clufactor_rational.cpp

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/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
/*                                                                           */
/*                  This file is part of the class library                   */
/*       SoPlex --- the Sequential object-oriented simPlex.                  */
/*                                                                           */
/*    Copyright (C) 1996-2015 Konrad-Zuse-Zentrum                            */
/*                            fuer Informationstechnik Berlin                */
/*                                                                           */
/*  SoPlex is distributed under the terms of the ZIB Academic Licence.       */
/*                                                                           */
/*  You should have received a copy of the ZIB Academic License              */
/*  along with SoPlex; see the file COPYING. If not email to soplex@zib.de.  */
/*                                                                           */
/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */

#include <assert.h>

#include "spxdefines.h"
#include "clufactor_rational.h"
#include "cring.h"
#include "spxalloc.h"
#include "exceptions.h"
#include "basevectors.h"
#include "rational.h"

namespace soplex
{
/* This number is used to decide wether a value is zero
 * or was explicitly set to zero.
 */
/*
#define MARKER     1e-100
*/

static const Real verySparseFactor = 0.001;
static const Real verySparseFactor4right = 0.2;
static const Real verySparseFactor4left  = 0.1;

/* generic heap management */
static void enQueueMax( int* heap, int* size, int elem )
{
   int i, j;

   j = ( *size )++;

   while ( j > 0 )
   {
      i = ( j - 1 ) / 2;

      if ( elem > heap[i] )
      {
         heap[j] = heap[i];
         j = i;
      }
      else
         break;
   }

   heap[j] = elem;

#ifdef SOPLEX_DEBUG
// no NDEBUG define, since this block is really expensive
   for ( i = 1; i < *size; ++i )
      for ( j = 0; j < i; ++j )
         assert( heap[i] != heap[j] );

#endif  /* SOPLEX_DEBUG */
}

static int deQueueMax( int* heap, int* size )
{
   int e, elem;
   int i, j, s;
   int e1, e2;

   elem = *heap;
   e = heap[s = --( *size )];
   --s;

   for ( j = 0, i = 1; i < s; i = 2 * j + 1 )
   {
      e1 = heap[i];
      e2 = heap[i + 1];

      if ( e1 > e2 )
      {
         if ( e < e1 )
         {
            heap[j] = e1;
            j = i;
         }
         else
         {
            heap[j] = e;
            return elem;
         }
      }
      else
      {
         if ( e < e2 )
         {
            heap[j] = e2;
            j = i + 1;
         }
         else
         {
            heap[j] = e;
            return elem;
         }
      }
   }

   if ( i < *size && e < heap[i] )
   {
      heap[j] = heap[i];
      j = i;
   }

   heap[j] = e;

   return elem;
}

static void enQueueMin( int* heap, int* size, int elem )
{
   int i, j;

   j = ( *size )++;

   while ( j > 0 )
   {
      i = ( j - 1 ) / 2;

      if ( elem < heap[i] )
      {
         heap[j] = heap[i];
         j = i;
      }
      else
         break;
   }

   heap[j] = elem;

#ifdef SOPLEX_DEBUG
// no NDEBUG define, since this block is really expensive
   for ( i = 1; i < *size; ++i )
      for ( j = 0; j < i; ++j )
         assert( heap[i] != heap[j] );

#endif  /* SOPLEX_DEBUG */
}

static int deQueueMin( int* heap, int* size )
{
   int e, elem;
   int i, j, s;
   int e1, e2;

   elem = *heap;
   e = heap[s = --( *size )];
   --s;

   for ( j = 0, i = 1; i < s; i = 2 * j + 1 )
   {
      e1 = heap[i];
      e2 = heap[i + 1];

      if ( e1 < e2 )
      {
         if ( e > e1 )
         {
            heap[j] = e1;
            j = i;
         }
         else
         {
            heap[j] = e;
            return elem;
         }
      }
      else
      {
         if ( e > e2 )
         {
            heap[j] = e2;
            j = i + 1;
         }
         else
         {
            heap[j] = e;
            return elem;
         }
      }
   }

   if ( i < *size && e > heap[i] )
   {
      heap[j] = heap[i];
      j = i;
   }

   heap[j] = e;

   return elem;
}

/************************************************************/
CLUFactorRational::Temp::Temp()
      : s_mark( 0 )
      , s_cact( 0 )
      , stage( 0 )
      , pivot_col( 0 )
      , pivot_colNZ( 0 )
      , pivot_row( 0 )
      , pivot_rowNZ( 0 )
{}

void CLUFactorRational::Temp::init( int p_dim )
{
   s_max.reDim(p_dim);
   spx_realloc( s_cact, p_dim );
   spx_realloc( s_mark, p_dim );
   stage = 0;
}

void CLUFactorRational::Temp::clear()
{
   if ( s_mark != 0 )
      spx_free( s_mark );

   if ( s_cact != 0 )
      spx_free( s_cact );

   s_max.reDim(0);

   if ( pivot_col != 0 )
      spx_free( pivot_col );

   if ( pivot_colNZ != 0 )
      spx_free( pivot_colNZ );

   if ( pivot_row != 0 )
      spx_free( pivot_row );

   if ( pivot_rowNZ != 0 )
      spx_free( pivot_rowNZ );
}

CLUFactorRational::Temp::~Temp()
{
   clear();
}

/************************************************************/
void CLUFactorRational::initPerm()
{

   for ( int i = 0; i < thedim; ++i )
      row.orig[i] = row.perm[i] = col.orig[i] = col.perm[i] = -1;
}

/*****************************************************************************/

void CLUFactorRational::setPivot( const int p_stage,
                          const int p_col,
                          const int p_row,
                          const Rational& val )
{
   assert( row.perm[p_row] < 0 );
   assert( col.perm[p_col] < 0 );

   row.orig[p_stage] = p_row;
   col.orig[p_stage] = p_col;
   row.perm[p_row]   = p_stage;
   col.perm[p_col]   = p_stage;
   diag[p_row]       = 1.0 / val;

   if ( spxAbs( diag[p_row] ) > maxabs )
      maxabs = spxAbs( diag[p_row] );
}

/*****************************************************************************/
/*
 *      Perform garbage collection on row file
 */
void CLUFactorRational::packRows()
{
   int n, i, j, l_row;
   Dring *ring, *list;

   int *l_ridx = u.row.idx;
   DVectorRational& l_rval = u.row.val;
   int *l_rlen = u.row.len;
   int *l_rmax = u.row.max;
   int *l_rbeg = u.row.start;

   n = 0;
   list = &( u.row.list );

   for ( ring = list->next; ring != list; ring = ring->next )
   {
      l_row = ring->idx;

      if ( l_rbeg[l_row] != n )
      {
         do
         {
            l_row = ring->idx;
            i = l_rbeg[l_row];
            assert( l_rlen[l_row] <= l_rmax[l_row] );
            l_rbeg[l_row] = n;
            l_rmax[l_row] = l_rlen[l_row];
            j = i + l_rlen[l_row];

            for ( ; i < j; ++i, ++n )
            {
               assert( n <= i );
               l_ridx[n] = l_ridx[i];
               l_rval[n] = l_rval[i];
            }

            ring = ring->next;
         }
         while ( ring != list );

         goto terminatePackRows;
      }

      n += l_rlen[l_row];

      l_rmax[l_row] = l_rlen[l_row];
   }

terminatePackRows:

   u.row.max[thedim] = 0;
   u.row.used = n;
}

/*****************************************************************************/
/*
 *      Perform garbage collection on column file
 */
void CLUFactorRational::forestPackColumns()
{
   int n, i, j, colno;
   Dring *ring, *list;

