isl_mat.c

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/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2014      Ecole Normale Superieure
 * Copyright 2017      Sven Verdoolaege
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
 * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
 */
 
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl/space.h>
#include <isl_seq.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl_space_private.h>
#include <isl_val_private.h>
 
isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat)
{
  return mat ? mat->ctx : NULL;
}
 
/* Return a hash value that digests "mat".
 */
uint32_t isl_mat_get_hash(__isl_keep isl_mat *mat)
{
  int i;
  uint32_t hash;
 
  if (!mat)
    return 0;
 
  hash = isl_hash_init();
  isl_hash_byte(hash, mat->n_row & 0xFF);
  isl_hash_byte(hash, mat->n_col & 0xFF);
  for (i = 0; i < mat->n_row; ++i) {
    uint32_t row_hash;
 
    row_hash = isl_seq_get_hash(mat->row[i], mat->n_col);
    isl_hash_hash(hash, row_hash);
  }
 
  return hash;
}
 
__isl_give isl_mat *isl_mat_alloc(isl_ctx *ctx,
  unsigned n_row, unsigned n_col)
{
  int i;
  struct isl_mat *mat;
 
  mat = isl_alloc_type(ctx, struct isl_mat);
  if (!mat)
    return NULL;
 
  mat->row = NULL;
  mat->block = isl_blk_alloc(ctx, n_row * n_col);
  if (isl_blk_is_error(mat->block))
    goto error;
  mat->row = isl_calloc_array(ctx, isl_int *, n_row);
  if (n_row && !mat->row)
    goto error;
 
  if (n_col != 0) {
    for (i = 0; i < n_row; ++i)
      mat->row[i] = mat->block.data + i * n_col;
  }
 
  mat->ctx = ctx;
  isl_ctx_ref(ctx);
  mat->ref = 1;
  mat->n_row = n_row;
  mat->n_col = n_col;
  mat->max_col = n_col;
  mat->flags = 0;
 
  return mat;
error:
  isl_blk_free(ctx, mat->block);
  free(mat);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_extend(__isl_take isl_mat *mat,
  unsigned n_row, unsigned n_col)
{
  int i;
  isl_int *old;
  isl_int **row;
 
  if (!mat)
    return NULL;
 
  if (mat->max_col >= n_col && mat->n_row >= n_row) {
    if (mat->n_col < n_col)
      mat->n_col = n_col;
    return mat;
  }
 
  if (mat->max_col < n_col) {
    struct isl_mat *new_mat;
 
    if (n_row < mat->n_row)
      n_row = mat->n_row;
    new_mat = isl_mat_alloc(mat->ctx, n_row, n_col);
    if (!new_mat)
      goto error;
    for (i = 0; i < mat->n_row; ++i)
      isl_seq_cpy(new_mat->row[i], mat->row[i], mat->n_col);
    isl_mat_free(mat);
    return new_mat;
  }
 
  mat = isl_mat_cow(mat);
  if (!mat)
    goto error;
 
  old = mat->block.data;
  mat->block = isl_blk_extend(mat->ctx, mat->block, n_row * mat->max_col);
  if (isl_blk_is_error(mat->block))
    goto error;
  row = isl_realloc_array(mat->ctx, mat->row, isl_int *, n_row);
  if (n_row && !row)
    goto error;
  mat->row = row;
 
  for (i = 0; i < mat->n_row; ++i)
    mat->row[i] = mat->block.data + (mat->row[i] - old);
  for (i = mat->n_row; i < n_row; ++i)
    mat->row[i] = mat->block.data + i * mat->max_col;
  mat->n_row = n_row;
  if (mat->n_col < n_col)
    mat->n_col = n_col;
 
  return mat;
error:
  isl_mat_free(mat);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_sub_alloc6(isl_ctx *ctx, isl_int **row,
  unsigned first_row, unsigned n_row, unsigned first_col, unsigned n_col)
{
  int i;
  struct isl_mat *mat;
 
  mat = isl_alloc_type(ctx, struct isl_mat);
  if (!mat)
    return NULL;
  mat->row = isl_alloc_array(ctx, isl_int *, n_row);
  if (n_row && !mat->row)
    goto error;
  for (i = 0; i < n_row; ++i)
    mat->row[i] = row[first_row+i] + first_col;
  mat->ctx = ctx;
  isl_ctx_ref(ctx);
  mat->ref = 1;
  mat->n_row = n_row;
  mat->n_col = n_col;
  mat->block = isl_blk_empty();
  mat->flags = ISL_MAT_BORROWED;
  return mat;
error:
  free(mat);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_sub_alloc(__isl_keep isl_mat *mat,
  unsigned first_row, unsigned n_row, unsigned first_col, unsigned n_col)
{
  if (!mat)
    return NULL;
  return isl_mat_sub_alloc6(mat->ctx, mat->row, first_row, n_row,
          first_col, n_col);
}
 
void isl_mat_sub_copy(struct isl_ctx *ctx, isl_int **dst, isl_int **src,
  unsigned n_row, unsigned dst_col, unsigned src_col, unsigned n_col)
{
  int i;
 
  for (i = 0; i < n_row; ++i)
    isl_seq_cpy(dst[i]+dst_col, src[i]+src_col, n_col);
}
 
void isl_mat_sub_neg(struct isl_ctx *ctx, isl_int **dst, isl_int **src,
  unsigned n_row, unsigned dst_col, unsigned src_col, unsigned n_col)
{
  int i;
 
  for (i = 0; i < n_row; ++i)
    isl_seq_neg(dst[i]+dst_col, src[i]+src_col, n_col);
}
 
__isl_give isl_mat *isl_mat_copy(__isl_keep isl_mat *mat)
{
  if (!mat)
    return NULL;
 
  mat->ref++;
  return mat;
}
 
__isl_give isl_mat *isl_mat_dup(__isl_keep isl_mat *mat)
{
  int i;
  struct isl_mat *mat2;
 
  if (!mat)
    return NULL;
  mat2 = isl_mat_alloc(mat->ctx, mat->n_row, mat->n_col);
  if (!mat2)
    return NULL;
  for (i = 0; i < mat->n_row; ++i)
    isl_seq_cpy(mat2->row[i], mat->row[i], mat->n_col);
  return mat2;
}
 
__isl_give isl_mat *isl_mat_cow(__isl_take isl_mat *mat)
{
  struct isl_mat *mat2;
  if (!mat)
    return NULL;
 
  if (mat->ref == 1 && !ISL_F_ISSET(mat, ISL_MAT_BORROWED))
    return mat;
 
  mat2 = isl_mat_dup(mat);
  isl_mat_free(mat);
  return mat2;
}
 
__isl_null isl_mat *isl_mat_free(__isl_take isl_mat *mat)
{
  if (!mat)
    return NULL;
 
  if (--mat->ref > 0)
    return NULL;
 
  if (!ISL_F_ISSET(mat, ISL_MAT_BORROWED))
    isl_blk_free(mat->ctx, mat->block);
  isl_ctx_deref(mat->ctx);
  free(mat->row);
  free(mat);
 
  return NULL;
}
 
isl_size isl_mat_rows(__isl_keep isl_mat *mat)
{
  return mat ? mat->n_row : isl_size_error;
}
 
isl_size isl_mat_cols(__isl_keep isl_mat *mat)
{
  return mat ? mat->n_col : isl_size_error;
}
 
/* Check that "col" is a valid column position for "mat".
 */
static isl_stat check_col(__isl_keep isl_mat *mat, int col)
{
  if (!mat)
    return isl_stat_error;
  if (col < 0 || col >= mat->n_col)
    isl_die(isl_mat_get_ctx(mat), isl_error_invalid,
      "column out of range", return isl_stat_error);
  return isl_stat_ok;
}
 
/* Check that "row" is a valid row position for "mat".
 */
static isl_stat check_row(__isl_keep isl_mat *mat, int row)
{
  if (!mat)
    return isl_stat_error;
  if (row < 0 || row >= mat->n_row)
    isl_die(isl_mat_get_ctx(mat), isl_error_invalid,
      "row out of range", return isl_stat_error);
  return isl_stat_ok;
}
 
