matrix_addon.c

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/** 
 * $Id: matrix_addon.c,v 1.17 2007/03/18 18:49:08 skimo Exp $
 * 
 * Polylib matrix addons
 * Mainly, deals with polyhedra represented as a matrix (implicit form)
 * @author Benoit Meister <meister@icps.u-strasbg.fr>
 * 
 */

#include <stdlib.h>
#include<polylib/polylib.h>
#include <polylib/matrix_addon.h>

/** Creates a view of the constraints of a polyhedron as a Matrix * */
Matrix * constraintsView(Polyhedron * P) {
  Matrix * view = (Matrix *)malloc(sizeof(Matrix));
  view->NbRows = P->NbConstraints;
  view->NbColumns = P->Dimension+2;
  view->p = P->Constraint;
  return view;
}

/** "Frees" a view of the constraints of a polyhedron */
void constraintsView_Free(Matrix * M) {
  free(M);
}

/** 
 * splits a matrix of constraints M into a matrix of equalities Eqs and a
 *  matrix of inequalities Ineqs allocs the new matrices. 
 * Allocates Eqs and Ineqs.
*/
void split_constraints(Matrix const * M, Matrix ** Eqs, Matrix **Ineqs) {
  unsigned int i, j, k_eq, k_ineq, nb_eqs=0;

  /* 1- count the number of equations */
  for (i=0; i< M->NbRows; i++)     
    if (value_zero_p(M->p[i][0])) nb_eqs++;

  /* 2- extract the two matrices of equations */
  (*Eqs) = Matrix_Alloc(nb_eqs, M->NbColumns);
  (*Ineqs) = Matrix_Alloc(M->NbRows-nb_eqs, M->NbColumns);

  k_eq = k_ineq = 0;
  for(i=0; i< M->NbRows; i++) {
    if (value_zero_p(M->p[i][0])) 
      {
        for(j=0; j< M->NbColumns; j++)
          value_assign((*Eqs)->p[k_eq][j], M->p[i][j]);
        k_eq++;
      }
    else
       {
        for(j=0; j< M->NbColumns; j++)
          value_assign((*Ineqs)->p[k_ineq][j], M->p[i][j]);
        k_ineq++;
      }
  }
}


/* returns the dim-dimensional identity matrix */
Matrix * Identity_Matrix(unsigned int dim) {
  Matrix * ret = Matrix_Alloc(dim, dim);
  unsigned int i,j;
  for (i=0; i< dim; i++) {
    for (j=0; j< dim; j++) {
      if (i==j) 
        { value_set_si(ret->p[i][j], 1); } 
      else value_set_si(ret->p[i][j], 0);
    }
  }
  return ret;
} /* Identity_Matrix */


/** 
 * returns the dim-dimensional identity matrix. 
 * If I is set to NULL, allocates it first. 
 * Else, assumes an existing, allocated Matrix.
*/
void Matrix_identity(unsigned int dim, Matrix ** I) {
  int i,j;
  if (*I==NULL) {
    (*I) = Identity_Matrix(dim);
  }
  else {
    assert((*I)->NbRows>=dim && (*I)->NbColumns>=dim);
    for (i=0; i< dim; i++) {
      for (j=0; j< dim; j++) {
        if (i==j) { 
            value_set_si((*I)->p[i][j], 1); 
          } 
        else {
          value_set_si((*I)->p[i][j], 0);
        }
      }
    }
  }
} /* Matrix_identity */


/** given a n x n integer transformation matrix transf, compute its inverse
    M/g, where M is a nxn integer matrix.  g is a common denominator for
    elements of (transf^{-1}) */
void mtransformation_inverse(Matrix * transf, Matrix ** inverse, Value * g) {
  Value factor;
  unsigned int i,j;
  Matrix *tmp, *inv;

  value_init(*g);
  value_set_si(*g,1);

  /* a - compute the inverse as usual (n x (n+1) matrix) */
  tmp = Matrix_Copy(transf);
  inv = Matrix_Alloc(transf->NbRows, transf->NbColumns+1);
  MatInverse(tmp, inv);
  Matrix_Free(tmp);