   DVectorRational& cval = u.col.val;
   int *cidx = u.col.idx;
   int *clen = u.col.len;
   int *cmax = u.col.max;
   int *cbeg = u.col.start;

   n = 0;
   list = &u.col.list;

   for ( ring = list->next; ring != list; ring = ring->next )
   {
      colno = ring->idx;

      if ( cbeg[colno] != n )
      {
         do
         {
            colno = ring->idx;
            i = cbeg[colno];
            cbeg[colno] = n;
            cmax[colno] = clen[colno];
            j = i + clen[colno];

            for ( ; i < j; ++i )
            {
               cval[n] = cval[i];
               cidx[n++] = cidx[i];
            }

            ring = ring->next;
         }
         while ( ring != list );

         goto terminatePackColumns;
      }

      n += clen[colno];

      cmax[colno] = clen[colno];
   }

terminatePackColumns :

   u.col.used = n;
   u.col.max[thedim] = 0;
}

/*
 *      Make row of fac large enough to hold len nonzeros.
 */
void CLUFactorRational::remaxRow( int p_row, int len )
{
   assert( u.row.max[p_row] < len );

   if ( u.row.elem[p_row].next == &( u.row.list ) ) /* last in row file */
   {
      int delta = len - u.row.max[p_row];

      if ( delta > u.row.val.dim() - u.row.used )
      {
         packRows();
         delta = len - u.row.max[p_row];  // packRows() changes u.row.max[] !

         if ( u.row.val.dim() < rowMemMult * u.row.used + len )
            minRowMem( 2 * u.row.used + len );

         /* minRowMem(rowMemMult * u.row.used + len); */
      }

      assert( delta <= u.row.val.dim() - u.row.used

              && "ERROR: could not allocate memory for row file" );

      u.row.used += delta;
      u.row.max[p_row] = len;
   }
   else                        /* row must be moved to end of row file */
   {
      int i, j, k;
      DVectorRational& val = u.row.val;
      Dring *ring;

      if ( len > u.row.val.dim() - u.row.used )
      {
         packRows();

         if ( u.row.val.dim() < rowMemMult * u.row.used + len )
            minRowMem( 2 * u.row.used + len );

         /* minRowMem(rowMemMult * u.row.used + len);*/
      }

      assert( len <= u.row.val.dim() - u.row.used

              && "ERROR: could not allocate memory for row file" );

      int *idx = u.row.idx;
      j = u.row.used;
      i = u.row.start[p_row];
      k = u.row.len[p_row] + i;
      u.row.start[p_row] = j;
      u.row.used += len;

      u.row.max[u.row.elem[p_row].prev->idx] += u.row.max[p_row];
      u.row.max[p_row] = len;
      removeDR( u.row.elem[p_row] );
      ring = u.row.list.prev;
      init2DR( u.row.elem[p_row], *ring );

      for ( ; i < k; ++i, ++j )
      {
         val[j] = val[i];
         idx[j] = idx[i];
      }
   }

   assert( u.row.start[u.row.list.prev->idx] + u.row.max[u.row.list.prev->idx]

           == u.row.used );
}

/*************************************************************************/
/*
 *      Perform garbage collection on column file
 */
void CLUFactorRational::packColumns()
{
   int n, i, j, l_col;
   Dring *ring, *list;

   int *l_cidx = u.col.idx;
   int *l_clen = u.col.len;
   int *l_cmax = u.col.max;
   int *l_cbeg = u.col.start;

   n = 0;
   list = &( u.col.list );

   for ( ring = list->next; ring != list; ring = ring->next )
   {
      l_col = ring->idx;

      if ( l_cbeg[l_col] != n )
      {
         do
         {
            l_col = ring->idx;
            i = l_cbeg[l_col];
            l_cbeg[l_col] = n;
            l_cmax[l_col] = l_clen[l_col];
            j = i + l_clen[l_col];

            for ( ; i < j; ++i )
               l_cidx[n++] = l_cidx[i];

            ring = ring->next;
         }
         while ( ring != list );

         goto terminatePackColumns;
      }

      n += l_clen[l_col];

      l_cmax[l_col] = l_clen[l_col];
   }

terminatePackColumns :

   u.col.used = n;
   u.col.max[thedim] = 0;
}

/*
 *      Make column col of fac large enough to hold len nonzeros.
 */
void CLUFactorRational::remaxCol( int p_col, int len )
{
   assert( u.col.max[p_col] < len );

   if ( u.col.elem[p_col].next == &( u.col.list ) ) /* last in column file */
   {
      int delta = len - u.col.max[p_col];

      if ( delta > u.col.size - u.col.used )
      {
         packColumns();
         delta = len - u.col.max[p_col];

         if ( u.col.size < colMemMult * u.col.used + len )
            minColMem( 2 * u.col.used + len );

         /* minColMem(colMemMult * u.col.used + len); */
      }

      assert( delta <= u.col.size - u.col.used

              && "ERROR: could not allocate memory for column file" );

      u.col.used += delta;
      u.col.max[p_col] = len;
   }

   else                        /* column must be moved to end of column file */
   {
      int i, j, k;
      int *idx;
      Dring *ring;

      if ( len > u.col.size - u.col.used )
      {
         packColumns();

         if ( u.col.size < colMemMult * u.col.used + len )
            minColMem( 2 * u.col.used + len );

         /* minColMem(colMemMult * u.col.used + len); */
      }

      assert( len <= u.col.size - u.col.used

              && "ERROR: could not allocate memory for column file" );

      j = u.col.used;
      i = u.col.start[p_col];
      k = u.col.len[p_col] + i;
      u.col.start[p_col] = j;
      u.col.used += len;

      u.col.max[u.col.elem[p_col].prev->idx] += u.col.max[p_col];
      u.col.max[p_col] = len;
      removeDR( u.col.elem[p_col] );
      ring = u.col.list.prev;
      init2DR( u.col.elem[p_col], *ring );

      idx = u.col.idx;

      for ( ; i < k; ++i )
         idx[j++] = idx[i];
   }
}

/*
 *      Make column col of fac large enough to hold len nonzeros.
 */
void CLUFactorRational::forestReMaxCol( int p_col, int len )
{
   assert( u.col.max[p_col] < len );

   if ( u.col.elem[p_col].next == &( u.col.list ) ) /* last in column file */
   {
      int delta = len - u.col.max[p_col];

      if ( delta > u.col.size - u.col.used )
      {
         forestPackColumns();
         delta = len - u.col.max[p_col];

         if ( u.col.size < colMemMult * u.col.used + len )
            forestMinColMem( int( colMemMult * u.col.used + len ) );
      }