/* Check that there are "n" columns starting at position "first" in "mat".
 */
static isl_stat check_col_range(__isl_keep isl_mat *mat, unsigned first,
  unsigned n)
{
  if (!mat)
    return isl_stat_error;
  if (first + n > mat->n_col || first + n < first)
    isl_die(isl_mat_get_ctx(mat), isl_error_invalid,
      "column position or range out of bounds",
      return isl_stat_error);
  return isl_stat_ok;
}
 
/* Check that there are "n" rows starting at position "first" in "mat".
 */
static isl_stat check_row_range(__isl_keep isl_mat *mat, unsigned first,
  unsigned n)
{
  if (!mat)
    return isl_stat_error;
  if (first + n > mat->n_row || first + n < first)
    isl_die(isl_mat_get_ctx(mat), isl_error_invalid,
      "row position or range out of bounds",
      return isl_stat_error);
  return isl_stat_ok;
}
 
int isl_mat_get_element(__isl_keep isl_mat *mat, int row, int col, isl_int *v)
{
  if (check_row(mat, row) < 0)
    return -1;
  if (check_col(mat, col) < 0)
    return -1;
  isl_int_set(*v, mat->row[row][col]);
  return 0;
}
 
/* Extract the element at row "row", oolumn "col" of "mat".
 */
__isl_give isl_val *isl_mat_get_element_val(__isl_keep isl_mat *mat,
  int row, int col)
{
  isl_ctx *ctx;
 
  if (check_row(mat, row) < 0)
    return NULL;
  if (check_col(mat, col) < 0)
    return NULL;
  ctx = isl_mat_get_ctx(mat);
  return isl_val_int_from_isl_int(ctx, mat->row[row][col]);
}
 
__isl_give isl_mat *isl_mat_set_element(__isl_take isl_mat *mat,
  int row, int col, isl_int v)
{
  mat = isl_mat_cow(mat);
  if (check_row(mat, row) < 0)
    return isl_mat_free(mat);
  if (check_col(mat, col) < 0)
    return isl_mat_free(mat);
  isl_int_set(mat->row[row][col], v);
  return mat;
}
 
__isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
  int row, int col, int v)
{
  mat = isl_mat_cow(mat);
  if (check_row(mat, row) < 0)
    return isl_mat_free(mat);
  if (check_col(mat, col) < 0)
    return isl_mat_free(mat);
  isl_int_set_si(mat->row[row][col], v);
  return mat;
}
 
/* Replace the element at row "row", column "col" of "mat" by "v".
 */
__isl_give isl_mat *isl_mat_set_element_val(__isl_take isl_mat *mat,
  int row, int col, __isl_take isl_val *v)
{
  if (!v)
    return isl_mat_free(mat);
  if (!isl_val_is_int(v))
    isl_die(isl_val_get_ctx(v), isl_error_invalid,
      "expecting integer value", goto error);
  mat = isl_mat_set_element(mat, row, col, v->n);
  isl_val_free(v);
  return mat;
error:
  isl_val_free(v);
  return isl_mat_free(mat);
}
 
__isl_give isl_mat *isl_mat_diag(isl_ctx *ctx, unsigned n_row, isl_int d)
{
  int i;
  struct isl_mat *mat;
 
  mat = isl_mat_alloc(ctx, n_row, n_row);
  if (!mat)
    return NULL;
  for (i = 0; i < n_row; ++i) {
    isl_seq_clr(mat->row[i], i);
    isl_int_set(mat->row[i][i], d);
    isl_seq_clr(mat->row[i]+i+1, n_row-(i+1));
  }
 
  return mat;
}
 
/* Create an "n_row" by "n_col" matrix with zero elements.
 */
__isl_give isl_mat *isl_mat_zero(isl_ctx *ctx, unsigned n_row, unsigned n_col)
{
  int i;
  isl_mat *mat;
 
  mat = isl_mat_alloc(ctx, n_row, n_col);
  if (!mat)
    return NULL;
  for (i = 0; i < n_row; ++i)
    isl_seq_clr(mat->row[i], n_col);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_identity(isl_ctx *ctx, unsigned n_row)
{
  if (!ctx)
    return NULL;
  return isl_mat_diag(ctx, n_row, ctx->one);
}
 
/* Is "mat" a (possibly scaled) identity matrix?
 */
isl_bool isl_mat_is_scaled_identity(__isl_keep isl_mat *mat)
{
  int i;
 
  if (!mat)
    return isl_bool_error;
  if (mat->n_row != mat->n_col)
    return isl_bool_false;
 
  for (i = 0; i < mat->n_row; ++i) {
    if (isl_seq_first_non_zero(mat->row[i], i) != -1)
      return isl_bool_false;
    if (isl_int_ne(mat->row[0][0], mat->row[i][i]))
      return isl_bool_false;
    if (isl_seq_first_non_zero(mat->row[i] + i + 1,
              mat->n_col - (i + 1)) != -1)
      return isl_bool_false;
  }
 
  return isl_bool_true;
}
 
__isl_give isl_vec *isl_mat_vec_product(__isl_take isl_mat *mat,
  __isl_take isl_vec *vec)
{
  int i;
  struct isl_vec *prod;
 
  if (!mat || !vec)
    goto error;
 
  isl_assert(mat->ctx, mat->n_col == vec->size, goto error);
 
  prod = isl_vec_alloc(mat->ctx, mat->n_row);
  if (!prod)
    goto error;
 
  for (i = 0; i < prod->size; ++i)
    isl_seq_inner_product(mat->row[i], vec->el, vec->size,
          &prod->block.data[i]);
  isl_mat_free(mat);
  isl_vec_free(vec);
  return prod;
error:
  isl_mat_free(mat);
  isl_vec_free(vec);
  return NULL;
}
 
__isl_give isl_vec *isl_mat_vec_inverse_product(__isl_take isl_mat *mat,
  __isl_take isl_vec *vec)
{
  struct isl_mat *vec_mat;
  int i;
 
  if (!mat || !vec)
    goto error;
  vec_mat = isl_mat_alloc(vec->ctx, vec->size, 1);
  if (!vec_mat)
    goto error;
  for (i = 0; i < vec->size; ++i)
    isl_int_set(vec_mat->row[i][0], vec->el[i]);
  vec_mat = isl_mat_inverse_product(mat, vec_mat);
  isl_vec_free(vec);
  if (!vec_mat)
    return NULL;
  vec = isl_vec_alloc(vec_mat->ctx, vec_mat->n_row);
  if (vec)
    for (i = 0; i < vec->size; ++i)
      isl_int_set(vec->el[i], vec_mat->row[i][0]);
  isl_mat_free(vec_mat);
  return vec;
error:
  isl_mat_free(mat);
  isl_vec_free(vec);
  return NULL;
}
 
__isl_give isl_vec *isl_vec_mat_product(__isl_take isl_vec *vec,
  __isl_take isl_mat *mat)
{
  int i, j;
  struct isl_vec *prod;
 
  if (!mat || !vec)
    goto error;
 
  isl_assert(mat->ctx, mat->n_row == vec->size, goto error);
 
  prod = isl_vec_alloc(mat->ctx, mat->n_col);
  if (!prod)
    goto error;
 
  for (i = 0; i < prod->size; ++i) {
    isl_int_set_si(prod->el[i], 0);
    for (j = 0; j < vec->size; ++j)
      isl_int_addmul(prod->el[i], vec->el[j], mat->row[j][i]);
  }
  isl_mat_free(mat);
  isl_vec_free(vec);
  return prod;
error:
  isl_mat_free(mat);
  isl_vec_free(vec);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_aff_direct_sum(__isl_take isl_mat *left,
  __isl_take isl_mat *right)
{
  int i;
  struct isl_mat *sum;
 
  if (!left || !right)
    goto error;
 
  isl_assert(left->ctx, left->n_row == right->n_row, goto error);
  isl_assert(left->ctx, left->n_row >= 1, goto error);
  isl_assert(left->ctx, left->n_col >= 1, goto error);
  isl_assert(left->ctx, right->n_col >= 1, goto error);
  isl_assert(left->ctx,
      isl_seq_first_non_zero(left->row[0]+1, left->n_col-1) == -1,
      goto error);
  isl_assert(left->ctx,
      isl_seq_first_non_zero(right->row[0]+1, right->n_col-1) == -1,
      goto error);
 
