matrix.c

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/* matrix.c 
     COPYRIGHT
          Both this software and its documentation are

              Copyright 1993 by IRISA /Universite de Rennes I -
              France, Copyright 1995,1996 by BYU, Provo, Utah
                         all rights reserved.

          Permission is granted to copy, use, and distribute
          for any commercial or noncommercial purpose under the terms
          of the GNU General Public license, version 2, June 1991
          (see file : LICENSING).
*/

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <ctype.h>
#include <polylib/polylib.h>

#ifdef mac_os
  #define abs __abs
#endif

/* 
 * Allocate space for matrix dimensioned by 'NbRows X NbColumns'.
 */
Matrix *Matrix_Alloc(unsigned NbRows,unsigned NbColumns) {
  
  Matrix *Mat;
  Value *p, **q;
  int i,j;

  Mat=(Matrix *)malloc(sizeof(Matrix));
  if(!Mat) {    
    errormsg1("Matrix_Alloc", "outofmem", "out of memory space");
    return 0;
  }
  Mat->NbRows=NbRows;
  Mat->NbColumns=NbColumns;
  if (NbRows==0 || NbColumns==0) {
      Mat->p = (Value **)0;
      Mat->p_Init= (Value *)0;
      Mat->p_Init_size = 0;
  } else {
      q = (Value **)malloc(NbRows * sizeof(*q));
      if(!q) {
        free(Mat);
        errormsg1("Matrix_Alloc", "outofmem", "out of memory space");
        return 0;
      }
      p = value_alloc(NbRows * NbColumns, &Mat->p_Init_size);
      if(!p) {
        free(q);
        free(Mat);
        errormsg1("Matrix_Alloc", "outofmem", "out of memory space");
        return 0;
      }
      Mat->p = q;
      Mat->p_Init = p;
      for (i=0;i<NbRows;i++) {
        *q++ = p;
        p += NbColumns;
      }
  }
  p = NULL;
  q = NULL;

  return Mat;
} /* Matrix_Alloc */

/* 
 * Free the memory space occupied by Matrix 'Mat' 
 */
void Matrix_Free(Matrix *Mat)
{ 
  if (Mat->p_Init)
    value_free(Mat->p_Init, Mat->p_Init_size);

  if (Mat->p)
    free(Mat->p);
  free(Mat);

} /* Matrix_Free */

void Matrix_Extend(Matrix *Mat, unsigned NbRows)
{
  Value *p, **q;
  int i,j;

  q = (Value **)realloc(Mat->p, NbRows * sizeof(*q));
  if(!q) {
    errormsg1("Matrix_Extend", "outofmem", "out of memory space");
    return;
  }
  Mat->p = q;
  if (Mat->p_Init_size < NbRows * Mat->NbColumns) {
    p = (Value *)realloc(Mat->p_Init, NbRows * Mat->NbColumns * sizeof(Value));
    if(!p) {
      errormsg1("Matrix_Extend", "outofmem", "out of memory space");
      return;
    }
    Mat->p_Init = p;
    Vector_Set(Mat->p_Init + Mat->NbRows*Mat->NbColumns, 0,
               Mat->p_Init_size - Mat->NbRows*Mat->NbColumns);
    for (i = Mat->p_Init_size; i < Mat->NbColumns*NbRows; ++i)
        value_init(Mat->p_Init[i]);
    Mat->p_Init_size = Mat->NbColumns*NbRows;
  } else
    Vector_Set(Mat->p_Init + Mat->NbRows*Mat->NbColumns, 0,
               (NbRows - Mat->NbRows) * Mat->NbColumns);
  for (i=0;i<NbRows;i++) {
    Mat->p[i] = Mat->p_Init + (i * Mat->NbColumns);
  }
  Mat->NbRows = NbRows;
}

/* 
 * Print the contents of the Matrix 'Mat'
 */
void Matrix_Print(FILE *Dst, const char *Format, Matrix *Mat)
{
  Value *p;
  int i, j;
  unsigned NbRows, NbColumns;
  
  fprintf(Dst,"%d %d\n", NbRows=Mat->NbRows, NbColumns=Mat->NbColumns);
  if (NbColumns ==0) {
    fprintf(Dst, "\n");
    return;
  }
  for (i=0;i<NbRows;i++) {
    p=*(Mat->p+i);
    for (j=0;j<NbColumns;j++) {
      if (!Format) {
        value_print(Dst," "P_VALUE_FMT" ",*p++);
      }
      else { 
        value_print(Dst,Format,*p++);
      } 
    }
    fprintf(Dst, "\n");
  }
} /* Matrix_Print */