      assert( delta <= u.col.size - u.col.used

              && "ERROR: could not allocate memory for column file" );

      u.col.used += delta;
      u.col.max[p_col] = len;
   }

   else                        /* column must be moved to end of column file */
   {
      int i, j, k;
      int *idx = u.col.idx;
      DVectorRational& val = u.col.val;
      Dring *ring;

      if ( len > u.col.size - u.col.used )
      {
         forestPackColumns();

         if ( u.col.size < colMemMult * u.col.used + len )
            forestMinColMem( int( colMemMult * u.col.used + len ) );
      }

      assert( len <= u.col.size - u.col.used

              && "ERROR: could not allocate memory for column file" );

      j = u.col.used;
      i = u.col.start[p_col];
      k = u.col.len[p_col] + i;
      u.col.start[p_col] = j;
      u.col.used += len;

      u.col.max[u.col.elem[p_col].prev->idx] += u.col.max[p_col];
      u.col.max[p_col] = len;
      removeDR( u.col.elem[p_col] );
      ring = u.col.list.prev;
      init2DR( u.col.elem[p_col], *ring );

      for ( ; i < k; ++i )
      {
         val[j] = val[i];
         idx[j++] = idx[i];
      }
   }
}

/*****************************************************************************/

/**
 *   \brief Performs the Forrest-Tomlin update of the LU factorization.
 *
 *   BH: I suppose this is implemented as described in UH Suhl, LM Suhl: A fast LU
 *       update for linear programming, Annals of OR 43, p. 33-47, 1993.
 *
 *   @param  p_col      Index of basis column to replace.
 *   @param  p_work     Dense vector to substitute in the basis.
 *   @param  num        Number of nonzeros in vector represented by p_work.
 *   @param  nonz       Indices of nonzero elements in vector p_work.
 *
 *   The parameters num and nonz are used for the following optimization: If both
 *   are nonzero, indices of the nonzero elements provided in nonz (num giving
 *   their number) allow to access only those nonzero entries.  Otherwise we have
 *   to go through the entire dense vector element by element.
 *
 *   After copying p_work into U, p_work is used to expand the row r, which is
 *   needed to restore the triangular structure of U.
 *
 *   Also num and nonz are used to maintain a heap if there are only very few
 *   nonzeros to be eliminated. This is plainly wrong if the method is called with
 *   nonz==0, see todo at the corresponding place below.
 *
 *   @throw SPxStatusException if the loaded matrix is singular
 *
 *   @todo Use an extra member variable as a buffer for working with the dense
 *         row instead of misusing p_work. I think that should be as efficient and
 *         much cleaner.
 */

void CLUFactorRational::forestUpdate( int p_col, Rational* p_work, int num, int *nonz )
{
   int i, j, k, h, m, n;
   int ll, c, r, rowno;
   Rational x;

   int *lbeg = l.start;

   DVectorRational& cval = u.col.val;
   int *cidx = u.col.idx;
   int *cmax = u.col.max;
   int *clen = u.col.len;
   int *cbeg = u.col.start;

   DVectorRational& rval = u.row.val;
   int *ridx = u.row.idx;
   int *rmax = u.row.max;
   int *rlen = u.row.len;
   int *rbeg = u.row.start;

   int *rperm = row.perm;
   int *rorig = row.orig;
   int *cperm = col.perm;
   int *corig = col.orig;

   Rational l_maxabs = maxabs;
   int dim = thedim;

   /*  Remove column p_col from U
    */
   j = cbeg[p_col];
   i = clen[p_col];
   nzCnt -= i;

   for ( i += j - 1; i >= j; --i )
   {
      m = cidx[i];          // remove column p_col from row m
      k = rbeg[m];
      h = --( rlen[m] ) + k;  // decrease length of row m

      while ( ridx[k] != p_col )
         ++k;

      assert( k <= h );     // k is the position of p_col, h is last position

      ridx[k] = ridx[h];    // store last index at the position of p_col

      rval[k] = rval[h];
   }

   /*  Insert new vector column p_col thereby determining the highest permuted
    *       row index r.
    *
    *       Distinguish between optimized call (num > 0, nonz != 0) and
    *       non-optimized one.
    */
   assert( num ); // otherwise the assert( nonz != 0 ) below should fail

   if ( num )
   {
      // Optimized call.
      assert( nonz != 0 );

      clen[p_col] = 0;

      if ( num > cmax[p_col] )
         forestReMaxCol( p_col, num );

      cidx = u.col.idx;

      cval = u.col.val;

      k = cbeg[p_col];

      r = 0;

      for ( j = 0; j < num; ++j )
      {
         i = nonz[j];
         x = p_work[i];
         p_work[i] = 0;

         if ( x != 0 )
         {
            if ( spxAbs( x ) > l_maxabs )
               l_maxabs = spxAbs( x );

            /* insert to column file */
            assert( k - cbeg[p_col] < cmax[p_col] );

            cval[k] = x;

            cidx[k++] = i;

            /* insert to row file */
            if ( rmax[i] <= rlen[i] )
            {
               remaxRow( i, rlen[i] + 1 );
               rval = u.row.val;
               ridx = u.row.idx;
            }

            h = rbeg[i] + ( rlen[i] )++;

            rval[h] = x;
            ridx[h] = p_col;

            /* check permuted row index */

            if ( rperm[i] > r )
               r = rperm[i];
         }
      }

      nzCnt += ( clen[p_col] = k - cbeg[p_col] );
   }
   else
   {
      // Non-optimized call: We have to access all elements of p_work.
      assert( nonz == 0 );

      /*
       *      clen[col] = 0;
       *      reMaxCol(fac, col, dim);
       */
      cidx = u.col.idx;
      cval = u.col.val;
      k = cbeg[p_col];
      j = k + cmax[p_col];
      r = 0;

      for ( i = 0; i < dim; ++i )
      {
         x = p_work[i];
         p_work[i] = 0;

         if ( x != 0 )
         {
            if ( spxAbs( x ) > l_maxabs )
               l_maxabs = spxAbs( x );

            /* insert to column file */
            if ( k >= j )
            {
               clen[p_col] = k - cbeg[p_col];
               forestReMaxCol( p_col, dim - i );
               cidx = u.col.idx;
               cval = u.col.val;
               k = cbeg[p_col];
               j = k + cmax[p_col];
               k += clen[p_col];
            }

            assert( k - cbeg[p_col] < cmax[p_col] );

            cval[k] = x;
            cidx[k++] = i;

            /* insert to row file */

            if ( rmax[i] <= rlen[i] )
            {
               remaxRow( i, rlen[i] + 1 );
               rval = u.row.val;
               ridx = u.row.idx;
            }

            h = rbeg[i] + ( rlen[i] )++;

            rval[h] = x;
            ridx[h] = p_col;