  sum = isl_mat_alloc(left->ctx, left->n_row, left->n_col + right->n_col - 1);
  if (!sum)
    goto error;
  isl_int_lcm(sum->row[0][0], left->row[0][0], right->row[0][0]);
  isl_int_divexact(left->row[0][0], sum->row[0][0], left->row[0][0]);
  isl_int_divexact(right->row[0][0], sum->row[0][0], right->row[0][0]);
 
  isl_seq_clr(sum->row[0]+1, sum->n_col-1);
  for (i = 1; i < sum->n_row; ++i) {
    isl_int_mul(sum->row[i][0], left->row[0][0], left->row[i][0]);
    isl_int_addmul(sum->row[i][0],
        right->row[0][0], right->row[i][0]);
    isl_seq_scale(sum->row[i]+1, left->row[i]+1, left->row[0][0],
        left->n_col-1);
    isl_seq_scale(sum->row[i]+left->n_col,
        right->row[i]+1, right->row[0][0],
        right->n_col-1);
  }
 
  isl_int_divexact(left->row[0][0], sum->row[0][0], left->row[0][0]);
  isl_int_divexact(right->row[0][0], sum->row[0][0], right->row[0][0]);
  isl_mat_free(left);
  isl_mat_free(right);
  return sum;
error:
  isl_mat_free(left);
  isl_mat_free(right);
  return NULL;
}
 
static void exchange(__isl_keep isl_mat *M, __isl_keep isl_mat **U,
  __isl_keep isl_mat **Q, unsigned row, unsigned i, unsigned j)
{
  int r;
  for (r = row; r < M->n_row; ++r)
    isl_int_swap(M->row[r][i], M->row[r][j]);
  if (U) {
    for (r = 0; r < (*U)->n_row; ++r)
      isl_int_swap((*U)->row[r][i], (*U)->row[r][j]);
  }
  if (Q)
    isl_mat_swap_rows(*Q, i, j);
}
 
static void subtract(__isl_keep isl_mat *M, __isl_keep isl_mat **U,
  __isl_keep isl_mat **Q, unsigned row, unsigned i, unsigned j, isl_int m)
{
  int r;
  for (r = row; r < M->n_row; ++r)
    isl_int_submul(M->row[r][j], m, M->row[r][i]);
  if (U) {
    for (r = 0; r < (*U)->n_row; ++r)
      isl_int_submul((*U)->row[r][j], m, (*U)->row[r][i]);
  }
  if (Q) {
    for (r = 0; r < (*Q)->n_col; ++r)
      isl_int_addmul((*Q)->row[i][r], m, (*Q)->row[j][r]);
  }
}
 
static void oppose(__isl_keep isl_mat *M, __isl_keep isl_mat **U,
  __isl_keep isl_mat **Q, unsigned row, unsigned col)
{
  int r;
  for (r = row; r < M->n_row; ++r)
    isl_int_neg(M->row[r][col], M->row[r][col]);
  if (U) {
    for (r = 0; r < (*U)->n_row; ++r)
      isl_int_neg((*U)->row[r][col], (*U)->row[r][col]);
  }
  if (Q)
    isl_seq_neg((*Q)->row[col], (*Q)->row[col], (*Q)->n_col);
}
 
/* Given matrix M, compute
 *
 *    M U = H
 *    M   = H Q
 *
 * with U and Q unimodular matrices and H a matrix in column echelon form
 * such that on each echelon row the entries in the non-echelon column
 * are non-negative (if neg == 0) or non-positive (if neg == 1)
 * and strictly smaller (in absolute value) than the entries in the echelon
 * column.
 * If U or Q are NULL, then these matrices are not computed.
 */
__isl_give isl_mat *isl_mat_left_hermite(__isl_take isl_mat *M, int neg,
  __isl_give isl_mat **U, __isl_give isl_mat **Q)
{
  isl_int c;
  int row, col;
 
  if (U)
    *U = NULL;
  if (Q)
    *Q = NULL;
  if (!M)
    goto error;
  if (U) {
    *U = isl_mat_identity(M->ctx, M->n_col);
    if (!*U)
      goto error;
  }
  if (Q) {
    *Q = isl_mat_identity(M->ctx, M->n_col);
    if (!*Q)
      goto error;
  }
 
  if (M->n_col == 0)
    return M;
 
  M = isl_mat_cow(M);
  if (!M)
    goto error;
 
  col = 0;
  isl_int_init(c);
  for (row = 0; row < M->n_row; ++row) {
    int first, i, off;
    first = isl_seq_abs_min_non_zero(M->row[row]+col, M->n_col-col);
    if (first == -1)
      continue;
    first += col;
    if (first != col)
      exchange(M, U, Q, row, first, col);
    if (isl_int_is_neg(M->row[row][col]))
      oppose(M, U, Q, row, col);
    first = col+1;
    while ((off = isl_seq_first_non_zero(M->row[row]+first,
                   M->n_col-first)) != -1) {
      first += off;
      isl_int_fdiv_q(c, M->row[row][first], M->row[row][col]);
      subtract(M, U, Q, row, col, first, c);
      if (!isl_int_is_zero(M->row[row][first]))
        exchange(M, U, Q, row, first, col);
      else
        ++first;
    }
    for (i = 0; i < col; ++i) {
      if (isl_int_is_zero(M->row[row][i]))
        continue;
      if (neg)
        isl_int_cdiv_q(c, M->row[row][i], M->row[row][col]);
      else
        isl_int_fdiv_q(c, M->row[row][i], M->row[row][col]);
      if (isl_int_is_zero(c))
        continue;
      subtract(M, U, Q, row, col, i, c);
    }
    ++col;
  }
  isl_int_clear(c);
 
  return M;
error:
  if (Q) {
    isl_mat_free(*Q);
    *Q = NULL;
  }
  if (U) {
    isl_mat_free(*U);
    *U = NULL;
  }
  isl_mat_free(M);
  return NULL;
}
 
/* Use row "row" of "mat" to eliminate column "col" from all other rows.
 */
static __isl_give isl_mat *eliminate(__isl_take isl_mat *mat, int row, int col)
{
  int k;
  isl_size nr, nc;
  isl_ctx *ctx;
 
  nr = isl_mat_rows(mat);
  nc = isl_mat_cols(mat);
  if (nr < 0 || nc < 0)
    return isl_mat_free(mat);
 
  ctx = isl_mat_get_ctx(mat);
 
  for (k = 0; k < nr; ++k) {
    if (k == row)
      continue;
    if (isl_int_is_zero(mat->row[k][col]))
      continue;
    mat = isl_mat_cow(mat);
    if (!mat)
      return NULL;
    isl_seq_elim(mat->row[k], mat->row[row], col, nc, NULL);
    isl_seq_normalize(ctx, mat->row[k], nc);
  }
 
  return mat;
}
 
/* Perform Gaussian elimination on the rows of "mat", but start
 * from the final row and the final column.
 * Any zero rows that result from the elimination are removed.
 *
 * In particular, for each column from last to first,
 * look for the last row with a non-zero coefficient in that column,
 * move it last (but before other rows moved last in previous steps) and
 * use it to eliminate the column from the other rows.
 */
__isl_give isl_mat *isl_mat_reverse_gauss(__isl_take isl_mat *mat)
{
  int k, row, last;
  isl_size nr, nc;
 
  nr = isl_mat_rows(mat);
  nc = isl_mat_cols(mat);
  if (nr < 0 || nc < 0)
    return isl_mat_free(mat);
 
  last = nc - 1;
  for (row = nr - 1; row >= 0; --row) {
    for (; last >= 0; --last) {
      for (k = row; k >= 0; --k)
        if (!isl_int_is_zero(mat->row[k][last]))
          break;
      if (k >= 0)
        break;
    }
    if (last < 0)
      break;
    if (k != row)
      mat = isl_mat_swap_rows(mat, k, row);
    if (!mat)
      return NULL;
    if (isl_int_is_neg(mat->row[row][last]))
      mat = isl_mat_row_neg(mat, row);
    mat = eliminate(mat, row, last);
    if (!mat)
      return NULL;
  }
  mat = isl_mat_drop_rows(mat, 0, row + 1);
 