/* 
 * Read the contents of the Matrix 'Mat' 
 */
void Matrix_Read_Input(Matrix *Mat) {
  
  Value *p;
  int i,j,n;
  char *c, s[1024],str[1024];
  
  p = Mat->p_Init;
  for (i=0;i<Mat->NbRows;i++) {
    do {
      c = fgets(s, 1024, stdin);
      while(isspace(*c) && *c!='\n')
        ++c;
    } while(c && (*c =='#' || *c== '\n'));
    
    if (!c) {
      errormsg1( "Matrix_Read", "baddim", "not enough rows" );
      break;
    }
    for (j=0;j<Mat->NbColumns;j++) {
      if(!c || *c=='\n' || *c=='#') {
        errormsg1("Matrix_Read", "baddim", "not enough columns");
        break;
      }
      if (sscanf(c,"%s%n",str,&n) == 0) {
        errormsg1( "Matrix_Read", "baddim", "not enough columns" );
        break;
      }
      value_read(*(p++),str);
      c += n;
    }
  }
} /* Matrix_Read_Input */

/* 
 * Read the contents of the matrix 'Mat' from standard input. 
 * A '#' in the first column is a comment line 
 */
Matrix *Matrix_Read(void) {
  
  Matrix *Mat;
  unsigned NbRows, NbColumns;
  char s[1024];
  
  if (fgets(s, 1024, stdin) == NULL)
    return NULL;
  while ((*s=='#' || *s=='\n') ||
         (sscanf(s, "%d %d", &NbRows, &NbColumns)<2)) {
    if (fgets(s, 1024, stdin) == NULL)
      return NULL;
  }
  Mat = Matrix_Alloc(NbRows,NbColumns);
  if(!Mat) {
    errormsg1("Matrix_Read", "outofmem", "out of memory space");
    return(NULL);
  }
  Matrix_Read_Input(Mat);
  return Mat;
} /* Matrix_Read */

/* 
 * Basic hermite engine 
 */
static int hermite(Matrix *H,Matrix *U,Matrix *Q) {
  
  int nc, nr, i, j, k, rank, reduced, pivotrow;
  Value pivot,x,aux;
  Value *temp1, *temp2;
  
  /*                     T                     -1   T */
  /* Computes form: A = Q H  and U A = H  and U  = Q  */
  
  if (!H) { 
    errormsg1("Domlib", "nullH", "hermite: ? Null H");
    return -1;
  }
  nc = H->NbColumns;
  nr = H->NbRows;
  temp1 = (Value *) malloc(nc * sizeof(Value));
  temp2 = (Value *) malloc(nr * sizeof(Value));
  if (!temp1 ||!temp2) {
    errormsg1("Domlib", "outofmem", "out of memory space");
    return -1;
  }
  
  /* Initialize all the 'Value' variables */
  value_init(pivot); value_init(x); 
  value_init(aux);   
  for(i=0;i<nc;i++)
    value_init(temp1[i]);
  for(i=0;i<nr;i++)
    value_init(temp2[i]);
  
#ifdef DEBUG
  fprintf(stderr,"Start  -----------\n");
  Matrix_Print(stderr,0,H);
#endif
  for (k=0, rank=0; k<nc && rank<nr; k=k+1) {
    reduced = 1;        /* go through loop the first time */
#ifdef DEBUG
    fprintf(stderr, "Working on col %d.  Rank=%d ----------\n", k+1, rank+1);
#endif
    while (reduced) {
      reduced=0;
      
      /* 1. find pivot row */
      value_absolute(pivot,H->p[rank][k]);
      
      /* the kth-diagonal element */
      pivotrow = rank;
      
      /* find the row i>rank with smallest nonzero element in col k */
      for (i=rank+1; i<nr; i++) {
        value_absolute(x,H->p[i][k]);
        if (value_notzero_p(x) &&
            (value_lt(x,pivot) || value_zero_p(pivot))) {
          value_assign(pivot,x);
          pivotrow = i;
        }
      }
      
      /* 2. Bring pivot to diagonal (exchange rows pivotrow and rank) */
      if (pivotrow != rank) {
        Vector_Exchange(H->p[pivotrow],H->p[rank],nc);
        if (U)
          Vector_Exchange(U->p[pivotrow],U->p[rank],nr);
        if (Q)
          Vector_Exchange(Q->p[pivotrow],Q->p[rank],nr);

#ifdef DEBUG
        fprintf(stderr,"Exchange rows %d and %d  -----------\n", rank+1, pivotrow+1);
        Matrix_Print(stderr,0,H);
#endif
      }
      value_assign(pivot,H->p[rank][k]);        /* actual ( no abs() ) pivot */
      