            /* check permuted row index */

            if ( rperm[i] > r )
               r = rperm[i];
         }
      }

      nzCnt += ( clen[p_col] = k - cbeg[p_col] );

      if ( cbeg[p_col] + cmax[p_col] == u.col.used )
      {
         u.col.used -= cmax[p_col];
         cmax[p_col] = clen[p_col];
         u.col.used += cmax[p_col];
      }
   }

   c = cperm[p_col];

   if ( r > c )                       /* Forest Tomlin update */
   {
      /*      update permutations
       */
      j = rorig[c];

      // memmove is more efficient than a for loop
      // for ( i = c; i < r; ++i )
      //    rorig[i] = rorig[i + 1];
      memmove(&rorig[c], &rorig[c + 1], (unsigned int)(r - c) * sizeof(int));

      rorig[r] = j;

      for ( i = c; i <= r; ++i )
         rperm[rorig[i]] = i;

      j = corig[c];

      // memmove is more efficient than a for loop
      // for ( i = c; i < r; ++i )
      //    corig[i] = corig[i + 1];
      memmove(&corig[c], &corig[c + 1], (unsigned int)(r - c) * sizeof(int));

      corig[r] = j;

      for ( i = c; i <= r; ++i )
         cperm[corig[i]] = i;


      rowno = rorig[r];

      j = rbeg[rowno];

      i = rlen[rowno];

      nzCnt -= i;

      if ( i < verySparseFactor * ( dim - c ) ) // few nonzeros to be eliminated
      {
         /**
          *          The following assert is obviously violated if this method is called
          *          with nonzero==0.
          *
          *          @todo Use an extra member variable as a buffer for the heap instead of
          *                misusing nonz and num. The method enQueueMin() seems to
          *                sort the nonzeros or something, for which it only needs
          *                some empty vector of size num.
          */
         assert( nonz != 0 );

         /*  move row r from U to p_work
          */
         num = 0;

         for ( i += j - 1; i >= j; --i )
         {
            k = ridx[i];
            p_work[k] = rval[i];
            enQueueMin( nonz, &num, cperm[k] );
            m = --( clen[k] ) + cbeg[k];

            for ( h = m; cidx[h] != rowno; --h )
               ;

            cidx[h] = cidx[m];

            cval[h] = cval[m];
         }


         /*  Eliminate row r from U to L file
          */
         ll = makeLvec( r - c, rowno );

         DVectorRational& lval = l.val;
         int *lidx = l.idx;

         assert(( num == 0 ) || ( nonz != 0 ) );

         /* for(i = c; i < r; ++i)       */
         while ( num )
         {
#ifndef NDEBUG
            // The numbers seem to be often 1e-100, is this ok ?

            for ( i = 0; i < num; ++i )
               assert( p_work[corig[nonz[i]]] != 0 );

#endif // NDEBUG
            i = deQueueMin( nonz, &num );

            if ( i == r )
               break;

            k = corig[i];

            assert( p_work[k] != 0 );

            n = rorig[i];

            x = p_work[k] * diag[n];

            lidx[ll] = n;

            lval[ll] = x;

            p_work[k] = 0;

            ll++;

            if ( spxAbs( x ) > l_maxabs )
               l_maxabs = spxAbs( x );

            j = rbeg[n];

            m = rlen[n] + j;

            for ( ; j < m; ++j )
            {
               int jj = ridx[j];
               Rational y = p_work[jj];

               if ( y == 0 )
                  enQueueMin( nonz, &num, cperm[jj] );

               y -= x * rval[j];

               //p_work[jj] = y + (( y == 0 ) ? MARKER : 0 );
               p_work[jj] = y;
            }
         }

         if ( lbeg[l.firstUnused - 1] == ll )
            ( l.firstUnused )--;
         else
            lbeg[l.firstUnused] = ll;


         /*  Set diagonal value
          */
         if ( i != r )
         {
            stat = SLinSolverRational::SINGULAR;
            throw SPxStatusException( "XFORE01 The loaded matrix is singular" );
         }

         k = corig[r];

         x = p_work[k];
         diag[rowno] = 1 / x;
         p_work[k] = 0;


         /*  make row large enough to fit all nonzeros.
          */

         if ( rmax[rowno] < num )
         {
            rlen[rowno] = 0;
            remaxRow( rowno, num );
            rval = u.row.val;
            ridx = u.row.idx;
         }

         nzCnt += num;

         /*  Insert work to updated row thereby clearing work;
          */
         n = rbeg[rowno];

         for ( i = 0; i < num; ++i )
         {
            j = corig[nonz[i]];
            x = p_work[j];

            // BH 2005-08-24: This if is very important. It may well happen that
            // during the elimination of row r a nonzero elements cancels out
            // and becomes zero. This would lead to an infinite loop in the
            // above elimination code, since the corresponding column index would
            // be enqueued for further elimination again and agian.

            if ( x != 0 )
            {
               if ( spxAbs( x ) > l_maxabs )
                  l_maxabs = spxAbs( x );

               ridx[n] = j;

               rval[n] = x;

               p_work[j] = 0;

               ++n;

               if ( clen[j] >= cmax[j] )
               {
                  forestReMaxCol( j, clen[j] + 1 );
                  cidx = u.col.idx;
                  cval = u.col.val;
               }

               cval[cbeg[j] + clen[j]] = x;

               cidx[cbeg[j] + clen[j]++] = rowno;
            }
         }

         rlen[rowno] = n - rbeg[rowno];
      }
      else            /* few nonzeros to be eliminated        */
      {
         /*  move row r from U to p_work
          */
         for ( i += j - 1; i >= j; --i )
         {
            k = ridx[i];
            p_work[k] = rval[i];
            m = --( clen[k] ) + cbeg[k];

            for ( h = m; cidx[h] != rowno; --h )
               ;

            cidx[h] = cidx[m];

            cval[h] = cval[m];
         }


         /*  Eliminate row r from U to L file
          */
         ll = makeLvec( r - c, rowno );

         DVectorRational& lval = l.val;
         int *lidx = l.idx;

         for ( i = c; i < r; ++i )
         {
            k = corig[i];

            if ( p_work[k] != 0 )
            {
               n = rorig[i];
               x = p_work[k] * diag[n];
               lidx[ll] = n;
               lval[ll] = x;
               p_work[k] = 0;
               ll++;

               if ( spxAbs( x ) > l_maxabs )
                  l_maxabs = spxAbs( x );

               j = rbeg[n];

               m = rlen[n] + j;

               for ( ; j < m; ++j )
                  p_work[ridx[j]] -= x * rval[j];
            }
         }

         if ( lbeg[l.firstUnused - 1] == ll )
            ( l.firstUnused )--;
         else
            lbeg[l.firstUnused] = ll;


         /*  Set diagonal value
          */
         k = corig[r];

         x = p_work[k];

         if ( x == 0 )
         {
            stat = SLinSolverRational::SINGULAR;
            throw SPxStatusException( "XFORE02 The loaded matrix is singular" );
            //            return;
         }

         diag[rowno] = 1 / x;

         p_work[k] = 0;


         /*  count remaining nonzeros in work and make row large enough
          *  to fit them all.
          */
         n = 0;

         for ( i = r + 1; i < dim; ++i )
            if ( p_work[corig[i]] != 0 )
               n++;

         if ( rmax[rowno] < n )
         {
            rlen[rowno] = 0;
            remaxRow( rowno, n );
            rval = u.row.val;
            ridx = u.row.idx;
         }

         nzCnt += n;

