *modulo
 758  * is set to 1 and *residue is set to 0.
 759  */
 760 int isl_set_dim_residue_class_val(__isl_keep isl_set *set,
 761         int pos, __isl_give isl_val **modulo, __isl_give isl_val **residue)
 762 {
 763         *modulo = NULL;
 764         *residue = NULL;
 765         if (!set)
 766                 return -1;
 767         *modulo = isl_val_alloc(isl_set_get_ctx(set));
 768         *residue = isl_val_alloc(isl_set_get_ctx(set));
 769         if (!*modulo || !*residue)
 770                 goto error;
 771         if (isl_set_dim_residue_class(set, pos,
 772                                         &(*modulo)->n, &(*residue)->n) < 0)
 773                 goto error;
 774         isl_int_set_si((*modulo)->d, 1);
 775         isl_int_set_si((*residue)->d, 1);
 776         return 0;
 777 error:
 778         isl_val_free(*modulo);
 779         isl_val_free(*residue);
 780         return -1;
 781 }