// Copyright (c) 1997-2000  Utrecht University (The Netherlands),
// ETH Zurich (Switzerland), Freie Universitaet Berlin (Germany),
// INRIA Sophia-Antipolis (France), Martin-Luther-University Halle-Wittenberg
// (Germany), Max-Planck-Institute Saarbruecken (Germany), RISC Linz (Austria),
// and Tel-Aviv University (Israel).  All rights reserved.
//
// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; version 2.1 of the License.
// See the file LICENSE.LGPL distributed with CGAL.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: svn+ssh://scm.gforge.inria.fr/svn/cgal/trunk/Kernel_d/include/CGAL/Kernel_d/Linear_algebraHd_impl.h $
// $Id: Linear_algebraHd_impl.h 56667 2010-06-09 07:37:13Z sloriot $
// 
//
// Author(s)     : Michael Seel <seel@mpi-sb.mpg.de>
//---------------------------------------------------------------------
// file generated by notangle from Linear_algebra.lw
// please debug or modify noweb file
// based on LEDA architecture by S. Naeher, C. Uhrig
// coding: K. Mehlhorn, M. Seel
// debugging and templatization: M. Seel
//---------------------------------------------------------------------
#ifndef CGAL_LINEAR_ALGEBRAHD_C
#define CGAL_LINEAR_ALGEBRAHD_C
namespace CGAL {

template <class RT_, class AL_>
bool Linear_algebraHd<RT_,AL_>::
linear_solver(const Matrix& A, const Vector& b, 
              Vector& x, RT& D, 
              Matrix& spanning_vectors, Vector& c)
{ 
  bool solvable = true; 
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }


  for(i = rank; i < rows && C(i,cols) == 0; ++i) {}
  if (i < rows) 
  { solvable = false; c = L.row(i); }

  if (solvable) { 
    x = Vector(cols); 
    D = denom; 
    for(i = rank - 1; i >= 0; i--) { 
      RT_ h = C(i,cols) * D; 
      for (j = i + 1; j < rank; j++) { 
        h -= C(i,j)*x[var[j]]; 
      }
      x[var[i]]= h / C(i,i); 
    }
    #ifdef CGAL_LA_SELFTEST
      CGAL_assertion( (M*x).is_zero() );
    #endif

    int defect = cols - rank; // dimension of kernel
    spanning_vectors = Matrix(cols,defect); 
     
    if (defect > 0) { 
      for(int l=0; l < defect; l++) { 
        spanning_vectors(var[rank + l],l)=D; 
        for(i = rank - 1; i >= 0 ; i--) { 
          RT_ h = - C(i,rank + l)*D; 
          for ( j= i + 1; j<rank; j++)  
            h -= C(i,j)*spanning_vectors(var[j],l); 
          spanning_vectors(var[i],l)= h / C(i,i); 
        }
    #ifdef CGAL_LA_SELFTEST
          /* we check whether the $l$ - th spanning vector is a solution 
             of the homogeneous system */
          CGAL_assertion( (M*spanning_vectors.column(l)).is_zero() );
    #endif
      }
    }

  }
  return solvable; 
}

template <class RT_, class AL_>
RT_ Linear_algebraHd<RT_,AL_>::
determinant(const Matrix& A)
{ 
  if (A.row_dimension() != A.column_dimension())
    CGAL_error_msg(      "determinant(): only square matrices are legal inputs."); 
  Vector b(A.row_dimension()); // zero - vector
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }



  if (rank < rows) 
    return 0; 
  else
    return RT(sign) * denom; 
}


template <class RT_, class AL_>
RT_ Linear_algebraHd<RT_,AL_>::
determinant(const Matrix& A, 
            Matrix& Ld, Matrix& Ud, 
            std::vector<int>& q, Vector& c) 
{ 
  if (A.row_dimension() != A.column_dimension())
    CGAL_error_msg(      "determinant(): only square matrices are legal inputs."); 
  Vector b(A.row_dimension()); // zero - vector
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }



  if (rank < rows) { 
    c = L.row(rows - 1); 
    return 0; 
  } else { 
    Ld = L; 
    Ud = Matrix(rows); // square
    for (i = 0; i < rows; i++) 
      for(j = 0; j <rows; j++) 
        Ud(i,j) = C(i,j); 
    q = var; 
    return RT(sign) * denom; 
  }
}

template <class RT_, class AL_>  
int  Linear_algebraHd<RT_,AL_>:: 
sign_of_determinant(const Matrix& M)
{ return CGAL_NTS sign(determinant(M)); }

template <class RT_, class AL_>  
bool Linear_algebraHd<RT_,AL_>:: 
verify_determinant(const Matrix& A, RT_ D, 
                   Matrix& L, Matrix& U, 
                   const std::vector<int>& q, Vector& c) 
{ 
  if ((int)q.size() != A.column_dimension())
    CGAL_error_msg(    "verify_determinant: q should be a permutation array \
    with index range [0,A.column_dimension() - 1]."); 
  int n = A.row_dimension(); 
  int i,j; 
  if (D == 0) { /* we have $c^T \cdot A = 0$  */
    Vector zero(n); 
    return  (transpose(A) * c == zero); 
  } else { 
    /* we check the conditions on |L| and |U| */
    if (L(0,0) != 1) return false; 
    for (i = 0; i<n; i++) { 
      for (j = 0; j < i; j++) 
        if (U(i,j) != 0) 
          return false; 

      if (i > 0 && L(i,i) != U(i - 1,i - 1)) 
        return false; 

      for (j = i + 1; j < n; j++) 
        if (L(i,j) != 0) 
          return false; 
    }