      /* 3. Invert the row 'rank' if pivot is negative */
      if (value_neg_p(pivot)) {
        value_oppose(pivot,pivot); /* pivot = -pivot */
        for (j=0; j<nc; j++)
          value_oppose(H->p[rank][j],H->p[rank][j]);
        
        /* H->p[rank][j] = -(H->p[rank][j]); */
        if (U)
          for (j=0; j<nr; j++)
            value_oppose(U->p[rank][j],U->p[rank][j]);
        
        /* U->p[rank][j] = -(U->p[rank][j]); */
        if (Q)
          for (j=0; j<nr; j++)
            value_oppose(Q->p[rank][j],Q->p[rank][j]);
        
        /* Q->p[rank][j] = -(Q->p[rank][j]); */
#ifdef DEBUG
        fprintf(stderr,"Negate row %d  -----------\n", rank+1);
        Matrix_Print(stderr,0,H);
#endif

      }      
      if (value_notzero_p(pivot)) {
        
        /* 4. Reduce the column modulo the pivot */
        /*    This eventually zeros out everything below the */
        /*    diagonal and produces an upper triangular matrix */
        
        for (i=rank+1;i<nr;i++) {
          value_assign(x,H->p[i][k]);
          if (value_notzero_p(x)) {         
            value_modulus(aux,x,pivot);
            
            /* floor[integer division] (corrected for neg x) */
            if (value_neg_p(x) && value_notzero_p(aux)) {
              
              /* x=(x/pivot)-1; */
              value_division(x,x,pivot);
              value_decrement(x,x);
            }   
            else 
              value_division(x,x,pivot);
            for (j=0; j<nc; j++) {
              value_multiply(aux,x,H->p[rank][j]);
              value_subtract(H->p[i][j],H->p[i][j],aux);
            }
            
            /* U->p[i][j] -= (x * U->p[rank][j]); */
            if (U)
              for (j=0; j<nr; j++) {
                value_multiply(aux,x,U->p[rank][j]);
                value_subtract(U->p[i][j],U->p[i][j],aux);
              }
            
            /* Q->p[rank][j] += (x * Q->p[i][j]); */
            if (Q)
              for(j=0;j<nr;j++) {
                value_addmul(Q->p[rank][j], x, Q->p[i][j]);
              }
            reduced = 1;

#ifdef DEBUG
            fprintf(stderr,
                    "row %d = row %d - %d row %d -----------\n", i+1, i+1, x, rank+1);
            Matrix_Print(stderr,0,H);
#endif
        
          } /* if (x) */
        } /* for (i) */
      } /* if (pivot != 0) */
    } /* while (reduced) */
    
    /* Last finish up this column */
    /* 5. Make pivot column positive (above pivot row) */
    /*    x should be zero for i>k */
    
    if (value_notzero_p(pivot)) {
      for (i=0; i<rank; i++) {
        value_assign(x,H->p[i][k]);
        if (value_notzero_p(x)) {         
          value_modulus(aux,x,pivot);
          
          /* floor[integer division] (corrected for neg x) */
          if (value_neg_p(x) && value_notzero_p(aux)) {
            value_division(x,x,pivot);
            value_decrement(x,x);
            
            /* x=(x/pivot)-1; */
          }
          else
            value_division(x,x,pivot);
          
          /* H->p[i][j] -= x * H->p[rank][j]; */
          for (j=0; j<nc; j++) {
            value_multiply(aux,x,H->p[rank][j]);
            value_subtract(H->p[i][j],H->p[i][j],aux);
          }
          
          /* U->p[i][j] -= x * U->p[rank][j]; */
          if (U)
            for (j=0; j<nr; j++) {
              value_multiply(aux,x,U->p[rank][j]);
              value_subtract(U->p[i][j],U->p[i][j],aux);
            }
          
          /* Q->p[rank][j] += x * Q->p[i][j]; */
          if (Q)
            for (j=0; j<nr; j++) {
              value_addmul(Q->p[rank][j], x, Q->p[i][j]);
            }  
#ifdef DEBUG
          fprintf(stderr,
                  "row %d = row %d - %d row %d -----------\n", i+1, i+1, x, rank+1);
          Matrix_Print(stderr,0,H);
#endif
        } /* if (x) */
      } /* for (i) */
      rank++;
    } /* if (pivot!=0) */
  } /* for (k) */
  