         /*  Insert p_work to updated row thereby clearing p_work;
          */
         n = rbeg[rowno];

         for ( i = r + 1; i < dim; ++i )
         {
            j = corig[i];
            x = p_work[j];

            if ( x != 0 )
            {
               if ( spxAbs( x ) > l_maxabs )
                  l_maxabs = spxAbs( x );

               ridx[n] = j;

               rval[n] = x;

               p_work[j] = 0;

               ++n;

               if ( clen[j] >= cmax[j] )
               {
                  forestReMaxCol( j, clen[j] + 1 );
                  cidx = u.col.idx;
                  cval = u.col.val;
               }

               cval[cbeg[j] + clen[j]] = x;

               cidx[cbeg[j] + clen[j]++] = rowno;
            }
         }

         rlen[rowno] = n - rbeg[rowno];
      }
   }

   else
      if ( r == c )
      {
         /*  Move diagonal element to diag.  Note, that it must be the last
          *  element, since it has just been inserted above.
          */
         rowno = rorig[r];
         i = rbeg[rowno] + --( rlen[rowno] );
         diag[rowno] = 1 / rval[i];

         for ( j = i = --( clen[p_col] ) + cbeg[p_col]; cidx[i] != rowno; --i )
            ;

         cidx[i] = cidx[j];

         cval[i] = cval[j];
      }
      else /* r < c */
      {
         stat = SLinSolverRational::SINGULAR;
         throw SPxStatusException( "XFORE03 The loaded matrix is singular" );
         //      return;
      }

   maxabs = l_maxabs;

   assert( isConsistent() );
   stat = SLinSolverRational::OK;
}

void CLUFactorRational::update( int p_col, Rational* p_work, const int* p_idx, int num )
{
   int ll, i, j;
   Rational x, rezi;

   assert( p_work[p_col] != 0 );
   rezi = 1 / p_work[p_col];
   p_work[p_col] = 0;

   ll = makeLvec( num, p_col );
   //   ll = fac->makeLvec(num, col);
   DVectorRational& lval = l.val;
   int* lidx = l.idx;

   for ( i = num - 1; ( j = p_idx[i] ) != p_col; --i )
   {
      lidx[ll] = j;
      lval[ll] = rezi * p_work[j];
      p_work[j] = 0;
      ++ll;
   }

   lidx[ll] = p_col;

   lval[ll] = 1 - rezi;
   ++ll;

   for ( --i; i >= 0; --i )
   {
      j = p_idx[i];
      lidx[ll] = j;
      lval[ll] = x = rezi * p_work[j];
      p_work[j] = 0;
      ++ll;

      if ( spxAbs( x ) > maxabs )
         maxabs = spxAbs( x );
   }

   stat = SLinSolverRational::OK;
}

void CLUFactorRational::updateNoClear(
   int p_col,
   const Rational* p_work,
   const int* p_idx,
   int num )
{
   int ll, i, j;
   Rational x, rezi;

   assert( p_work[p_col] != 0 );
   rezi = 1 / p_work[p_col];
   ll = makeLvec( num, p_col );
   //ll = fac->makeLvec(num, col);
   DVectorRational& lval = l.val;
   int* lidx = l.idx;

   for ( i = num - 1; ( j = p_idx[i] ) != p_col; --i )
   {
      lidx[ll] = j;
      lval[ll] = rezi * p_work[j];
      ++ll;
   }

   lidx[ll] = p_col;

   lval[ll] = 1 - rezi;
   ++ll;

   for ( --i; i >= 0; --i )
   {
      j = p_idx[i];
      lidx[ll] = j;
      lval[ll] = x = rezi * p_work[j];
      ++ll;

      if ( spxAbs( x ) > maxabs )
         maxabs = spxAbs( x );
   }

   stat = SLinSolverRational::OK;
}

/*****************************************************************************/
/*
 *      Temporary data structures.
 */

/*
 *        For the i=th column the situation might look like this:
 *
 * \begin{verbatim}
 *        idx     = ....................iiiIIIIII-----..............
 *        cbeg[i] =                     ^
 *        cact[i] =                        +----+
 *        clen[i] =                     +-------+
 *        cmax[i] =                     +------------+
 *
 *        Indices clen[i]-cbeg[i]:      ^^^
 * \end{verbatim}
 *        belong to column i, but have already been pivotal and don't belong to
 *        the active submatrix.
 */

/****************************************************************************/
/*
 *      Initialize row and column file of working matrix and
 *      mark column singletons.
 */
void CLUFactorRational::initFactorMatrix( const SVectorRational** vec )
{

   Rational x;
   int m;
   int tot;
   Dring *rring, *lastrring;
   Dring *cring, *lastcring;
   const SVectorRational *psv;
   int *sing = temp.s_mark;

   /*  Initialize:
    *  - column file thereby remembering column singletons in |sing|.
    *  - nonzeros counts per row
    *  - total number of nonzeros
    */

   for ( int i = 0; i < thedim; i++ )
      u.row.max[i] = u.row.len[i] = 0;

   tot = 0;

   for ( int i = 0; i < thedim; i++ )
   {
      int k;

      psv = vec[i];
      k = psv->size();

      if ( k > 1 )
      {
         tot += k;

         for ( int j = 0; j < k; ++j )
            u.row.max[psv->index( j )]++;
      }
      else
         if ( k == 0 )
         {
            stat = SLinSolverRational::SINGULAR;
            return;
         }
   }

   /*  Resize nonzero memory if necessary
    */
   minRowMem( int( rowMemMult * tot ) );

   minColMem( int( colMemMult * tot ) );

   minLMem( int( lMemMult * tot ) );


   /*  Initialize:
    *  - row ring lists
    *  - row vectors in file
    *  - column ring lists
    */
   u.row.start[0] = 0;

   rring = u.row.elem;

   lastrring = &( u.row.list );

   lastrring->idx = thedim;

   lastrring->next = rring;

   cring = u.col.elem;

   lastcring = &( u.col.list );

   lastcring->idx = thedim;

   lastcring->next = cring;

   m = 0;

   for ( int i = 0; i < thedim; i++ )
   {
      u.row.start[i] = m;
      m += u.row.max[i];

      rring->idx = i;
      rring->prev = lastrring;
      lastrring->next = rring;
      lastrring = rring;
      ++rring;

      cring->idx = i;
      cring->prev = lastcring;
      lastcring->next = cring;
      lastcring = cring;
      ++cring;
   }

   u.row.start[thedim]       = 0;

   u.row.max[thedim]       = 0;
   u.row.used = m;

   lastrring->next = &( u.row.list );
   lastrring->next->prev = lastrring;

   lastcring->next = &( u.col.list );
   lastcring->next->prev = lastcring;

   /*  Copy matrix to row and column file
    *  excluding and marking column singletons!
    */
   m = 0;
   temp.stage = 0;

   initMaxabs = 0;

   for ( int i = 0; i < thedim; i++ )
   {
      int nnonzeros;

      psv = vec[i];
      u.col.start[i] = m;