  return mat;
}
 
/* Negate the lexicographically negative rows of "mat" such that
 * all rows in the result are lexicographically non-negative.
 */
__isl_give isl_mat *isl_mat_lexnonneg_rows(__isl_take isl_mat *mat)
{
  int i;
  isl_size nr, nc;
 
  nr = isl_mat_rows(mat);
  nc = isl_mat_cols(mat);
  if (nr < 0 || nc < 0)
    return isl_mat_free(mat);
 
  for (i = 0; i < nr; ++i) {
    int pos;
 
    pos = isl_seq_first_non_zero(mat->row[i], nc);
    if (pos < 0)
      continue;
    if (isl_int_is_nonneg(mat->row[i][pos]))
      continue;
    mat = isl_mat_row_neg(mat, i);
    if (!mat)
      return NULL;
  }
 
  return mat;
}
 
/* Given a matrix "H" is column echelon form, what is the first
 * zero column?  That is how many initial columns are non-zero?
 * Start looking at column "first_col" and only consider
 * the columns to be of size "n_row".
 * "H" is assumed to be non-NULL.
 *
 * Since "H" is in column echelon form, the first non-zero entry
 * in a column is always in a later position compared to the previous column.
 */
static int hermite_first_zero_col(__isl_keep isl_mat *H, int first_col,
  int n_row)
{
  int row, col;
 
  for (col = first_col, row = 0; col < H->n_col; ++col) {
    for (; row < n_row; ++row)
      if (!isl_int_is_zero(H->row[row][col]))
        break;
    if (row == n_row)
      return col;
  }
 
  return H->n_col;
}
 
/* Return the rank of "mat", or isl_size_error in case of error.
 */
isl_size isl_mat_rank(__isl_keep isl_mat *mat)
{
  int rank;
  isl_mat *H;
 
  H = isl_mat_left_hermite(isl_mat_copy(mat), 0, NULL, NULL);
  if (!H)
    return isl_size_error;
 
  rank = hermite_first_zero_col(H, 0, H->n_row);
  isl_mat_free(H);
 
  return rank;
}
 
__isl_give isl_mat *isl_mat_right_kernel(__isl_take isl_mat *mat)
{
  int rank;
  struct isl_mat *U = NULL;
  struct isl_mat *K;
 
  mat = isl_mat_left_hermite(mat, 0, &U, NULL);
  if (!mat || !U)
    goto error;
 
  rank = hermite_first_zero_col(mat, 0, mat->n_row);
  K = isl_mat_alloc(U->ctx, U->n_row, U->n_col - rank);
  if (!K)
    goto error;
  isl_mat_sub_copy(K->ctx, K->row, U->row, U->n_row, 0, rank, U->n_col-rank);
  isl_mat_free(mat);
  isl_mat_free(U);
  return K;
error:
  isl_mat_free(mat);
  isl_mat_free(U);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_lin_to_aff(__isl_take isl_mat *mat)
{
  int i;
  struct isl_mat *mat2;
 
  if (!mat)
    return NULL;
  mat2 = isl_mat_alloc(mat->ctx, 1+mat->n_row, 1+mat->n_col);
  if (!mat2)
    goto error;
  isl_int_set_si(mat2->row[0][0], 1);
  isl_seq_clr(mat2->row[0]+1, mat->n_col);
  for (i = 0; i < mat->n_row; ++i) {
    isl_int_set_si(mat2->row[1+i][0], 0);
    isl_seq_cpy(mat2->row[1+i]+1, mat->row[i], mat->n_col);
  }
  isl_mat_free(mat);
  return mat2;
error:
  isl_mat_free(mat);
  return NULL;
}
 
/* Given two matrices M1 and M2, return the block matrix
 *
 *  [ M1  0  ]
 *  [ 0   M2 ]
 */
__isl_give isl_mat *isl_mat_diagonal(__isl_take isl_mat *mat1,
  __isl_take isl_mat *mat2)
{
  int i;
  isl_mat *mat;
 
  if (!mat1 || !mat2)
    goto error;
 
  mat = isl_mat_alloc(mat1->ctx, mat1->n_row + mat2->n_row,
               mat1->n_col + mat2->n_col);
  if (!mat)
    goto error;
  for (i = 0; i < mat1->n_row; ++i) {
    isl_seq_cpy(mat->row[i], mat1->row[i], mat1->n_col);
    isl_seq_clr(mat->row[i] + mat1->n_col, mat2->n_col);
  }
  for (i = 0; i < mat2->n_row; ++i) {
    isl_seq_clr(mat->row[mat1->n_row + i], mat1->n_col);
    isl_seq_cpy(mat->row[mat1->n_row + i] + mat1->n_col,
                mat2->row[i], mat2->n_col);
  }
  isl_mat_free(mat1);
  isl_mat_free(mat2);
  return mat;
error:
  isl_mat_free(mat1);
  isl_mat_free(mat2);
  return NULL;
}
 
static int row_first_non_zero(isl_int **row, unsigned n_row, unsigned col)
{
  int i;
 
  for (i = 0; i < n_row; ++i)
    if (!isl_int_is_zero(row[i][col]))
      return i;
  return -1;
}
 
static int row_abs_min_non_zero(isl_int **row, unsigned n_row, unsigned col)
{
  int i, min = row_first_non_zero(row, n_row, col);
  if (min < 0)
    return -1;
  for (i = min + 1; i < n_row; ++i) {
    if (isl_int_is_zero(row[i][col]))
      continue;
    if (isl_int_abs_lt(row[i][col], row[min][col]))
      min = i;
  }
  return min;
}
 
static isl_stat inv_exchange(__isl_keep isl_mat **left,
  __isl_keep isl_mat **right, unsigned i, unsigned j)
{
  *left = isl_mat_swap_rows(*left, i, j);
  *right = isl_mat_swap_rows(*right, i, j);
 
  if (!*left || !*right)
    return isl_stat_error;
  return isl_stat_ok;
}
 
static void inv_oppose(
  __isl_keep isl_mat *left, __isl_keep isl_mat *right, unsigned row)
{
  isl_seq_neg(left->row[row]+row, left->row[row]+row, left->n_col-row);
  isl_seq_neg(right->row[row], right->row[row], right->n_col);
}
 
static void inv_subtract(__isl_keep isl_mat *left, __isl_keep isl_mat *right,
  unsigned row, unsigned i, isl_int m)
{
  isl_int_neg(m, m);
  isl_seq_combine(left->row[i]+row,
      left->ctx->one, left->row[i]+row,
      m, left->row[row]+row,
      left->n_col-row);
  isl_seq_combine(right->row[i], right->ctx->one, right->row[i],
      m, right->row[row], right->n_col);
}
 
/* Compute inv(left)*right
 */
__isl_give isl_mat *isl_mat_inverse_product(__isl_take isl_mat *left,
  __isl_take isl_mat *right)
{
  int row;
  isl_int a, b;
 
  if (!left || !right)
    goto error;
 
  isl_assert(left->ctx, left->n_row == left->n_col, goto error);
  isl_assert(left->ctx, left->n_row == right->n_row, goto error);
 
  if (left->n_row == 0) {
    isl_mat_free(left);
    return right;
  }
 
  left = isl_mat_cow(left);
  right = isl_mat_cow(right);
  if (!left || !right)
    goto error;
 
