__isl_give isl_point *isl_point_alloc(__isl_take isl_dim *dim,
__isl_take isl_vec *vec)
{
struct isl_point *pnt;
if (!dim || !vec)
goto error;
if (vec->size > 1 + isl_dim_total(dim)) {
vec = isl_vec_cow(vec);
if (!vec)
goto error;
vec->size = 1 + isl_dim_total(dim);
}
pnt = isl_alloc_type(dim->ctx, struct isl_point);
if (!pnt)
goto error;
pnt->ref = 1;
pnt->dim = dim;
pnt->vec = vec;
return pnt;
error:
isl_dim_free(dim);
isl_vec_free(vec);
return NULL;
}
__isl_give isl_point *isl_point_sub_ui(__isl_take isl_point *pnt,
enum isl_dim_type type, int pos, unsigned val)
{
if (!pnt || isl_point_is_void(pnt))
return pnt;
pnt = isl_point_cow(pnt);
if (!pnt)
return NULL;
pnt->vec = isl_vec_cow(pnt->vec);
if (!pnt->vec)
goto error;
if (type == isl_dim_set)
pos += isl_dim_size(pnt->dim, isl_dim_param);
isl_int_sub_ui(pnt->vec->el[1 + pos], pnt->vec->el[1 + pos], val);
return pnt;
error:
isl_point_free(pnt);
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''
*
* with y'' integer, describes the set of solutions.
*
* We first remove the common factors of each row.
* In particular if gcd(A_i,d_i) != 1, then we divide the whole
* row i (including d_i) by this common factor. If afterwards gcd(A_i) != 1,
* then we divide this row of A by the common factor, unless gcd(A_i) = 0.
* In the later case, we simply drop the row (in both A and d).
*
* If there are no rows left in A, then G is the identity matrix. Otherwise,
* for each row i, we now determine the lattice of integer vectors
* that satisfies this row. Let U_i be the unimodular extension of the
* row A_i. This unimodular extension exists because gcd(A_i) = 1.
* The first component of
*
* y' = U_i y
*
* needs to be a multiple of d_i. Let y' = diag(d_i, 1, ..., 1) y''.
* Then,
*
* y = U_i^{-1} diag(d_i, 1, ..., 1) y''
*
* for arbitrary integer vectors y''. That is, y belongs to the lattice
* generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
* If there is only one row, then G = L_1.
*
* If there is more than one row left, we need to compute the intersection
* of the lattices. That is, we need to compute an L such that
*
* L = L_i L_i' for all i
*
* with L_i' some integer matrices. Let A be constructed as follows
*
* A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
*
* and computed the Hermite Normal Form of A = [ H 0 ] U
* Then,
*
* L_i^{-T} = H U_{1,i}
*
* or
*
* H^{-T} = L_i U_{1,i}^T
*
* In other words G = L = H^{-T}.
* To ensure that G is lower triangular, we compute and use its Hermite
* normal form.
*
* The affine transformation matrix returned is then
*
* [ 1 0 ]
* [ y_0 G ]
*
* as any y = y_0 + G y' with y' integer is a solution to the original
* modulo constraints.
*/
struct isl_mat *isl_mat_parameter_compression(
struct isl_mat *B, struct isl_vec *d)
{
int i;
struct isl_mat *cst = NULL;
struct isl_mat *T = NULL;
isl_int D;
if (!B || !d)
goto error;
isl_assert(B->ctx, B->n_row == d->size, goto error);
cst = particular_solution(B, d);
if (!cst)
goto error;
if (cst->n_col == 0) {
T = isl_mat_alloc(B->ctx, B->n_col, 0);
isl_mat_free(cst);
isl_mat_free(B);
isl_vec_free(d);
return T;
}
isl_int_init(D);
/* Replace a*g*row = 0 mod g*m by row = 0 mod m */
for (i = 0; i < B->n_row; ++i) {
isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
if (isl_int_is_one(D))
continue;
if (isl_int_is_zero(D)) {
B = isl_mat_drop_rows(B, i, 1);
d = isl_vec_cow(d);
if (!B || !d)
goto error2;
isl_seq_cpy(d->block.data+i, d->block.data+i+1,
d->size - (i+1));
d->size--;
i--;
continue;
}
B = isl_mat_cow(B);
if (!B)
goto error2;
isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
isl_int_gcd(D, D, d->block.data[i]);
d = isl_vec_cow(d);
if (!d)
goto error2;
isl_int_divexact(d->block.data[i], d->block.data[i], D);
}
isl_int_clear(D);
if (B->n_row == 0)
T = isl_mat_identity(B->ctx, B->n_col);
else if (B->n_row == 1)
T = parameter_compression_1(B, d);
else
T = parameter_compression_multi(B, d);
T = isl_mat_left_hermite(T, 0, NULL, NULL);
if (!T)
goto error;
isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
isl_mat_free(cst);
isl_mat_free(B);
isl_vec_free(d);
return T;
error2:
isl_int_clear(D);
error:
isl_mat_free(cst);
isl_mat_free(B);
isl_vec_free(d);
return NULL;
}