      /* check whether number of nonzeros above tolerance is 0, 1 or >= 2 */
      nnonzeros = 0;

      for ( int j = 0; j < psv->size() && nnonzeros <= 1; j++ )
      {
         if ( psv->value(j) != 0 )
            nnonzeros++;
      }

      /* basis is singular due to empty column */
      if ( nnonzeros == 0 )
      {
         stat = SLinSolverRational::SINGULAR;
         return;
      }

      /* exclude column singletons */
      else
         if ( nnonzeros == 1 )
         {
            int j;

            /* find nonzero */

            for ( j = 0; psv->value(j) == 0; j++ )
               ;

            assert( j < psv->size() );

            /* basis is singular due to two linearly dependent column singletons */
            if ( row.perm[psv->index( j )] >= 0 )
            {
               stat = SLinSolverRational::SINGULAR;
               return;
            }

            /* update maximum absolute nonzero value */
            x = psv->value( j );

            if ( spxAbs( x ) > initMaxabs )
               initMaxabs = spxAbs( x );

            /* permute to front and mark as singleton */
            setPivot( temp.stage, i, psv->index( j ), x );

            sing[temp.stage] = i;

            temp.stage++;

            /* set column length to zero */
            temp.s_cact[i] = u.col.len[i] = u.col.max[i] = 0;
         }

      /* add to active matrix if not a column singleton */
         else
         {
            int end;
            int k;

            /* go through all nonzeros in column */
            assert( nnonzeros >= 2 );
            nnonzeros = 0;

            for ( int j = 0; j < psv->size(); j++ )
            {
               x = psv->value( j );

               if ( x != 0 )
               {
                  /* add to column array */
                  k = psv->index( j );
                  u.col.idx[m] = k;
                  m++;

                  /* add to row array */
                  end = u.row.start[k] + u.row.len[k];
                  u.row.idx[end] = i;
                  u.row.val[end] = x;
                  u.row.len[k]++;

                  /* update maximum absolute nonzero value */

                  if ( spxAbs( x ) > initMaxabs )
                     initMaxabs = spxAbs( x );

                  nnonzeros++;
               }
            }

            assert( nnonzeros >= 2 );

            /* set column length */
            temp.s_cact[i] = u.col.len[i] = u.col.max[i] = nnonzeros;
         }
   }

   u.col.used = m;
}



/*****************************************************************************/
/*
 *      Remove column singletons
 */

void CLUFactorRational::colSingletons()
{
   int i, j, k, n;
   int len;
   int p_col, p_row, newrow;
   int *idx;
   int *rorig = row.orig;
   int *rperm = row.perm;
   int *sing = temp.s_mark;


   /*  Iteratively update column counts due to removed column singletons
    *  thereby removing new arising columns singletons
    *  and computing the index of the first row singleton (-1)
    *  until no more can be found.
    */

   for ( i = 0; i < temp.stage; ++i )
   {
      p_row = rorig[i];
      assert( p_row >= 0 );
      idx = &( u.row.idx[u.row.start[p_row]] );
      len = u.row.len[p_row];

      for ( j = 0; j < len; ++j )
      {
         /*  Move pivotal nonzeros to front of column.
          */
         p_col = idx[j];
         assert( temp.s_cact[p_col] > 0 );

         n = u.col.start[p_col] + u.col.len[p_col] - temp.s_cact[p_col];

         for ( k = n; u.col.idx[k] != p_row; ++k )
            ;

         assert( k < u.col.start[p_col] + u.col.len[p_col] );

         u.col.idx[k] = u.col.idx[n];

         u.col.idx[n] = p_row;

         n = --( temp.s_cact[p_col] );        /* column nonzeros of ACTIVE matrix */

         if ( n == 1 )                /* Here is another singleton */
         {
            newrow = u.col.idx[--u.col.len[p_col] + u.col.start[p_col]];

            /*      Ensure, matrix not singular
             */

            if ( rperm[newrow] >= 0 )
            {
               stat = SLinSolverRational::SINGULAR;
               return;
            }

            /*      Find singleton in row.
             */
            n = u.row.start[newrow] + ( --( u.row.len[newrow] ) );

            for ( k = n; u.row.idx[k] != p_col; --k )
               ;

            /*      Remove singleton from column.
             */
            setPivot( temp.stage, p_col, newrow, u.row.val[k] );

            sing[temp.stage++] = p_col;

            /*      Move pivot element to diag.
             */
            u.row.val[k] = u.row.val[n];

            u.row.idx[k] = u.row.idx[n];
         }
         else
            if ( n == 0 )
            {
               stat = SLinSolverRational::SINGULAR;
               return;
            }
      }
   }

   assert( temp.stage <= thedim );
}


/*****************************************************************************/
/*
 *      Remove row singletons
 */
void CLUFactorRational::rowSingletons()
{
   Rational pval;
   int i, j, k, ll, r;
   int p_row, p_col, len, rs, lk;
   int *idx;
   int *rperm = row.perm;
   int *sing = temp.s_mark;

   /*  Mark row singletons
    */
   rs = temp.stage;

   for ( i = 0; i < thedim; ++i )
   {
      if ( rperm[i] < 0 && u.row.len[i] == 1 )
         sing[temp.stage++] = i;
   }

   /*  Eliminate row singletons
    *  thereby marking newly arising ones
    *  until no more can be found.
    */
   for ( ; rs < temp.stage; ++rs )
   {
      /*      Move pivot element from row file to diag
       */
      p_row = sing[rs];
      j = u.row.start[p_row];
      p_col = u.row.idx[j];
      pval = u.row.val[j];
      setPivot( rs, p_col, p_row, pval );
      u.row.len[p_row] = 0;

      /*      Remove pivot column form workingmatrix
       *      thereby building up L vector.
       */
      idx = &( u.col.idx[u.col.start[p_col]] );
      i = temp.s_cact[p_col];                /* nr. nonzeros of new L vector */
      lk = makeLvec( i - 1, p_row );
      len = u.col.len[p_col];
      i = ( u.col.len[p_col] -= i );       /* remove pivot column from U */

      for ( ; i < len; ++i )
      {
         r = idx[i];

         if ( r != p_row )
         {
            /*      Find pivot column in row.
             */
            ll = --( u.row.len[r] );
            k = u.row.start[r] + ll;

            for ( j = k; u.row.idx[j] != p_col; --j )
               ;

            assert( k >= u.row.start[r] );

            /*      Initialize L vector
             */
            l.idx[lk] = r;

            l.val[lk] = u.row.val[j] / pval;

            ++lk;

            /*      Remove pivot column from row.
             */
            u.row.idx[j] = u.row.idx[k];

            u.row.val[j] = u.row.val[k];

            /*      Check new row length.
             */
            if ( ll == 1 )
               sing[temp.stage++] = r;
            else
               if ( ll == 0 )
               {
                  stat = SLinSolverRational::SINGULAR;
                  return;
               }
         }
      }
   }
}