  isl_int_init(a);
  isl_int_init(b);
  for (row = 0; row < left->n_row; ++row) {
    int pivot, first, i, off;
    pivot = row_abs_min_non_zero(left->row+row, left->n_row-row, row);
    if (pivot < 0) {
      isl_int_clear(a);
      isl_int_clear(b);
      isl_assert(left->ctx, pivot >= 0, goto error);
    }
    pivot += row;
    if (pivot != row)
      if (inv_exchange(&left, &right, pivot, row) < 0)
        goto error;
    if (isl_int_is_neg(left->row[row][row]))
      inv_oppose(left, right, row);
    first = row+1;
    while ((off = row_first_non_zero(left->row+first,
          left->n_row-first, row)) != -1) {
      first += off;
      isl_int_fdiv_q(a, left->row[first][row],
          left->row[row][row]);
      inv_subtract(left, right, row, first, a);
      if (!isl_int_is_zero(left->row[first][row])) {
        if (inv_exchange(&left, &right, row, first) < 0)
          goto error;
      } else {
        ++first;
      }
    }
    for (i = 0; i < row; ++i) {
      if (isl_int_is_zero(left->row[i][row]))
        continue;
      isl_int_gcd(a, left->row[row][row], left->row[i][row]);
      isl_int_divexact(b, left->row[i][row], a);
      isl_int_divexact(a, left->row[row][row], a);
      isl_int_neg(b, b);
      isl_seq_combine(left->row[i] + i,
          a, left->row[i] + i,
          b, left->row[row] + i,
          left->n_col - i);
      isl_seq_combine(right->row[i], a, right->row[i],
          b, right->row[row], right->n_col);
    }
  }
  isl_int_clear(b);
 
  isl_int_set(a, left->row[0][0]);
  for (row = 1; row < left->n_row; ++row)
    isl_int_lcm(a, a, left->row[row][row]);
  if (isl_int_is_zero(a)){
    isl_int_clear(a);
    isl_assert(left->ctx, 0, goto error);
  }
  for (row = 0; row < left->n_row; ++row) {
    isl_int_divexact(left->row[row][row], a, left->row[row][row]);
    if (isl_int_is_one(left->row[row][row]))
      continue;
    isl_seq_scale(right->row[row], right->row[row],
        left->row[row][row], right->n_col);
  }
  isl_int_clear(a);
 
  isl_mat_free(left);
  return right;
error:
  isl_mat_free(left);
  isl_mat_free(right);
  return NULL;
}
 
void isl_mat_col_scale(__isl_keep isl_mat *mat, unsigned col, isl_int m)
{
  int i;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_int_mul(mat->row[i][col], mat->row[i][col], m);
}
 
void isl_mat_col_combine(__isl_keep isl_mat *mat, unsigned dst,
  isl_int m1, unsigned src1, isl_int m2, unsigned src2)
{
  int i;
  isl_int tmp;
 
  isl_int_init(tmp);
  for (i = 0; i < mat->n_row; ++i) {
    isl_int_mul(tmp, m1, mat->row[i][src1]);
    isl_int_addmul(tmp, m2, mat->row[i][src2]);
    isl_int_set(mat->row[i][dst], tmp);
  }
  isl_int_clear(tmp);
}
 
__isl_give isl_mat *isl_mat_right_inverse(__isl_take isl_mat *mat)
{
  struct isl_mat *inv;
  int row;
  isl_int a, b;
 
  mat = isl_mat_cow(mat);
  if (!mat)
    return NULL;
 
  inv = isl_mat_identity(mat->ctx, mat->n_col);
  inv = isl_mat_cow(inv);
  if (!inv)
    goto error;
 
  isl_int_init(a);
  isl_int_init(b);
  for (row = 0; row < mat->n_row; ++row) {
    int pivot, first, i, off;
    pivot = isl_seq_abs_min_non_zero(mat->row[row]+row, mat->n_col-row);
    if (pivot < 0) {
      isl_int_clear(a);
      isl_int_clear(b);
      isl_assert(mat->ctx, pivot >= 0, goto error);
    }
    pivot += row;
    if (pivot != row)
      exchange(mat, &inv, NULL, row, pivot, row);
    if (isl_int_is_neg(mat->row[row][row]))
      oppose(mat, &inv, NULL, row, row);
    first = row+1;
    while ((off = isl_seq_first_non_zero(mat->row[row]+first,
                mat->n_col-first)) != -1) {
      first += off;
      isl_int_fdiv_q(a, mat->row[row][first],
                mat->row[row][row]);
      subtract(mat, &inv, NULL, row, row, first, a);
      if (!isl_int_is_zero(mat->row[row][first]))
        exchange(mat, &inv, NULL, row, row, first);
      else
        ++first;
    }
    for (i = 0; i < row; ++i) {
      if (isl_int_is_zero(mat->row[row][i]))
        continue;
      isl_int_gcd(a, mat->row[row][row], mat->row[row][i]);
      isl_int_divexact(b, mat->row[row][i], a);
      isl_int_divexact(a, mat->row[row][row], a);
      isl_int_neg(a, a);
      isl_mat_col_combine(mat, i, a, i, b, row);
      isl_mat_col_combine(inv, i, a, i, b, row);
    }
  }
  isl_int_clear(b);
 
  isl_int_set(a, mat->row[0][0]);
  for (row = 1; row < mat->n_row; ++row)
    isl_int_lcm(a, a, mat->row[row][row]);
  if (isl_int_is_zero(a)){
    isl_int_clear(a);
    goto error;
  }
  for (row = 0; row < mat->n_row; ++row) {
    isl_int_divexact(mat->row[row][row], a, mat->row[row][row]);
    if (isl_int_is_one(mat->row[row][row]))
      continue;
    isl_mat_col_scale(inv, row, mat->row[row][row]);
  }
  isl_int_clear(a);
 
  isl_mat_free(mat);
 
  return inv;
error:
  isl_mat_free(mat);
  isl_mat_free(inv);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_transpose(__isl_take isl_mat *mat)
{
  struct isl_mat *transpose = NULL;
  int i, j;
 
  if (!mat)
    return NULL;
 
  if (mat->n_col == mat->n_row) {
    mat = isl_mat_cow(mat);
    if (!mat)
      return NULL;
    for (i = 0; i < mat->n_row; ++i)
      for (j = i + 1; j < mat->n_col; ++j)
        isl_int_swap(mat->row[i][j], mat->row[j][i]);
    return mat;
  }
  transpose = isl_mat_alloc(mat->ctx, mat->n_col, mat->n_row);
  if (!transpose)
    goto error;
  for (i = 0; i < mat->n_row; ++i)
    for (j = 0; j < mat->n_col; ++j)
      isl_int_set(transpose->row[j][i], mat->row[i][j]);
  isl_mat_free(mat);
  return transpose;
error:
  isl_mat_free(mat);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_swap_cols(__isl_take isl_mat *mat,
  unsigned i, unsigned j)
{
  int r;
 
  mat = isl_mat_cow(mat);
  if (check_col_range(mat, i, 1) < 0 ||
      check_col_range(mat, j, 1) < 0)
    return isl_mat_free(mat);
 
  for (r = 0; r < mat->n_row; ++r)
    isl_int_swap(mat->row[r][i], mat->row[r][j]);
  return mat;
}
 
__isl_give isl_mat *isl_mat_swap_rows(__isl_take isl_mat *mat,
  unsigned i, unsigned j)
{
  isl_int *t;
 
  if (!mat)
    return NULL;
  mat = isl_mat_cow(mat);
  if (check_row_range(mat, i, 1) < 0 ||
      check_row_range(mat, j, 1) < 0)
    return isl_mat_free(mat);
 
  t = mat->row[i];
  mat->row[i] = mat->row[j];
  mat->row[j] = t;
  return mat;
}
 
/* Calculate the product of two matrices.
 *
 * This function is optimized for operand matrices that contain many zeros and
 * skips multiplications where we know one of the operands is zero.
 */
__isl_give isl_mat *isl_mat_product(__isl_take isl_mat *left,
  __isl_take isl_mat *right)
{
  int i, j, k;
  struct isl_mat *prod;
 
  if (!left || !right)
    goto error;
  isl_assert(left->ctx, left->n_col == right->n_row, goto error);
  prod = isl_mat_alloc(left->ctx, left->n_row, right->n_col);
  if (!prod)
    goto error;
  if (left->n_col == 0) {
    for (i = 0; i < prod->n_row; ++i)
      isl_seq_clr(prod->row[i], prod->n_col);
    isl_mat_free(left);
    isl_mat_free(right);
    return prod;
  }
  for (i = 0; i < prod->n_row; ++i) {
    for (j = 0; j < prod->n_col; ++j)
      isl_int_mul(prod->row[i][j],
            left->row[i][0], right->row[0][j]);
    for (k = 1; k < left->n_col; ++k) {
      if (isl_int_is_zero(left->row[i][k]))
        continue;
      for (j = 0; j < prod->n_col; ++j)
        isl_int_addmul(prod->row[i][j],
              left->row[i][k], right->row[k][j]);
    }
  }
  isl_mat_free(left);
  isl_mat_free(right);
  return prod;
error:
  isl_mat_free(left);
  isl_mat_free(right);
  return NULL;
}
 