  /* Clear all the 'Value' variables */
  value_clear(pivot); value_clear(x); 
  value_clear(aux); 
  for(i=0;i<nc;i++)
    value_clear(temp1[i]);
  for(i=0;i<nr;i++)
    value_clear(temp2[i]);
  free(temp2);
  free(temp1);
  return rank;
} /* Hermite */ 

void right_hermite(Matrix *A,Matrix **Hp,Matrix **Up,Matrix **Qp) {
  
  Matrix *H, *Q, *U;
  int i, j, nr, nc, rank;
  Value tmp;
  
  /* Computes form: A = QH , UA = H */  
  nc = A->NbColumns;
  nr = A->NbRows;
  
  /* H = A */
  *Hp = H = Matrix_Alloc(nr,nc);
  if (!H) { 
    errormsg1("DomRightHermite", "outofmem", "out of memory space");
    return;
  }
  
  /* Initialize all the 'Value' variables */
  value_init(tmp);
  
  Vector_Copy(A->p_Init,H->p_Init,nr*nc);
  
  /* U = I */
  if (Up) {
    *Up = U = Matrix_Alloc(nr, nr);
    if (!U) {
      errormsg1("DomRightHermite", "outofmem", "out of memory space");
      value_clear(tmp);
      return;
    }
    Vector_Set(U->p_Init,0,nr*nr);             /* zero's */
    for(i=0;i<nr;i++)                          /* with diagonal of 1's */
      value_set_si(U->p[i][i],1);
  }
  else
    U = (Matrix *)0;
  
  /* Q = I */
  /* Actually I compute Q transpose... its easier */
  if (Qp) {
    *Qp = Q = Matrix_Alloc(nr,nr);
    if (!Q) {
      errormsg1("DomRightHermite", "outofmem", "out of memory space");
      value_clear(tmp);
      return;
    }
    Vector_Set(Q->p_Init,0,nr*nr);            /* zero's */
    for (i=0;i<nr;i++)                      /* with diagonal of 1's */
      value_set_si(Q->p[i][i],1);
  }
  else
    Q = (Matrix *)0;
  
  rank = hermite(H,U,Q);
  
  /* Q is returned transposed */ 
  /* Transpose Q */
  if (Q) {
    for (i=0; i<nr; i++) {
      for (j=i+1; j<nr; j++) {
        value_assign(tmp,Q->p[i][j]);
        value_assign(Q->p[i][j],Q->p[j][i] );
        value_assign(Q->p[j][i],tmp);
      }
    }
  }
  value_clear(tmp);
  return;
} /* right_hermite */

void left_hermite(Matrix *A,Matrix **Hp,Matrix **Qp,Matrix **Up) {
  
  Matrix *H, *HT, *Q, *U;
  int i, j, nc, nr, rank;
  Value tmp;
  
  /* Computes left form: A = HQ , AU = H , 
                        T    T T    T T   T
     using right form  A  = Q H  , U A = H */
  
  nr = A->NbRows;
  nc = A->NbColumns;
  
  /* HT = A transpose */
  HT = Matrix_Alloc(nc, nr);
  if (!HT) {
    errormsg1("DomLeftHermite", "outofmem", "out of memory space");
    return;
  }
  value_init(tmp);
  for (i=0; i<nr; i++)
    for (j=0; j<nc; j++)
      value_assign(HT->p[j][i],A->p[i][j]);
  
  /* U = I */
  if (Up) {
    *Up = U = Matrix_Alloc(nc,nc);
    if (!U) {
      errormsg1("DomLeftHermite", "outofmem", "out of memory space");
      value_clear(tmp);
      return;
    }
    Vector_Set(U->p_Init,0,nc*nc);            /* zero's */
    for (i=0;i<nc;i++)                        /* with diagonal of 1's */
      value_set_si(U->p[i][i],1);
  }
  else U=(Matrix *)0;
  
  /* Q = I */
  if (Qp) {
    *Qp = Q = Matrix_Alloc(nc, nc);
    if (!Q) {
      errormsg1("DomLeftHermite", "outofmem", "out of memory space");
      value_clear(tmp);
      return;
    }
    Vector_Set(Q->p_Init,0,nc*nc);            /* zero's */
    for (i=0;i<nc;i++)                        /* with diagonal of 1's */
      value_set_si(Q->p[i][i],1);
  }
  else Q=(Matrix *)0;
  rank = hermite(HT,U,Q);
  
  /* H = HT transpose */
  *Hp = H = Matrix_Alloc(nr,nc);
  if (!H) {
    errormsg1("DomLeftHermite", "outofmem", "out of memory space");
    value_clear(tmp);
    return;
  }
  for (i=0; i<nr; i++)
    for (j=0;j<nc;j++)
      value_assign(H->p[i][j],HT->p[j][i]);
  Matrix_Free(HT);
  