    /* check whether $L \cdot A \cdot Q = U$ */
   Matrix LA = L * A; 
   for (j = 0; j < n; j++)
     if (LA.column(q[j]) != U.column(j)) 
       return false; 

    /* compute sign |s| of |Q| */
    int sign = 1; 

    /* we chase the cycles of |q|. An even length cycle contributes - 1
       and vice versa */

    std::vector<bool> already_considered(n); 

    for (i = 0; i < n; i++)  
      already_considered[i] = false; 

    for (i = 0; i < n; i++)  
      already_considered[q[i]] = true; 

    for (i = 0; i < n; i++) 
      if (! already_considered[i])
        CGAL_error_msg("verify_determinant:q is not a permutation."); 
      else 
        already_considered[i] = false; 

    for (i = 0; i < n; i++) { 
      if (already_considered[i]) continue; 

      /* we have found a new cycle with minimal element $i$. */
      int k = q[i]; 
      already_considered[i] =true; 

      while (k != i) { 
        sign = - sign; 
        already_considered[k]= true; 
        k = q[k]; 
      }
    }
    return (D == RT(sign) * U(n - 1,n - 1)); 
  }
}

template <class RT_, class AL_> 
int Linear_algebraHd<RT_,AL_>::
independent_columns(const Matrix& A, 
                    std::vector<int>& columns) 
{ 
  Vector b(A.row_dimension()); // zero - vector
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }


  /* at this point we have:
     |C| has an $rank \times rank$ upper triangular matrix 
     in its left upper corner; 
     |var| tells us the columns of |A| corresponding to the
     dependent variables; */
     
  columns = std::vector<int>(rank); 
  for(i = 0; i < rank; i++) 
    columns[i] = var[i]; 
  return rank;
}


template <class RT_, class AL_>  
int  Linear_algebraHd<RT_,AL_>::
rank(const Matrix& A)
{ 
  Vector b(A.row_dimension()); // zero - vector
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }


  return rank; 
}

template <class RT_, class AL_>  
bool  Linear_algebraHd<RT_,AL_>::
inverse(const Matrix& A, Matrix& inverse, 
        RT_& D, Vector& c)
{ 
  if (A.row_dimension() != A.column_dimension())
    CGAL_error_msg("inverse: only square matrices are legal inputs."); 
  Vector b(A.row_dimension()); // zero - vector
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }


  
  if (rank < rows) { 
    // matrix is singular; return a vector $c$ with $c^T \cdot A = 0$.*/
    c = L.row(rows-1);
    return false; 
  }
  D = denom; 
  inverse = Matrix(rows); //square
  RT_ h; 
  for(i = 0; i <rows; i++) 
  {  // $i$-th column of inverse
    for (j = rows - 1; j >= 0; j--) { 
      h = L (j,i) * D; 
      for (int l = j + 1; l<rows; l++)  
        h -= C(j,l)*inverse(var[l],i); 
      inverse(var[j],i) = h / C(j,j); 
    }
  }  

  #ifdef CGAL_LA_SELFTEST
  if (A*inverse != Matrix(rows,Matrix::RT_val(1))*D)
    CGAL_error_msg("inverse(): matrix inverse computed incorrectly."); 
  #endif      

  return true; 
}


template <class RT_, class AL_>  
int  Linear_algebraHd<RT_,AL_>::
homogeneous_linear_solver(const Matrix &A, 
                          Matrix& spanning_vectors)
/* returns the dimension of the solution space of the homogeneous system 
   $Ax = 0$. The columns of spanning\_vectors span the solution space. */
{ 
  Vector b(A.row_dimension()); // zero - vector
  RT_ D; 
  int i,j,k; // indices to step through the matrix
  int rows = A.row_dimension(); 
  int cols = A.column_dimension(); 

  /* at this point one might want to check whether the computation can
     be carried out with doubles, see section \ref{ optimization }. */

  CGAL_assertion_msg(b.dimension() == rows, 
    "linear_solver: b has wrong dimension");

  Matrix C(rows,cols + 1); 
  // the matrix in which we will calculate ($C = (A \vert b)$)
    
  /* copy |A| and |b| into |C| and L becomes the identity matrix */
  Matrix L(rows); // zero initialized
  for(i=0; i<rows; i++) { 
    for(j=0; j<cols; j++) 
      C(i,j)=A(i,j); 
    C(i,cols)=b[i]; 
    L(i,i) = 1; // diagonal elements are 1
  }


  std::vector<int> var(cols); 
  // column $j$ of |C| represents the |var[j]| - th variable
  // the array is indexed between |0| and |cols - 1|

  for(j=0; j<cols; j++)
    var[j]= j; // at the beginning, variable $x_j$ stands in column $j$