/* Replace the variables x in the rows q by x' given by x = M x',
 * with M the matrix mat.
 *
 * If the number of new variables is greater than the original
 * number of variables, then the rows q have already been
 * preextended.  If the new number is smaller, then the coefficients
 * of the divs, which are not changed, need to be shifted down.
 * The row q may be the equalities, the inequalities or the
 * div expressions.  In the latter case, has_div is true and
 * we need to take into account the extra denominator column.
 */
static int preimage(struct isl_ctx *ctx, isl_int **q, unsigned n,
  unsigned n_div, int has_div, struct isl_mat *mat)
{
  int i;
  struct isl_mat *t;
  int e;
 
  if (mat->n_col >= mat->n_row)
    e = 0;
  else
    e = mat->n_row - mat->n_col;
  if (has_div)
    for (i = 0; i < n; ++i)
      isl_int_mul(q[i][0], q[i][0], mat->row[0][0]);
  t = isl_mat_sub_alloc6(mat->ctx, q, 0, n, has_div, mat->n_row);
  t = isl_mat_product(t, mat);
  if (!t)
    return -1;
  for (i = 0; i < n; ++i) {
    isl_seq_swp_or_cpy(q[i] + has_div, t->row[i], t->n_col);
    isl_seq_cpy(q[i] + has_div + t->n_col,
          q[i] + has_div + t->n_col + e, n_div);
    isl_seq_clr(q[i] + has_div + t->n_col + n_div, e);
  }
  isl_mat_free(t);
  return 0;
}
 
/* Replace the variables x in bset by x' given by x = M x', with
 * M the matrix mat.
 *
 * If there are fewer variables x' then there are x, then we perform
 * the transformation in place, which means that, in principle,
 * this frees up some extra variables as the number
 * of columns remains constant, but we would have to extend
 * the div array too as the number of rows in this array is assumed
 * to be equal to extra.
 */
__isl_give isl_basic_set *isl_basic_set_preimage(
  __isl_take isl_basic_set *bset, __isl_take isl_mat *mat)
{
  struct isl_ctx *ctx;
 
  if (!bset || !mat)
    goto error;
 
  ctx = bset->ctx;
  bset = isl_basic_set_cow(bset);
  if (isl_basic_set_check_no_params(bset) < 0)
    goto error;
 
  isl_assert(ctx, 1+bset->dim->n_out == mat->n_row, goto error);
  isl_assert(ctx, mat->n_col > 0, goto error);
 
  if (mat->n_col > mat->n_row) {
    bset = isl_basic_set_add_dims(bset, isl_dim_set,
            mat->n_col - mat->n_row);
    if (!bset)
      goto error;
  } else if (mat->n_col < mat->n_row) {
    bset->dim = isl_space_cow(bset->dim);
    if (!bset->dim)
      goto error;
    bset->dim->n_out -= mat->n_row - mat->n_col;
  }
 
  if (preimage(ctx, bset->eq, bset->n_eq, bset->n_div, 0,
      isl_mat_copy(mat)) < 0)
    goto error;
 
  if (preimage(ctx, bset->ineq, bset->n_ineq, bset->n_div, 0,
      isl_mat_copy(mat)) < 0)
    goto error;
 
  if (preimage(ctx, bset->div, bset->n_div, bset->n_div, 1, mat) < 0)
    goto error2;
 
  ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
  ISL_F_CLR(bset, ISL_BASIC_SET_NO_REDUNDANT);
  ISL_F_CLR(bset, ISL_BASIC_SET_SORTED);
  ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED_DIVS);
  ISL_F_CLR(bset, ISL_BASIC_SET_ALL_EQUALITIES);
 
  bset = isl_basic_set_simplify(bset);
  bset = isl_basic_set_finalize(bset);
 
  return bset;
error:
  isl_mat_free(mat);
error2:
  isl_basic_set_free(bset);
  return NULL;
}
 
__isl_give isl_set *isl_set_preimage(
  __isl_take isl_set *set, __isl_take isl_mat *mat)
{
  int i;
 
  set = isl_set_cow(set);
  if (!set)
    goto error;
 
  for (i = 0; i < set->n; ++i) {
    set->p[i] = isl_basic_set_preimage(set->p[i],
                isl_mat_copy(mat));
    if (!set->p[i])
      goto error;
  }
  if (mat->n_col != mat->n_row) {
    set->dim = isl_space_cow(set->dim);
    if (!set->dim)
      goto error;
    set->dim->n_out += mat->n_col;
    set->dim->n_out -= mat->n_row;
  }
  isl_mat_free(mat);
  ISL_F_CLR(set, ISL_SET_NORMALIZED);
  return set;
error:
  isl_set_free(set);
  isl_mat_free(mat);
  return NULL;
}
 
/* Replace the variables x starting at "first_col" in the rows "rows"
 * of some coefficient matrix by x' with x = M x' with M the matrix mat.
 * That is, replace the corresponding coefficients c by c M.
 */
isl_stat isl_mat_sub_transform(isl_int **row, unsigned n_row,
  unsigned first_col, __isl_take isl_mat *mat)
{
  int i;
  isl_ctx *ctx;
  isl_mat *t;
 
  if (!mat)
    return isl_stat_error;
  ctx = isl_mat_get_ctx(mat);
  t = isl_mat_sub_alloc6(ctx, row, 0, n_row, first_col, mat->n_row);
  t = isl_mat_product(t, mat);
  if (!t)
    return isl_stat_error;
  for (i = 0; i < n_row; ++i)
    isl_seq_swp_or_cpy(row[i] + first_col, t->row[i], t->n_col);
  isl_mat_free(t);
  return isl_stat_ok;
}
 
void isl_mat_print_internal(__isl_keep isl_mat *mat, FILE *out, int indent)
{
  int i, j;
 
  if (!mat) {
    fprintf(out, "%*snull mat\n", indent, "");
    return;
  }
 
  if (mat->n_row == 0)
    fprintf(out, "%*s[]\n", indent, "");
 
  for (i = 0; i < mat->n_row; ++i) {
    if (!i)
      fprintf(out, "%*s[[", indent, "");
    else
      fprintf(out, "%*s[", indent+1, "");
    for (j = 0; j < mat->n_col; ++j) {
      if (j)
          fprintf(out, ",");
      isl_int_print(out, mat->row[i][j], 0);
    }
    if (i == mat->n_row-1)
      fprintf(out, "]]\n");
    else
      fprintf(out, "]\n");
  }
}
 
void isl_mat_dump(__isl_keep isl_mat *mat)
{
  isl_mat_print_internal(mat, stderr, 0);
}
 
__isl_give isl_mat *isl_mat_drop_cols(__isl_take isl_mat *mat,
  unsigned col, unsigned n)
{
  int r;
 
  if (n == 0)
    return mat;
 
  mat = isl_mat_cow(mat);
  if (check_col_range(mat, col, n) < 0)
    return isl_mat_free(mat);
 
  if (col != mat->n_col-n) {
    for (r = 0; r < mat->n_row; ++r)
      isl_seq_cpy(mat->row[r]+col, mat->row[r]+col+n,
          mat->n_col - col - n);
  }
  mat->n_col -= n;
  return mat;
}
 