  /* Transpose U */
  if (U) {
    for (i=0; i<nc; i++) {
      for (j=i+1; j<nc; j++) {
        value_assign(tmp,U->p[i][j]);
        value_assign(U->p[i][j],U->p[j][i] );
        value_assign(U->p[j][i],tmp);
      }
    }
  }
  value_clear(tmp);
} /* left_hermite */

/*
 * Given a integer matrix 'Mat'(k x k), compute its inverse rational matrix 
 * 'MatInv' k x (k+1). The last column of each row in matrix MatInv is used 
 * to store the common denominator of the entries in a row. The output is 1,
 * if 'Mat' is non-singular (invertible), otherwise the output is 0. Note:: 
 * (1) Matrix 'Mat' is modified during the inverse operation.
 * (2) Matrix 'MatInv' must be preallocated before passing into this function.
 */
int MatInverse(Matrix *Mat,Matrix *MatInv ) {
  
  int i, k, j, c;
  Value x, gcd, piv;
  Value m1,m2;
  
  if(Mat->NbRows != Mat->NbColumns) {
   fprintf(stderr,"Trying to invert a non-square matrix !\n");
    return 0;
  }
  
  /* Initialize all the 'Value' variables */
  value_init(x);  value_init(gcd); value_init(piv);
  value_init(m1); value_init(m2);

  k = Mat->NbRows; 

  /* Initialise MatInv */
  Vector_Set(MatInv->p[0],0,k*(k+1));

  /* Initialize 'MatInv' to Identity matrix form. Each diagonal entry is set*/
  /* to 1. Last column of each row (denominator of each entry in a row) is  */
  /* also set to 1.                                                         */ 
  for(i=0;i<k;++i) {
    value_set_si(MatInv->p[i][i],1);    
    value_set_si(MatInv->p[i][k],1);    /* denum */
  }  
  /* Apply Gauss-Jordan elimination method on the two matrices 'Mat' and  */
  /* 'MatInv' in parallel.                                                */
  for(i=0;i<k;++i) {
    
    /* Check if the diagonal entry (new pivot) is non-zero or not */
    if(value_zero_p(Mat->p[i][i])) {    
      
      /* Search for a non-zero pivot down the column(i) */
      for(j=i;j<k;++j)      
        if(value_notzero_p(Mat->p[j][i]))
          break;
      
      /* If no non-zero pivot is found, the matrix 'Mat' is non-invertible */
      /* Return 0.                                                         */
      if(j==k) {
        
        /* Clear all the 'Value' variables */
        value_clear(x);  value_clear(gcd); value_clear(piv);
        value_clear(m1); value_clear(m2);
        return 0;
      } 
      
      /* Exchange the rows, row(i) and row(j) so that the diagonal element */
      /* Mat->p[i][i] (pivot) is non-zero. Repeat the same operations on    */
      /* matrix 'MatInv'.                                                   */
      for(c=0;c<k;++c) {

        /* Interchange rows, row(i) and row(j) of matrix 'Mat'    */
        value_assign(x,Mat->p[j][c]);
        value_assign(Mat->p[j][c],Mat->p[i][c]);
        value_assign(Mat->p[i][c],x);
        
        /* Interchange rows, row(i) and row(j) of matrix 'MatInv' */
        value_assign(x,MatInv->p[j][c]);
        value_assign(MatInv->p[j][c],MatInv->p[i][c]);
        value_assign(MatInv->p[i][c],x);
      }
    }
    
    /* Make all the entries in column(i) of matrix 'Mat' zero except the */
    /* diagonal entry. Repeat the same sequence of operations on matrix  */
    /* 'MatInv'.                                                         */
    for(j=0;j<k;++j) {
      if (j==i) continue;                /* Skip the pivot */
      value_assign(x,Mat->p[j][i]);
      if(value_notzero_p(x)) {
        value_assign(piv,Mat->p[i][i]);
        value_gcd(gcd, x, piv);
        if (value_notone_p(gcd) ) {
          value_divexact(x, x, gcd);
          value_divexact(piv, piv, gcd);
        }
        for(c=((j>i)?i:0);c<k;++c) {
          value_multiply(m1,piv,Mat->p[j][c]);
          value_multiply(m2,x,Mat->p[i][c]);
          value_subtract(Mat->p[j][c],m1,m2); 
        }
        for(c=0;c<k;++c) {
          value_multiply(m1,piv,MatInv->p[j][c]);
          value_multiply(m2,x,MatInv->p[i][c]);
          value_subtract(MatInv->p[j][c],m1,m2);
        }
              