  /* b - as it is rational, put it to the same denominator*/
  (*inverse) = Matrix_Alloc(transf->NbRows, transf->NbRows);
  for (i=0; i< inv->NbRows; i++) 
    value_lcm(*g, *g, inv->p[i][inv->NbColumns-1]);
  for (i=0; i< inv->NbRows; i++) {
    value_division(factor, *g, inv->p[i][inv->NbColumns-1]);
    for (j=0; j< (*inverse)->NbColumns; j++) 
      value_multiply((*inverse)->p[i][j], inv->p[i][j],  factor);
  }

  /* c- clean up */
  value_clear(factor);
  Matrix_Free(inv);
} /* mtransformation_inverse */


/** takes a transformation matrix, and expands it to a higher dimension with
    the identity matrix regardless of it homogeneousness */
Matrix * mtransformation_expand_left_to_dim(Matrix * M, int new_dim) {
  Matrix * ret = Identity_Matrix(new_dim);
  int offset = new_dim-M->NbRows;
  unsigned int i,j;

  assert(new_dim>=M->NbColumns);
  assert(M->NbRows==M->NbColumns);

  for (i=0; i< M->NbRows; i++)
    for (j=0; j< M->NbRows; j++)
      value_assign(ret->p[offset+i][offset+j], M->p[i][j]);
  return ret;
} /* mtransformation_expand_left_to_dim */


/** simplify a matrix seen as a polyhedron, by dividing its rows by the gcd of
   their elements. */
void mpolyhedron_simplify(Matrix * polyh) {
  int i, j;
  Value cur_gcd;
  value_init(cur_gcd);
  for (i=0; i< polyh->NbRows; i++) {
    Vector_Gcd(polyh->p[i]+1, polyh->NbColumns-1, &cur_gcd);
    printf(" gcd[%d] = ", i); 
    value_print(stdout, VALUE_FMT, cur_gcd);printf("\n");
    Vector_AntiScale(polyh->p[i]+1, polyh->p[i]+1, cur_gcd, polyh->NbColumns-1);
  }
  value_clear(cur_gcd);
} /* mpolyhedron_simplify */


/** inflates a polyhedron (represented as a matrix) P, so that the apx of its
    Ehrhart Polynomial is an upper bound of the Ehrhart polynomial of P
    WARNING: this inflation is supposed to be applied on full-dimensional
    polyhedra. */
void mpolyhedron_inflate(Matrix * polyh, unsigned int nb_parms) {
  unsigned int i,j;
  unsigned nb_vars = polyh->NbColumns-nb_parms-2;
  Value infl;
  value_init(infl);
  /* subtract the sum of the negative coefficients of each inequality */
  for (i=0; i< polyh->NbRows; i++) {
    value_set_si(infl, 0);
    for (j=0; j< nb_vars; j++) {
      if (value_neg_p(polyh->p[i][1+j]))
        value_addto(infl, infl, polyh->p[i][1+j]);
    }
    /* here, we subtract a negative value */
    value_subtract(polyh->p[i][polyh->NbColumns-1], 
                   polyh->p[i][polyh->NbColumns-1], infl);
  }
  value_clear(infl);
} /* mpolyhedron_inflate */


/** deflates a polyhedron (represented as a matrix) P, so that the apx of its
    Ehrhart Polynomial is a lower bound of the Ehrhart polynomial of P WARNING:
    this deflation is supposed to be applied on full-dimensional polyhedra. */
void mpolyhedron_deflate(Matrix * polyh, unsigned int nb_parms) {
  unsigned int i,j;
  unsigned nb_vars = polyh->NbColumns-nb_parms-2;
  Value defl;
  value_init(defl);
  /* substract the sum of the negative coefficients of each inequality */
  for (i=0; i< polyh->NbRows; i++) {
    value_set_si(defl, 0);
    for (j=0; j< nb_vars; j++) {
      if (value_pos_p(polyh->p[i][1+j]))
        value_addto(defl, defl, polyh->p[i][1+j]);
    }
    /* here, we substract a negative value */
    value_subtract(polyh->p[i][polyh->NbColumns-1], 
                   polyh->p[i][polyh->NbColumns-1], defl);
  }
  value_clear(defl);
} /* mpolyhedron_deflate */