__isl_give isl_mat *isl_mat_drop_rows(__isl_take isl_mat *mat,
  unsigned row, unsigned n)
{
  int r;
 
  mat = isl_mat_cow(mat);
  if (check_row_range(mat, row, n) < 0)
    return isl_mat_free(mat);
 
  for (r = row; r+n < mat->n_row; ++r)
    mat->row[r] = mat->row[r+n];
 
  mat->n_row -= n;
  return mat;
}
 
__isl_give isl_mat *isl_mat_insert_cols(__isl_take isl_mat *mat,
        unsigned col, unsigned n)
{
  isl_mat *ext;
 
  if (check_col_range(mat, col, 0) < 0)
    return isl_mat_free(mat);
  if (n == 0)
    return mat;
 
  ext = isl_mat_alloc(mat->ctx, mat->n_row, mat->n_col + n);
  if (!ext)
    goto error;
 
  isl_mat_sub_copy(mat->ctx, ext->row, mat->row, mat->n_row, 0, 0, col);
  isl_mat_sub_copy(mat->ctx, ext->row, mat->row, mat->n_row,
        col + n, col, mat->n_col - col);
 
  isl_mat_free(mat);
  return ext;
error:
  isl_mat_free(mat);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_insert_zero_cols(__isl_take isl_mat *mat,
  unsigned first, unsigned n)
{
  int i;
 
  if (!mat)
    return NULL;
  mat = isl_mat_insert_cols(mat, first, n);
  if (!mat)
    return NULL;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_seq_clr(mat->row[i] + first, n);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_add_zero_cols(__isl_take isl_mat *mat, unsigned n)
{
  if (!mat)
    return NULL;
 
  return isl_mat_insert_zero_cols(mat, mat->n_col, n);
}
 
__isl_give isl_mat *isl_mat_insert_rows(__isl_take isl_mat *mat,
        unsigned row, unsigned n)
{
  isl_mat *ext;
 
  if (check_row_range(mat, row, 0) < 0)
    return isl_mat_free(mat);
  if (n == 0)
    return mat;
 
  ext = isl_mat_alloc(mat->ctx, mat->n_row + n, mat->n_col);
  if (!ext)
    goto error;
 
  isl_mat_sub_copy(mat->ctx, ext->row, mat->row, row, 0, 0, mat->n_col);
  isl_mat_sub_copy(mat->ctx, ext->row + row + n, mat->row + row,
        mat->n_row - row, 0, 0, mat->n_col);
 
  isl_mat_free(mat);
  return ext;
error:
  isl_mat_free(mat);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_add_rows(__isl_take isl_mat *mat, unsigned n)
{
  if (!mat)
    return NULL;
 
  return isl_mat_insert_rows(mat, mat->n_row, n);
}
 
__isl_give isl_mat *isl_mat_insert_zero_rows(__isl_take isl_mat *mat,
  unsigned row, unsigned n)
{
  int i;
 
  mat = isl_mat_insert_rows(mat, row, n);
  if (!mat)
    return NULL;
  
  for (i = 0; i < n; ++i)
    isl_seq_clr(mat->row[row + i], mat->n_col);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_add_zero_rows(__isl_take isl_mat *mat, unsigned n)
{
  if (!mat)
    return NULL;
 
  return isl_mat_insert_zero_rows(mat, mat->n_row, n);
}
 
void isl_mat_col_submul(__isl_keep isl_mat *mat,
      int dst_col, isl_int f, int src_col)
{
  int i;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_int_submul(mat->row[i][dst_col], f, mat->row[i][src_col]);
}
 
void isl_mat_col_add(__isl_keep isl_mat *mat, int dst_col, int src_col)
{
  int i;
 
  if (!mat)
    return;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_int_add(mat->row[i][dst_col],
          mat->row[i][dst_col], mat->row[i][src_col]);
}
 
void isl_mat_col_mul(__isl_keep isl_mat *mat, int dst_col, isl_int f,
  int src_col)
{
  int i;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_int_mul(mat->row[i][dst_col], f, mat->row[i][src_col]);
}
 
/* Add "f" times column "src_col" to column "dst_col" of "mat" and
 * return the result.
 */
__isl_give isl_mat *isl_mat_col_addmul(__isl_take isl_mat *mat, int dst_col,
  isl_int f, int src_col)
{
  int i;
 
  if (check_col(mat, dst_col) < 0 || check_col(mat, src_col) < 0)
    return isl_mat_free(mat);
 
  for (i = 0; i < mat->n_row; ++i) {
    if (isl_int_is_zero(mat->row[i][src_col]))
      continue;
    mat = isl_mat_cow(mat);
    if (!mat)
      return NULL;
    isl_int_addmul(mat->row[i][dst_col], f, mat->row[i][src_col]);
  }
 
  return mat;
}
 
/* Negate column "col" of "mat" and return the result.
 */
__isl_give isl_mat *isl_mat_col_neg(__isl_take isl_mat *mat, int col)
{
  int i;
 
  if (check_col(mat, col) < 0)
    return isl_mat_free(mat);
 
  for (i = 0; i < mat->n_row; ++i) {
    if (isl_int_is_zero(mat->row[i][col]))
      continue;
    mat = isl_mat_cow(mat);
    if (!mat)
      return NULL;
    isl_int_neg(mat->row[i][col], mat->row[i][col]);
  }
 
  return mat;
}
 
/* Negate row "row" of "mat" and return the result.
 */
__isl_give isl_mat *isl_mat_row_neg(__isl_take isl_mat *mat, int row)
{
  if (check_row(mat, row) < 0)
    return isl_mat_free(mat);
  if (isl_seq_first_non_zero(mat->row[row], mat->n_col) == -1)
    return mat;
  mat = isl_mat_cow(mat);
  if (!mat)
    return NULL;
  isl_seq_neg(mat->row[row], mat->row[row], mat->n_col);
  return mat;
}
 
__isl_give isl_mat *isl_mat_unimodular_complete(__isl_take isl_mat *M, int row)
{
  int r;
  struct isl_mat *H = NULL, *Q = NULL;
 
  if (!M)
    return NULL;
 
  isl_assert(M->ctx, M->n_row == M->n_col, goto error);
  M->n_row = row;
  H = isl_mat_left_hermite(isl_mat_copy(M), 0, NULL, &Q);
  M->n_row = M->n_col;
  if (!H)
    goto error;
  for (r = 0; r < row; ++r)
    isl_assert(M->ctx, isl_int_is_one(H->row[r][r]), goto error);
  for (r = row; r < M->n_row; ++r)
    isl_seq_cpy(M->row[r], Q->row[r], M->n_col);
  isl_mat_free(H);
  isl_mat_free(Q);
  return M;
error:
  isl_mat_free(H);
  isl_mat_free(Q);
  isl_mat_free(M);
  return NULL;
}
 
__isl_give isl_mat *isl_mat_concat(__isl_take isl_mat *top,
  __isl_take isl_mat *bot)
{
  struct isl_mat *mat;
 
  if (!top || !bot)
    goto error;
 
  isl_assert(top->ctx, top->n_col == bot->n_col, goto error);
  if (top->n_row == 0) {
    isl_mat_free(top);
    return bot;
  }
  if (bot->n_row == 0) {
    isl_mat_free(bot);
    return top;
  }
 
  mat = isl_mat_alloc(top->ctx, top->n_row + bot->n_row, top->n_col);
  if (!mat)
    goto error;
  isl_mat_sub_copy(mat->ctx, mat->row, top->row, top->n_row,
       0, 0, mat->n_col);
  isl_mat_sub_copy(mat->ctx, mat->row + top->n_row, bot->row, bot->n_row,
       0, 0, mat->n_col);
  isl_mat_free(top);
  isl_mat_free(bot);
  return mat;
error:
  isl_mat_free(top);
  isl_mat_free(bot);
  return NULL;
}
 
isl_bool isl_mat_is_equal(__isl_keep isl_mat *mat1, __isl_keep isl_mat *mat2)
{
  int i;
 
  if (!mat1 || !mat2)
    return isl_bool_error;
 
  if (mat1->n_row != mat2->n_row)
    return isl_bool_false;
 
  if (mat1->n_col != mat2->n_col)
    return isl_bool_false;
 
  for (i = 0; i < mat1->n_row; ++i)
    if (!isl_seq_eq(mat1->row[i], mat2->row[i], mat1->n_col))
      return isl_bool_false;
 
  return isl_bool_true;
}
 
__isl_give isl_mat *isl_mat_from_row_vec(__isl_take isl_vec *vec)
{
  struct isl_mat *mat;
 
  if (!vec)
    return NULL;
  mat = isl_mat_alloc(vec->ctx, 1, vec->size);
  if (!mat)
    goto error;
 
  isl_seq_cpy(mat->row[0], vec->el, vec->size);
 
  isl_vec_free(vec);
  return mat;
error:
  isl_vec_free(vec);
  return NULL;
}
 
/* Return a copy of row "row" of "mat" as an isl_vec.
 */
__isl_give isl_vec *isl_mat_get_row(__isl_keep isl_mat *mat, unsigned row)
{
  isl_vec *v;
 
  if (!mat)
    return NULL;
  if (row >= mat->n_row)
    isl_die(mat->ctx, isl_error_invalid, "row out of range",
      return NULL);
 
  v = isl_vec_alloc(isl_mat_get_ctx(mat), mat->n_col);
  if (!v)
    return NULL;
  isl_seq_cpy(v->el, mat->row[row], mat->n_col);
 
  return v;
}
 
__isl_give isl_mat *isl_mat_vec_concat(__isl_take isl_mat *top,
  __isl_take isl_vec *bot)
{
  return isl_mat_concat(top, isl_mat_from_row_vec(bot));
}
 