        /* Simplify row(j) of the two matrices 'Mat' and 'MatInv' by */
        /* dividing the rows with the common GCD.                     */
        Vector_Gcd(&MatInv->p[j][0],k,&m1);
        Vector_Gcd(&Mat->p[j][0],k,&m2);
        value_gcd(gcd, m1, m2);
        if(value_notone_p(gcd)) {
          for(c=0;c<k;++c) {
            value_divexact(Mat->p[j][c], Mat->p[j][c], gcd);
            value_divexact(MatInv->p[j][c], MatInv->p[j][c], gcd);
          }
        }
      }
    }
  }
  
  /* Simplify every row so that 'Mat' reduces to Identity matrix. Perform  */
  /* the same sequence of operations on the matrix 'MatInv'.               */
  for(j=0;j<k;++j) {
    value_assign(MatInv->p[j][k],Mat->p[j][j]);
    
    /* Make the last column (denominator of each entry) of every row greater */
    /* than zero.                                                            */
    Vector_Normalize_Positive(&MatInv->p[j][0],k+1,k); 
  }
  
  /* Clear all the 'Value' variables */
  value_clear(x);  value_clear(gcd); value_clear(piv);
  value_clear(m1); value_clear(m2);

  return 1;
} /* Mat_Inverse */

/*
 * Given (m x n) integer matrix 'X' and n x (k+1) rational matrix 'P', compute
 * the rational m x (k+1) rational matrix  'S'. The last column in each row of
 * the rational matrices is used to store the common denominator of elements
 * in a row.                              
 */
void rat_prodmat(Matrix *S,Matrix *X,Matrix *P) {
  
  int i,j,k;
  int last_column_index = P->NbColumns - 1;
  Value lcm, old_lcm,gcd,last_column_entry,s1;
  Value m1,m2;
  
  /* Initialize all the 'Value' variables */
  value_init(lcm); value_init(old_lcm); value_init(gcd);
  value_init(last_column_entry); value_init(s1); 
  value_init(m1); value_init(m2);

  /* Compute the LCM of last column entries (denominators) of rows */
  value_assign(lcm,P->p[0][last_column_index]); 
  for(k=1;k<P->NbRows;++k) {
    value_assign(old_lcm,lcm);
    value_assign(last_column_entry,P->p[k][last_column_index]);
    value_gcd(gcd, lcm, last_column_entry);
    value_divexact(m1, last_column_entry, gcd);
    value_multiply(lcm,lcm,m1);
  }
  
  /* S[i][j] = Sum(X[i][k] * P[k][j] where Sum is extended over k = 1..nbrows*/
  for(i=0;i<X->NbRows;++i)
    for(j=0;j<P->NbColumns-1;++j) {
      
      /* Initialize s1 to zero. */
      value_set_si(s1,0);
      for(k=0;k<P->NbRows;++k) {
        
        /* If the LCM of last column entries is one, simply add the products */
        if(value_one_p(lcm)) {
          value_addmul(s1, X->p[i][k], P->p[k][j]);
        }  
        
        /* Numerator (num) and denominator (denom) of S[i][j] is given by :- */
        /* numerator  = Sum(X[i][k]*P[k][j]*lcm/P[k][last_column_index]) and */
        /* denominator= lcm where Sum is extended over k = 1..nbrows.        */
        else {
          value_multiply(m1,X->p[i][k],P->p[k][j]);
          value_division(m2,lcm,P->p[k][last_column_index]);
          value_addmul(s1, m1, m2);
        }
      } 
      value_assign(S->p[i][j],s1);
    }
  
  for(i=0;i<S->NbRows;++i) {
    value_assign(S->p[i][last_column_index],lcm);

    /* Normalize the rows so that last element >=0 */
    Vector_Normalize_Positive(&S->p[i][0],S->NbColumns,S->NbColumns-1);
  }
  
  /* Clear all the 'Value' variables */
  value_clear(lcm); value_clear(old_lcm); value_clear(gcd);
  value_clear(last_column_entry); value_clear(s1); 
  value_clear(m1); value_clear(m2);
 
  return;
} /* rat_prodmat */

/*
 * Given a matrix 'Mat' and vector 'p1', compute the matrix-vector product 
 * and store the result in vector 'p2'. 
 */
void Matrix_Vector_Product(Matrix *Mat,Value *p1,Value *p2) {

  int NbRows, NbColumns, i, j;
  Value **cm, *q, *cp1, *cp2;
  