/*****************************************************************************/
/*
 *      Init nonzero number Ring lists
 *      and required entries of arrays max and mark
 */

void CLUFactorRational::initFactorRings()
{
   int i;
   int *rperm = row.perm;
   int *cperm = col.perm;
   CLUFactorRational::Pring *ring;

   assert( thedim >= 0 );
   spx_alloc( temp.pivot_col,   thedim + 1 );
   spx_alloc( temp.pivot_colNZ, thedim + 1 );
   spx_alloc( temp.pivot_row,   thedim + 1 );
   spx_alloc( temp.pivot_rowNZ, thedim + 1 );

   for ( i = thedim - temp.stage; i >= 0; --i )
   {
      initDR( temp.pivot_colNZ[i] );
      initDR( temp.pivot_rowNZ[i] );
   }

   for ( i = 0; i < thedim; ++i )
   {
      if ( rperm[i] < 0 )
      {
         if ( u.row.len[i] <= 0 )
         {
            stat = SLinSolverRational::SINGULAR;
            return;
         }

         ring = &( temp.pivot_rowNZ[u.row.len[i]] );

         init2DR( temp.pivot_row[i], *ring );
         temp.pivot_row[i].idx = i;
         temp.s_max[i] = -1;
      }

      if ( cperm[i] < 0 )
      {
         if ( temp.s_cact[i] <= 0 )
         {
            stat = SLinSolverRational::SINGULAR;
            return;
         }

         ring = &( temp.pivot_colNZ[temp.s_cact[i]] );

         init2DR( temp.pivot_col[i], *ring );
         temp.pivot_col[i].idx = i;
         temp.s_mark[i] = 0;
      }
   }
}

void CLUFactorRational::freeFactorRings( void )
{

   if ( temp.pivot_col )
      spx_free( temp.pivot_col );

   if ( temp.pivot_colNZ )
      spx_free( temp.pivot_colNZ );

   if ( temp.pivot_row )
      spx_free( temp.pivot_row );

   if ( temp.pivot_rowNZ )
      spx_free( temp.pivot_rowNZ );
}


/*
 *      Eliminate all row singletons from nucleus.
 *      A row singleton may well be column singleton at the same time!
 */
void CLUFactorRational::eliminateRowSingletons()
{
   int i, j, k, ll, r;
   int len, lk;
   int pcol, prow;
   Rational pval;
   int *idx;
   CLUFactorRational::Pring *sing;

   for ( sing = temp.pivot_rowNZ[1].prev; sing != &( temp.pivot_rowNZ[1] ); sing = sing->prev )
   {
      prow = sing->idx;
      i = u.row.start[prow];
      pcol = u.row.idx[i];
      pval = u.row.val[i];
      setPivot( temp.stage++, pcol, prow, pval );
      u.row.len[prow] = 0;
      removeDR( temp.pivot_col[pcol] );

      /*      Eliminate pivot column and build L vector.
       */
      i = temp.s_cact[pcol];

      if ( i > 1 )
      {
         idx = &( u.col.idx[u.col.start[pcol]] );
         len = u.col.len[pcol];
         lk = makeLvec( i - 1, prow );
         i = u.col.len[pcol] -= i;

         for ( ; ( r = idx[i] ) != prow; ++i )
         {
            /*      Find pivot column in row.
             */
            ll = --( u.row.len[r] );
            k = u.row.start[r] + ll;

            for ( j = k; u.row.idx[j] != pcol; --j )
               ;

            assert( j >= u.row.start[r] );

            /*      Initialize L vector
             */
            l.idx[lk] = r;

            l.val[lk] = u.row.val[j] / pval;

            ++lk;

            /*      Remove pivot column from row.
             */
            u.row.idx[j] = u.row.idx[k];

            u.row.val[j] = u.row.val[k];

            /*      Move column to appropriate nonzero ring.
             */
            removeDR( temp.pivot_row[r] );

            init2DR( temp.pivot_row[r], temp.pivot_rowNZ[ll] );

            assert( row.perm[r] < 0 );

            temp.s_max[r] = -1;
         }

         /* skip pivot element */
         assert( i < len && "ERROR: pivot column does not contain pivot row" );

         for ( ++i; i < len; ++i )
         {
            /*      Find pivot column in row.
             */
            r = idx[i];
            ll = --( u.row.len[r] );
            k = u.row.start[r] + ll;

            for ( j = k; u.row.idx[j] != pcol; --j )
               ;

            assert( j >= u.row.start[r] );

            /*      Initialize L vector
             */
            l.idx[lk] = r;

            l.val[lk] = u.row.val[j] / pval;

            ++lk;

            /*      Remove pivot column from row.
             */
            u.row.idx[j] = u.row.idx[k];

            u.row.val[j] = u.row.val[k];

            /*      Move column to appropriate nonzero ring.
             */
            removeDR( temp.pivot_row[r] );

            init2DR( temp.pivot_row[r], temp.pivot_rowNZ[ll] );

            assert( row.perm[r] < 0 );

            temp.s_max[r] = -1;
         }
      }
      else
         u.col.len[pcol] -= i;
   }

   initDR( temp.pivot_rowNZ[1] );  /* Remove all row singletons from list */
}



/*
 *      Eliminate all column singletons from nucleus.
 *      A column singleton must not be row singleton at the same time!
 */
void CLUFactorRational::eliminateColSingletons()
{
   int i, j, k, m, c;
   int pcol, prow;
   CLUFactorRational::Pring *sing;

   for ( sing = temp.pivot_colNZ[1].prev;
         sing != &( temp.pivot_colNZ[1] );
         sing = sing->prev )
   {
      /*      Find pivot value
       */
      pcol = sing->idx;
      j = --( u.col.len[pcol] ) + u.col.start[pcol]; /* remove pivot column */
      prow = u.col.idx[j];
      removeDR( temp.pivot_row[prow] );

      j = --( u.row.len[prow] ) + u.row.start[prow];

      for ( i = j; ( c = u.row.idx[i] ) != pcol; --i )
      {
         m = u.col.len[c] + u.col.start[c] - ( temp.s_cact[c] )--;

         for ( k = m; u.col.idx[k] != prow; ++k )
            ;

         u.col.idx[k] = u.col.idx[m];

         u.col.idx[m] = prow;

         m = temp.s_cact[c];

         removeDR( temp.pivot_col[c] );

         init2DR( temp.pivot_col[c], temp.pivot_colNZ[m] );

         assert( col.perm[c] < 0 );
      }

      /*      remove pivot element from pivot row
       */
      setPivot( temp.stage++, pcol, prow, u.row.val[i] );

      u.row.idx[i] = u.row.idx[j];

      u.row.val[i] = u.row.val[j];

      j = u.row.start[prow];

      for ( --i; i >= j; --i )
      {
         c = u.row.idx[i];
         m = u.col.len[c] + u.col.start[c] - ( temp.s_cact[c] )--;

         for ( k = m; u.col.idx[k] != prow; ++k )
            ;

         u.col.idx[k] = u.col.idx[m];

         u.col.idx[m] = prow;

         m = temp.s_cact[c];

         removeDR( temp.pivot_col[c] );