__isl_give isl_mat *isl_mat_move_cols(__isl_take isl_mat *mat,
  unsigned dst_col, unsigned src_col, unsigned n)
{
  isl_mat *res;
 
  if (!mat)
    return NULL;
  if (n == 0 || dst_col == src_col)
    return mat;
 
  res = isl_mat_alloc(mat->ctx, mat->n_row, mat->n_col);
  if (!res)
    goto error;
 
  if (dst_col < src_col) {
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         0, 0, dst_col);
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         dst_col, src_col, n);
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         dst_col + n, dst_col, src_col - dst_col);
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         src_col + n, src_col + n,
         res->n_col - src_col - n);
  } else {
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         0, 0, src_col);
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         src_col, src_col + n, dst_col - src_col);
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         dst_col, src_col, n);
    isl_mat_sub_copy(res->ctx, res->row, mat->row, mat->n_row,
         dst_col + n, dst_col + n,
         res->n_col - dst_col - n);
  }
  isl_mat_free(mat);
 
  return res;
error:
  isl_mat_free(mat);
  return NULL;
}
 
/* Return the gcd of the elements in row "row" of "mat" in *gcd.
 * Return isl_stat_ok on success and isl_stat_error on failure.
 */
isl_stat isl_mat_row_gcd(__isl_keep isl_mat *mat, int row, isl_int *gcd)
{
  if (check_row(mat, row) < 0)
    return isl_stat_error;
 
  isl_seq_gcd(mat->row[row], mat->n_col, gcd);
 
  return isl_stat_ok;
}
 
void isl_mat_gcd(__isl_keep isl_mat *mat, isl_int *gcd)
{
  int i;
  isl_int g;
 
  isl_int_set_si(*gcd, 0);
  if (!mat)
    return;
 
  isl_int_init(g);
  for (i = 0; i < mat->n_row; ++i) {
    isl_seq_gcd(mat->row[i], mat->n_col, &g);
    isl_int_gcd(*gcd, *gcd, g);
  }
  isl_int_clear(g);
}
 
/* Return the result of scaling "mat" by a factor of "m".
 */
__isl_give isl_mat *isl_mat_scale(__isl_take isl_mat *mat, isl_int m)
{
  int i;
 
  if (isl_int_is_one(m))
    return mat;
 
  mat = isl_mat_cow(mat);
  if (!mat)
    return NULL;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_seq_scale(mat->row[i], mat->row[i], m, mat->n_col);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_scale_down(__isl_take isl_mat *mat, isl_int m)
{
  int i;
 
  if (isl_int_is_one(m))
    return mat;
 
  mat = isl_mat_cow(mat);
  if (!mat)
    return NULL;
 
  for (i = 0; i < mat->n_row; ++i)
    isl_seq_scale_down(mat->row[i], mat->row[i], m, mat->n_col);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_scale_down_row(__isl_take isl_mat *mat, int row,
  isl_int m)
{
  if (isl_int_is_one(m))
    return mat;
 
  mat = isl_mat_cow(mat);
  if (!mat)
    return NULL;
 
  isl_seq_scale_down(mat->row[row], mat->row[row], m, mat->n_col);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_normalize(__isl_take isl_mat *mat)
{
  isl_int gcd;
 
  if (!mat)
    return NULL;
 
  isl_int_init(gcd);
  isl_mat_gcd(mat, &gcd);
  mat = isl_mat_scale_down(mat, gcd);
  isl_int_clear(gcd);
 
  return mat;
}
 
__isl_give isl_mat *isl_mat_normalize_row(__isl_take isl_mat *mat, int row)
{
  mat = isl_mat_cow(mat);
  if (!mat)
    return NULL;
 
  isl_seq_normalize(mat->ctx, mat->row[row], mat->n_col);
 
  return mat;
}
 
/* Number of initial non-zero columns.
 */
int isl_mat_initial_non_zero_cols(__isl_keep isl_mat *mat)
{
  int i;
 
  if (!mat)
    return -1;
 
  for (i = 0; i < mat->n_col; ++i)
    if (row_first_non_zero(mat->row, mat->n_row, i) < 0)
      break;
 
  return i;
}
 
/* Return a basis for the space spanned by the rows of "mat".
 * Any basis will do, so simply perform Gaussian elimination and
 * remove the empty rows.
 */
__isl_give isl_mat *isl_mat_row_basis(__isl_take isl_mat *mat)
{
  return isl_mat_reverse_gauss(mat);
}
 
/* Return rows that extend a basis of "mat1" to one
 * that covers both "mat1" and "mat2".
 * The Hermite normal form of the concatenation of the two matrices is
 *
 *                       [ Q1 ]
 *  [ M1 ] = [ H1 0  0 ] [ Q2 ]
 *  [ M2 ] = [ H2 H3 0 ] [ Q3 ]
 *
 * The number of columns in H1 and H3 determine the number of rows
 * in Q1 and Q2.  Q1 is a basis for M1, while Q2 extends this basis
 * to also cover M2.
 */
__isl_give isl_mat *isl_mat_row_basis_extension(
  __isl_take isl_mat *mat1, __isl_take isl_mat *mat2)
{
  isl_size n_row;
  int r1, r;
  isl_size n1;
  isl_mat *H, *Q;
 
  n1 = isl_mat_rows(mat1);
  H = isl_mat_concat(mat1, mat2);
  H = isl_mat_left_hermite(H, 0, NULL, &Q);
  if (n1 < 0 || !H || !Q)
    goto error;
 
  r1 = hermite_first_zero_col(H, 0, n1);
  r = hermite_first_zero_col(H, r1, H->n_row);
  n_row = isl_mat_rows(Q);
  if (n_row < 0)
    goto error;
  Q = isl_mat_drop_rows(Q, r, n_row - r);
  Q = isl_mat_drop_rows(Q, 0, r1);
 
  isl_mat_free(H);
  return Q;
error:
  isl_mat_free(H);
  isl_mat_free(Q);
  return NULL;
}
 
/* Are the rows of "mat1" linearly independent of those of "mat2"?
 * That is, is there no linear dependence among the combined rows
 * that is not already present in either "mat1" or "mat2"?
 * In other words, is the rank of "mat1" and "mat2" combined equal
 * to the sum of the ranks of "mat1" and "mat2"?
 */
isl_bool isl_mat_has_linearly_independent_rows(__isl_keep isl_mat *mat1,
  __isl_keep isl_mat *mat2)
{
  isl_size r1, r2, r;
  isl_mat *mat;
 
  r1 = isl_mat_rank(mat1);
  if (r1 < 0)
    return isl_bool_error;
  if (r1 == 0)
    return isl_bool_true;
  r2 = isl_mat_rank(mat2);
  if (r2 < 0)
    return isl_bool_error;
  if (r2 == 0)
    return isl_bool_true;
 
  mat = isl_mat_concat(isl_mat_copy(mat1), isl_mat_copy(mat2));
  r = isl_mat_rank(mat);
  isl_mat_free(mat);
  if (r < 0)
    return isl_bool_error;
  return isl_bool_ok(r == r1 + r2);
}