+/* Given a set of modulo constraints
+ *
+ * c + A y = 0 mod d
+ *
+ * this function computes a particular solution y_0
+ *
+ * The input is given as a matrix B = [ c A ] and a vector d.
+ *
+ * The output is matrix containing the solution y_0 or
+ * a zero-column matrix if the constraints admit no integer solution.
+ *
+ * The given set of constrains is equivalent to
+ *
+ * c + A y = -D x
+ *
+ * with D = diag d and x a fresh set of variables.
+ * Reducing both c and A modulo d does not change the
+ * value of y in the solution and may lead to smaller coefficients.
+ * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
+ * Then
+ * [ x ]
+ * M [ y ] = - c
+ * and so
+ * [ x ]
+ * [ H 0 ] U^{-1} [ y ] = - c
+ * Let
+ * [ A ] [ x ]
+ * [ B ] = U^{-1} [ y ]
+ * then
+ * H A + 0 B = -c
+ *
+ * so B may be chosen arbitrarily, e.g., B = 0, and then
+ *
+ * [ x ] = [ -c ]
+ * U^{-1} [ y ] = [ 0 ]
+ * or
+ * [ x ] [ -c ]
+ * [ y ] = U [ 0 ]
+ * specifically,
+ *
+ * y = U_{2,1} (-c)
+ *
+ * If any of the coordinates of this y are non-integer
+ * then the constraints admit no integer solution and
+ * a zero-column matrix is returned.
+ */
+static struct isl_mat *particular_solution(struct isl_mat *B, struct isl_vec *d)
+{
+ int i, j;
+ struct isl_mat *M = NULL;
+ struct isl_mat *C = NULL;
+ struct isl_mat *U = NULL;
+ struct isl_mat *H = NULL;
+ struct isl_mat *cst = NULL;
+ struct isl_mat *T = NULL;
+
+ M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
+ C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
+ if (!M || !C)
+ goto error;
+ isl_int_set_si(C->row[0][0], 1);
+ for (i = 0; i < B->n_row; ++i) {
+ isl_seq_clr(M->row[i], B->n_row);
+ isl_int_set(M->row[i][i], d->block.data[i]);
+ isl_int_neg(C->row[1 + i][0], B->row[i][0]);
+ isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
+ for (j = 0; j < B->n_col - 1; ++j)
+ isl_int_fdiv_r(M->row[i][B->n_row + j],
+ B->row[i][1 + j], M->row[i][i]);
+ }
+ M = isl_mat_left_hermite(M, 0, &U, NULL);
+ if (!M || !U)
+ goto error;
+ H = isl_mat_sub_alloc(M, 0, B->n_row, 0, B->n_row);
+ H = isl_mat_lin_to_aff(H);
+ C = isl_mat_inverse_product(H, C);
+ if (!C)
+ goto error;
+ for (i = 0; i < B->n_row; ++i) {
+ if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
+ break;
+ isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
+ }
+ if (i < B->n_row)
+ cst = isl_mat_alloc(B->ctx, B->n_row, 0);
+ else
+ cst = isl_mat_sub_alloc(C, 1, B->n_row, 0, 1);
+ T = isl_mat_sub_alloc(U, B->n_row, B->n_col - 1, 0, B->n_row);
+ cst = isl_mat_product(T, cst);
+ isl_mat_free(M);
+ isl_mat_free(C);
+ isl_mat_free(U);
+ return cst;
+error:
+ isl_mat_free(M);
+ isl_mat_free(C);
+ isl_mat_free(U);
+ return NULL;
+}
+
+/* Compute and return the matrix
+ *
+ * U_1^{-1} diag(d_1, 1, ..., 1)
+ *
+ * with U_1 the unimodular completion of the first (and only) row of B.
+ * The columns of this matrix generate the lattice that satisfies
+ * the single (linear) modulo constraint.