/** use an eliminator row to eliminate a variable in a victim row (without
 * changing the sign of the victim row -> important if it is an inequality).
 * @param Eliminator the matrix containing the eliminator row
 * @param eliminator_row the index of the eliminator row in <tt>Eliminator</tt>
 * @param Victim the matrix containing the row to be eliminated
 * @param victim_row the row to be eliminated in <tt>Victim</tt>
 * @param var_to_elim the variable to be eliminated.
 */
void eliminate_var_with_constr(Matrix * Eliminator, 
                               unsigned int eliminator_row, Matrix * Victim, 
                               unsigned int victim_row, 
                               unsigned int var_to_elim) {
  Value cur_lcm, mul_a, mul_b;
  Value tmp, tmp2;
  int k; 

  value_init(cur_lcm); 
  value_init(mul_a); 
  value_init(mul_b); 
  value_init(tmp); 
  value_init(tmp2);
  /* if the victim coefficient is not zero */
  if (value_notzero_p(Victim->p[victim_row][var_to_elim+1])) {
    value_lcm(cur_lcm, Eliminator->p[eliminator_row][var_to_elim+1], 
              Victim->p[victim_row][var_to_elim+1]);
    /* multiplication factors */
    value_division(mul_a, cur_lcm, 
                   Eliminator->p[eliminator_row][var_to_elim+1]);
    value_division(mul_b, cur_lcm, 
                   Victim->p[victim_row][var_to_elim+1]);
    /* the multiplicator for the vitim row has to be positive */
    if (value_pos_p(mul_b)) {
      value_oppose(mul_a, mul_a);
    }
    else {
      value_oppose(mul_b, mul_b);
    }
    value_clear(cur_lcm); 
    /* now we have a.mul_a = -(b.mul_b) and mul_a > 0 */
    for (k=1; k<Victim->NbColumns; k++) {
      value_multiply(tmp, Eliminator->p[eliminator_row][k], mul_a);
      value_multiply(tmp2, Victim->p[victim_row][k], mul_b);
      value_addto(Victim->p[victim_row][k], tmp, tmp2);
    }
  }
  value_clear(mul_a); 
  value_clear(mul_b); 
  value_clear(tmp); 
  value_clear(tmp2);
}
/* eliminate_var_with_constr */


/* STUFF WITH PARTIAL MAPPINGS (Mappings to a subset of the
   variables/parameters) : on the first or last variables/parameters */

/** compress the last vars/pars of the polyhedron M expressed as a polylib
    matrix
 - adresses the full-rank compressions only
 - modfies M */
void mpolyhedron_compress_last_vars(Matrix * M, Matrix * compression) {
  unsigned int i, j, k;
  unsigned int offset = M->NbColumns - compression->NbRows; 
  /* the computations on M will begin on column "offset" */

  Matrix * M_tmp = Matrix_Alloc(1, M->NbColumns);
  assert(compression->NbRows==compression->NbColumns);
  /* basic matrix multiplication (using a temporary row instead of a whole
     temporary matrix), but with a column offset */
  for(i=0; i< M->NbRows; i++) {
    for (j=0; j< compression->NbRows; j++) {
      value_set_si(M_tmp->p[0][j], 0);
      for (k=0; k< compression->NbRows; k++) {
        value_addmul(M_tmp->p[0][j], M->p[i][k+offset],compression->p[k][j]);
      }
    }
    for (j=0; j< compression->NbRows; j++) 
      value_assign(M->p[i][j+offset], M_tmp->p[0][j]);
  }
  Matrix_Free(M_tmp);
} /* mpolyhedron_compress_last_vars */