  NbRows=Mat->NbRows;
  NbColumns=Mat->NbColumns;
  
  cm = Mat->p;
  cp2 = p2;
  for(i=0;i<NbRows;i++) {
    q = *cm++;
    cp1 = p1;
    value_multiply(*cp2,*q,*cp1);
    q++;
    cp1++;
    
    /* *cp2 = *q++ * *cp1++ */
    for(j=1;j<NbColumns;j++) {
      value_addmul(*cp2, *q, *cp1);
      q++;
      cp1++;
    }
    cp2++;
  }
  return;
} /* Matrix_Vector_Product */

/*
 * Given a vector 'p1' and a matrix 'Mat', compute the vector-matrix product 
 * and store the result in vector 'p2'
 */
void Vector_Matrix_Product(Value *p1,Matrix *Mat,Value *p2) {
  
  int NbRows, NbColumns, i, j;
  Value **cm, *cp1, *cp2;
  
  NbRows=Mat->NbRows;
  NbColumns=Mat->NbColumns;
  cp2 = p2;
  cm  = Mat->p;
  for (j=0;j<NbColumns;j++) {
    cp1 = p1;
    value_multiply(*cp2,*(*cm+j),*cp1);
    cp1++;
    
    /* *cp2= *(*cm+j) * *cp1++; */
    for (i=1;i<NbRows;i++) {
      value_addmul(*cp2, *(*(cm+i)+j), *cp1);
      cp1++;
    }
    cp2++;
  }
  return;
} /* Vector_Matrix_Product */

/* 
 * Given matrices 'Mat1' and 'Mat2', compute the matrix product and store in 
 * matrix 'Mat3' 
 */
void Matrix_Product(Matrix *Mat1,Matrix *Mat2,Matrix *Mat3) {
  
  int Size, i, j, k;
  unsigned NbRows, NbColumns;
  Value **q1, **q2, *p1, *p3,sum;
  
  NbRows    = Mat1->NbRows;
  NbColumns = Mat2->NbColumns;
  
  Size      = Mat1->NbColumns;
  if(Mat2->NbRows!=Size||Mat3->NbRows!=NbRows||Mat3->NbColumns!=NbColumns) {
    fprintf(stderr, "? Matrix_Product : incompatable matrix dimension\n");
    return;
  }     
  value_init(sum); 
  p3 = Mat3->p_Init;
  q1 = Mat1->p;
  q2 = Mat2->p;
  
  /* Mat3[i][j] = Sum(Mat1[i][k]*Mat2[k][j] where sum is over k = 1..nbrows */
  for (i=0;i<NbRows;i++) {
    for (j=0;j<NbColumns;j++) {
      p1 = *(q1+i);
      value_set_si(sum,0);
      for (k=0;k<Size;k++) {
        value_addmul(sum, *p1, *(*(q2+k)+j));
        p1++;
      }
      value_assign(*p3,sum);
      p3++;
    }
  }
  value_clear(sum); 
  return;
} /* Matrix_Product */
  
/*
 * Given a rational matrix 'Mat'(k x k), compute its inverse rational matrix 
 * 'MatInv' k x k.
 * The output is 1,
 * if 'Mat' is non-singular (invertible), otherwise the output is 0. Note:: 
 * (1) Matrix 'Mat' is modified during the inverse operation.
 * (2) Matrix 'MatInv' must be preallocated before passing into this function.
 */
int Matrix_Inverse(Matrix *Mat,Matrix *MatInv ) {
  
  int i, k, j, c;
  Value x, gcd, piv;
  Value m1,m2;
  Value *den;
  
  if(Mat->NbRows != Mat->NbColumns) {
   fprintf(stderr,"Trying to invert a non-square matrix !\n");
    return 0;
  }
  
  /* Initialize all the 'Value' variables */
  value_init(x);  value_init(gcd); value_init(piv);
  value_init(m1); value_init(m2);

  k = Mat->NbRows; 

  /* Initialise MatInv */
  Vector_Set(MatInv->p[0],0,k*k);

  /* Initialize 'MatInv' to Identity matrix form. Each diagonal entry is set*/
  /* to 1. Last column of each row (denominator of each entry in a row) is  */
  /* also set to 1.                                                         */ 
  for(i=0;i<k;++i) {
    value_set_si(MatInv->p[i][i],1);    
    /* value_set_si(MatInv->p[i][k],1); /* denum */
  }  
  /* Apply Gauss-Jordan elimination method on the two matrices 'Mat' and  */
  /* 'MatInv' in parallel.                                                */
  for(i=0;i<k;++i) {
    
    /* Check if the diagonal entry (new pivot) is non-zero or not */
    if(value_zero_p(Mat->p[i][i])) {    
      