+ */
+static struct isl_mat *parameter_compression_1(
+ struct isl_mat *B, struct isl_vec *d)
+{
+ struct isl_mat *U;
+
+ U = isl_mat_alloc(B->ctx, B->n_col - 1, B->n_col - 1);
+ if (!U)
+ return NULL;
+ isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
+ U = isl_mat_unimodular_complete(U, 1);
+ U = isl_mat_right_inverse(U);
+ if (!U)
+ return NULL;
+ isl_mat_col_mul(U, 0, d->block.data[0], 0);
+ U = isl_mat_lin_to_aff(U);
+ return U;
+}
+
+/* Compute a common lattice of solutions to the linear modulo
+ * constraints specified by B and d.
+ * See also the documentation of isl_mat_parameter_compression.
+ * We put the matrix
+ *
+ * A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
+ *
+ * on a common denominator. This denominator D is the lcm of modulos d.
+ * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
+ * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
+ * Putting this on the common denominator, we have
+ * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
+ */
+static struct isl_mat *parameter_compression_multi(
+ struct isl_mat *B, struct isl_vec *d)
+{
+ int i, j, k;
+ isl_int D;
+ struct isl_mat *A = NULL, *U = NULL;
+ struct isl_mat *T;
+ unsigned size;
+
+ isl_int_init(D);
+
+ isl_vec_lcm(d, &D);
+
+ size = B->n_col - 1;
+ A = isl_mat_alloc(B->ctx, size, B->n_row * size);
+ U = isl_mat_alloc(B->ctx, size, size);
+ if (!U || !A)
+ goto error;
+ for (i = 0; i < B->n_row; ++i) {
+ isl_seq_cpy(U->row[0], B->row[i] + 1, size);
+ U = isl_mat_unimodular_complete(U, 1);
+ if (!U)
+ goto error;
+ isl_int_divexact(D, D, d->block.data[i]);
+ for (k = 0; k < U->n_col; ++k)
+ isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
+ isl_int_mul(D, D, d->block.data[i]);
+ for (j = 1; j < U->n_row; ++j)
+ for (k = 0; k < U->n_col; ++k)
+ isl_int_mul(A->row[k][i*size+j],
+ D, U->row[j][k]);
+ }
+ A = isl_mat_left_hermite(A, 0, NULL, NULL);
+ T = isl_mat_sub_alloc(A, 0, A->n_row, 0, A->n_row);
+ T = isl_mat_lin_to_aff(T);
+ if (!T)
+ goto error;
+ isl_int_set(T->row[0][0], D);
+ T = isl_mat_right_inverse(T);
+ if (!T)
+ goto error;
+ isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
+ T = isl_mat_transpose(T);
+ isl_mat_free(A);
+ isl_mat_free(U);
+
+ isl_int_clear(D);
+ return T;
+error:
+ isl_mat_free(A);
+ isl_mat_free(U);
+ isl_int_clear(D);
+ return NULL;
+}
+
+/* Given a set of modulo constraints
+ *
+ * c + A y = 0 mod d
+ *
+ * this function returns an affine transformation T,
+ *
+ * y = T y'
+ *
+ * that bijectively maps the integer vectors y' to integer
+ * vectors y that satisfy the modulo constraints.
+ *
+ * This function is inspired by Section 2.5.3
+ * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
+ * Model. Applications to Program Analysis and Optimization".
+ * However, the implementation only follows the algorithm of that
+ * section for computing a particular solution and not for computing
+ * a general homogeneous solution. The latter is incomplete and
+ * may remove some valid solutions.
+ * Instead, we use an adaptation of the algorithm in Section 7 of
+ * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
+ * Model: Bringing the Power of Quasi-Polynomials to the Masses".
+ *
+ * The input is given as a matrix B = [ c A ] and a vector d.
+ * Each element of the vector d corresponds to a row in B.
+ * The output is a lower triangular matrix.
+ * If no integer vector y satisfies the given constraints then
+ * a matrix with zero columns is returned.
+ *
+ * We first compute a particular solution y_0 to the given set of
+ * modulo constraints in particular_solution. If no such solution
+ * exists, then we return a zero-columned transformation matrix.
+ * Otherwise, we compute the generic solution to
+ *
+ * A y = 0 mod d
+ *
+ * That is we want to compute G such that
+ *
+ * y = G y''