/** use a set of m equalities Eqs to eliminate m variables in the polyhedron
    Ineqs represented as a matrix
 eliminates the m first variables
 - assumes that Eqs allow to eliminate the m equalities
 - modifies Eqs and Ineqs */
unsigned int mpolyhedron_eliminate_first_variables(Matrix * Eqs, 
                                                   Matrix *Ineqs) {
  unsigned int i, j, k;
  /* eliminate one variable (index i) after each other */
  for (i=0; i< Eqs->NbRows; i++) {
    /* find j, the first (non-marked) row of Eqs with a non-zero coefficient */
    for (j=0; j<Eqs->NbRows && (Eqs->p[j][i+1]==0 || 
                                ( !value_cmp_si(Eqs->p[j][0],2) )); 
         j++);
    /* if no row is found in Eqs that allows to eliminate variable i, return an
       error code (0) */
    if (j==Eqs->NbRows) return 0;
    /* else, eliminate variable i in Eqs and Ineqs with the j^th row of Eqs
       (and mark this row so we don't use it again for an elimination) */
    for (k=j+1; k<Eqs->NbRows; k++)
      eliminate_var_with_constr(Eqs, j, Eqs, k, i);
    for (k=0; k< Ineqs->NbRows; k++)
      eliminate_var_with_constr(Eqs, j, Ineqs, k, i);
    /* mark the row */
    value_set_si(Eqs->p[j][0],2);
  }
  /* un-mark all the rows */
  for (i=0; i< Eqs->NbRows; i++) value_set_si(Eqs->p[i][0],0);
  return 1;
} /* mpolyhedron_eliminate_first_variables */


/** returns a contiguous submatrix of a matrix.
 * @param M the input matrix
 * @param sr the index of the starting row
 * @param sc the index of the starting column
 * @param er the index ofthe ending row (excluded)
 * @param ec the ined of the ending colummn (excluded)
 * @param sub (returned), the submatrix. Allocated if set to NULL, assumed to
 * be already allocated else.
 */
void Matrix_subMatrix(Matrix * M, unsigned int sr, unsigned int sc, 
                          unsigned int er, unsigned int ec, Matrix ** sub) {
  int i;
  int nbR = er-sr;
  int nbC = ec-sc;
  assert (er<=M->NbRows && ec<=M->NbColumns);
  if ((*sub)==NULL) {
    (*sub) = Matrix_Alloc(nbR, nbC);
  }
  if (nbR==0 || nbC==0) return;
  for (i=0; i< nbR; i++) {
    Vector_Copy(&(M->p[i+sr][sc]), (*sub)->p[i], nbC);
  }
}/* Matrix_subMatrix */


/**
 * Cloning function. Similar to Matrix_Copy() but allocates the target matrix
 * if it is set to NULL.
 */
void Matrix_clone(Matrix * M, Matrix ** Cl) {
  Matrix_subMatrix(M, 0,0, M->NbRows, M->NbColumns, Cl);
} 


/**
 * Copies a contiguous submatrix of M1 into M2, at the indicated position.
 * M1 and M2 are assumed t be allocated already.
 * @param M1 the source matrix
 * @param sr1 the starting source row
 * @param sc1 the starting source column
 * @param nbR the number of rows
 * @param nbC the number of columns
 * @param M2 the target matrix
 * @param sr2 the starting target row
 * @param sc2 the starting target column
*/
void Matrix_copySubMatrix(Matrix *M1,
                          unsigned int sr1, unsigned int sc1,
                          unsigned int nbR, unsigned int nbC,
                          Matrix * M2,
                          unsigned int sr2, unsigned int sc2) {
  int i;
  for (i=0; i< nbR; i++) {
    Vector_Copy(&(M1->p[i+sr1][sc1]), &(M2->p[i+sr2][sc2]), nbC);
  }
} /* Matrix_copySubMatrix */


/** 
 * transforms a matrix M into -M
 */
void Matrix_oppose(Matrix * M) {
  int i,j;
  for (i=0; i<M->NbRows; i++) {
    for (j=0; j< M->NbColumns; j++) {
      value_oppose(M->p[i][j], M->p[i][j]);
    }
  }
}

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