  RT_ denom = 1; // the determinant of an empty matrix is 1
  int sign = 1; // no interchanges yet
  int rank = 0; // we have not seen any non-zero row yet

  /* here comes the main loop */
  for(k=0; k<rows; k++) {   
    bool non_zero_found = false; 
    for(i = k; i < rows; i++) { // step through rows $k$ to |rows - 1|
      for (j = k ; j < cols && C(i,j) == 0; j++) {}
      // step through columns |k| to |cols - 1|
      if (j < cols) {  
        non_zero_found = true; 
        break; 
      }
    }


    if ( non_zero_found ) { 
      rank++; //increase the rank
      if (i != k) { 
        sign = -sign; 
        /* we interchange rows |k| and |i| of |L| and |C| */
        L.swap_rows(i,k); C.swap_rows(i,k);
      }
      if (j != k) { 
        /* We interchange columns |k| and |j|, and
           store the exchange of variables in |var| */
        sign = - sign; 
        C.swap_columns(j,k);
        std::swap(var[k],var[j]); 
      }
       
      for(i = k + 1; i < rows; i++)  
        for (j = 0; j <  rows; j++)  //and all columns of |L|
          L(i,j) = (L(i,j)*C(k,k) - C(i,k)*L(k,j))/denom; 

      for(i = k + 1; i < rows; i++) {      
        /* the following iteration uses and changes |C(i,k)| */
        RT temp = C(i,k); 
        for (j = k; j <= cols; j++)
          C(i,j) = (C(i,j)*C(k,k) - temp*C(k,j))/denom; 
      }
      denom = C(k,k); 
      #ifdef CGAL_LA_SELFTEST
      for(i = 0; i < rows; i++) { 
        for (j = 0; j < cols; j++) { 
          RT Sum = 0; 
          for (int l = 0; l < rows; l++) 
            Sum += L(i,l)*A(l, var[j]); 
          CGAL_assertion_msg( Sum  == C(i,j), 
            "linear_solver: L*A*P different from C"); 
        }
        RT Sum = 0; 
        for (int l = 0; l < rows; l++) 
          Sum += L(i,l)*b[l];
          CGAL_assertion_msg( Sum  == C(i,cols), 
            "linear_solver: L*A*P different from C"); 
      }
      #endif


    }
    else 
      break; 
  }


  Vector x; 
  x = Vector(cols); 
  D = denom; 
  for(i = rank - 1; i >= 0; i--) { 
    RT_ h = C(i,cols) * D; 
    for (j = i + 1; j < rank; j++) { 
      h -= C(i,j)*x[var[j]]; 
    }
    x[var[i]]= h / C(i,i); 
  }
  #ifdef CGAL_LA_SELFTEST
    CGAL_assertion( (M*x).is_zero() );
  #endif

  int defect = cols - rank; // dimension of kernel
  spanning_vectors = Matrix(cols,defect); 
   
  if (defect > 0) { 
    for(int l=0; l < defect; l++) { 
      spanning_vectors(var[rank + l],l)=D; 
      for(i = rank - 1; i >= 0 ; i--) { 
        RT_ h = - C(i,rank + l)*D; 
        for ( j= i + 1; j<rank; j++)  
          h -= C(i,j)*spanning_vectors(var[j],l); 
        spanning_vectors(var[i],l)= h / C(i,i); 
      }
  #ifdef CGAL_LA_SELFTEST
        /* we check whether the $l$ - th spanning vector is a solution 
           of the homogeneous system */
        CGAL_assertion( (M*spanning_vectors.column(l)).is_zero() );
  #endif
    }
  }
; 
  return defect; 
}

template <class RT_, class AL_>  
bool  Linear_algebraHd<RT_,AL_>::
homogeneous_linear_solver(const Matrix& A, Vector& x)
/* returns true if the homogeneous system $Ax = 0$ has a non - trivial
   solution and false otherwise. */
{  
  x = Vector(A.row_dimension());
  Matrix spanning_vectors; 
  int dimension = homogeneous_linear_solver(A,spanning_vectors); 
  if (dimension == 0) return false; 

  /* return first column of |spanning_vectors| */
  for (int i = 0; i < spanning_vectors.row_dimension(); i++)  
    x[i] = spanning_vectors(i,0); 
  return true; 
}


template <class RT_, class AL_>  
typename Linear_algebraHd<RT_,AL_>::Matrix 
Linear_algebraHd<RT_,AL_>::transpose(const Matrix& M)
{ 
  int d1 = M.row_dimension(); 
  int d2 = M.column_dimension(); 
  Matrix result(d2,d1); 
  for(int i = 0; i < d2; i++)
    for(int j = 0; j < d1; j++)
      result(i,j) = M(j,i); 
  return result; 
}


} //namespace CGAL
#endif //CGAL_LINEAR_ALGEBRAHD_C