      /* Search for a non-zero pivot down the column(i) */
      for(j=i;j<k;++j)      
        if(value_notzero_p(Mat->p[j][i]))
          break;
      
      /* If no non-zero pivot is found, the matrix 'Mat' is non-invertible */
      /* Return 0.                                                         */
      if(j==k) {
        
        /* Clear all the 'Value' variables */
        value_clear(x);  value_clear(gcd); value_clear(piv);
        value_clear(m1); value_clear(m2);
        return 0;
      } 
      
      /* Exchange the rows, row(i) and row(j) so that the diagonal element */
      /* Mat->p[i][i] (pivot) is non-zero. Repeat the same operations on    */
      /* matrix 'MatInv'.                                                   */
      for(c=0;c<k;++c) {

        /* Interchange rows, row(i) and row(j) of matrix 'Mat'    */
        value_assign(x,Mat->p[j][c]);
        value_assign(Mat->p[j][c],Mat->p[i][c]);
        value_assign(Mat->p[i][c],x);
        
        /* Interchange rows, row(i) and row(j) of matrix 'MatInv' */
        value_assign(x,MatInv->p[j][c]);
        value_assign(MatInv->p[j][c],MatInv->p[i][c]);
        value_assign(MatInv->p[i][c],x);
      }
    }
    
    /* Make all the entries in column(i) of matrix 'Mat' zero except the */
    /* diagonal entry. Repeat the same sequence of operations on matrix  */
    /* 'MatInv'.                                                         */
    for(j=0;j<k;++j) {
      if (j==i) continue;                /* Skip the pivot */
      value_assign(x,Mat->p[j][i]);
      if(value_notzero_p(x)) {
        value_assign(piv,Mat->p[i][i]);
        value_gcd(gcd, x, piv);
        if (value_notone_p(gcd) ) {
          value_divexact(x, x, gcd);
          value_divexact(piv, piv, gcd);
        }
        for(c=((j>i)?i:0);c<k;++c) {
          value_multiply(m1,piv,Mat->p[j][c]);
          value_multiply(m2,x,Mat->p[i][c]);
          value_subtract(Mat->p[j][c],m1,m2); 
        }
        for(c=0;c<k;++c) {
          value_multiply(m1,piv,MatInv->p[j][c]);
          value_multiply(m2,x,MatInv->p[i][c]);
          value_subtract(MatInv->p[j][c],m1,m2);
        }
              
        /* Simplify row(j) of the two matrices 'Mat' and 'MatInv' by */
        /* dividing the rows with the common GCD.                     */
        Vector_Gcd(&MatInv->p[j][0],k,&m1);
        Vector_Gcd(&Mat->p[j][0],k,&m2);
        value_gcd(gcd, m1, m2);
        if(value_notone_p(gcd)) {
          for(c=0;c<k;++c) {
            value_divexact(Mat->p[j][c], Mat->p[j][c], gcd);
            value_divexact(MatInv->p[j][c], MatInv->p[j][c], gcd);
          }
        }
      }
    }
  }
  
  /* Find common denom for each row */ 
   den = (Value *)malloc(k*sizeof(Value));
   value_set_si(x,1);
   for(j=0 ; j<k ; ++j) {
     value_init(den[j]);
     value_assign(den[j],Mat->p[j][j]);
     
     /* gcd is always positive */
     Vector_Gcd(&MatInv->p[j][0],k,&gcd);
     value_gcd(gcd, gcd, den[j]);
     if (value_neg_p(den[j])) 
       value_oppose(gcd,gcd); /* make denominator positive */
     if (value_notone_p(gcd)) {
       for (c=0; c<k; c++) 
         value_divexact(MatInv->p[j][c], MatInv->p[j][c], gcd); /* normalize */
       value_divexact(den[j], den[j], gcd);
     }  
     value_gcd(gcd, x, den[j]);
     value_divexact(m1, den[j], gcd);
     value_multiply(x,x,m1);
   }
   if (value_notone_p(x)) 
     for(j=0 ; j<k ; ++j) {       
       for (c=0; c<k; c++) {
         value_division(m1,x,den[j]);
         value_multiply(MatInv->p[j][c],MatInv->p[j][c],m1);  /* normalize */
       }
     }

   /* Clear all the 'Value' variables */
   for(j=0 ; j<k ; ++j) {
     value_clear(den[j]);
   }  
   value_clear(x);  value_clear(gcd); value_clear(piv);
   value_clear(m1); value_clear(m2);
   free(den);
   
   return 1;
} /* Matrix_Inverse */

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