         init2DR( temp.pivot_col[c], temp.pivot_colNZ[m] );

         assert( col.perm[c] < 0 );
      }
   }

   initDR( temp.pivot_colNZ[1] );  /* Remove all column singletons from list */
}

/*
 * No singletons available: Select pivot elements.
 */
void CLUFactorRational::selectPivots( const Rational& threshold )
{
   int ii;
   int i;
   int j;
   int k;
   int ll = -1; // This value should never be used.
   int kk;
   int m;
   int count;
   int num;
   int rw = -1; // This value should never be used.
   int cl = -1; // This value should never be used.
   int len;
   int beg;
   Rational l_maxabs;
   Rational x = 0; // This value should never be used.
   int mkwtz;
   int candidates;

   candidates = thedim - temp.stage - 1;

   if ( candidates > 4 )
      candidates = 4;

   num = 0;

   count = 2;

   for ( ;; )
   {
      ii = -1;

      if ( temp.pivot_rowNZ[count].next != &( temp.pivot_rowNZ[count] ) )
      {
         rw = temp.pivot_rowNZ[count].next->idx;
         beg = u.row.start[rw];
         len = u.row.len[rw] + beg - 1;

         /*  set l_maxabs to maximum absolute value in row
          *  (compute it if necessary).
          */

         if( temp.s_max[rw] < 0 )
         {
            l_maxabs = spxAbs( u.row.val[len] );

            for ( i = len - 1; i >= beg; --i )
               if ( l_maxabs < spxAbs( u.row.val[i] ) )
                  l_maxabs = spxAbs( u.row.val[i] );

            temp.s_max[rw] = l_maxabs;               /* ##### */
         }
         else
            l_maxabs = temp.s_max[rw];

         l_maxabs *= threshold;

         /*  select pivot element with lowest markowitz number in row
          */
         mkwtz = thedim + 1;

         for ( i = len; i >= beg; --i )
         {
            k = u.row.idx[i];
            j = temp.s_cact[k];
            x = u.row.val[i];

            if ( j < mkwtz && spxAbs( x ) > l_maxabs )
            {
               mkwtz = j;
               cl = k;
               ii = i;

               if ( j <= count )            /* ##### */
                  break;
            }
         }
      }
      else
         if ( temp.pivot_colNZ[count].next != &( temp.pivot_colNZ[count] ) )
         {
            cl = temp.pivot_colNZ[count].next->idx;
            beg = u.col.start[cl];
            len = u.col.len[cl] + beg - 1;
            beg = len - temp.s_cact[cl] + 1;
            assert( count == temp.s_cact[cl] );

            /*  select pivot element with lowest markowitz number in column
             */
            mkwtz = thedim + 1;

            for ( i = len; i >= beg; --i )
            {
               k = u.col.idx[i];
               j = u.row.len[k];

               if ( j < mkwtz )
               {
                  /*  ensure that element (cl,k) is stable.
                   */
                  if ( temp.s_max[k] > 0 )
                  {
                     /*  case 1: l_maxabs is known
                      */
                     for ( m = u.row.start[k], kk = m + u.row.len[k] - 1;
                           kk >= m; --kk )
                     {
                        if ( u.row.idx[kk] == cl )
                        {
                           x = u.row.val[kk];
                           ll = kk;
                           break;
                        }
                     }

                     l_maxabs = temp.s_max[k];
                  }
                  else
                  {
                     /*  case 2: l_maxabs needs to be computed
                      */
                     m = u.row.start[k];
                     l_maxabs = spxAbs( u.row.val[m] );

                     for ( kk = m + u.row.len[k] - 1; kk >= m; --kk )
                     {
                        if ( l_maxabs < spxAbs( u.row.val[kk] ) )
                           l_maxabs = spxAbs( u.row.val[kk] );

                        if ( u.row.idx[kk] == cl )
                        {
                           x = u.row.val[kk];
                           ll = kk;
                           break;
                        }
                     }

                     for ( --kk; kk > m; --kk )
                     {
                        if ( l_maxabs < spxAbs( u.row.val[kk] ) )
                           l_maxabs = spxAbs( u.row.val[kk] );
                     }

                     temp.s_max[k] = l_maxabs;
                  }

                  l_maxabs *= threshold;

                  if ( spxAbs( x ) > l_maxabs )
                  {
                     mkwtz = j;
                     rw = k;
                     ii = ll;

                     if ( j <= count + 1 )
                        break;
                  }
               }
            }
         }
         else
         {
            ++count;
            continue;
         }

      assert( cl >= 0 );

      removeDR( temp.pivot_col[cl] );
      initDR( temp.pivot_col[cl] );

      if ( ii >= 0 )
      {
         /*  Initialize selected pivot element
          */
         CLUFactorRational::Pring *pr;
         temp.pivot_row[rw].pos = ii - u.row.start[rw];
         temp.pivot_row[rw].mkwtz = mkwtz = ( mkwtz - 1 ) * ( count - 1 );
         // ??? mkwtz originally was long,
         // maybe to avoid an overflow in this instruction?

         for ( pr = temp.pivots.next; pr->idx >= 0; pr = pr->next )
         {
            if ( pr->idx == rw || pr->mkwtz >= mkwtz )
               break;
         }

         pr = pr->prev;

         if ( pr->idx != rw )
         {
            removeDR( temp.pivot_row[rw] );
            init2DR( temp.pivot_row[rw], *pr );
         }

         num++;

         if ( num >= candidates )
            break;
      }
   }

   /*
    *     while(temp.temp.next->mkwtz < temp.temp.prev->mkwtz)
    *     {
    *     Pring   *pr;
    *     pr = temp.temp.prev;
    *     removeDR(*pr);
    *     init2DR (*pr, rowNZ[u.row.len[pr->idx]]);
    }
    */

   assert( row.perm[rw] < 0 );

   assert( col.perm[cl] < 0 );
}


/*
 *      Perform L and update loop for row r
 */
int CLUFactorRational::updateRow( int r,
                          int lv,
                          int prow,
                          int pcol,
                          const Rational& pval )
{
   int fill;
   Rational x, lx;
   int c, i, j, k, ll, m, n;

   n = u.row.start[r];
   m = --( u.row.len[r] ) + n;

   /*  compute L vector entry and
    *  and remove pivot column form row file
    */

   for ( j = m; u.row.idx[j] != pcol; --j )
      ;

   lx = u.row.val[j] / pval;

   l.val[lv] = lx;

   l.idx[lv] = r;

   ++lv;

   u.row.idx[j] = u.row.idx[m];

   u.row.val[j] = u.row.val[m];


   /*  update loop (I) and
    *  computing expected fill
    */
   fill = u.row.len[prow];

   for ( j = m - 1; j >= n; --j )
   {
      c = u.row.idx[j];

      if ( temp.s_mark[c] )
      {
         /*  count fill elements.
          */
         temp.s_mark[c] = 0;
         --fill;

         /*  update row values
          */
         x = u.row.val[j] -= work[c] * lx;

         if ( x == 0 )
         {
            /* Eliminate zero from row r
             */
            --u.row.len[r];
            --m;
            u.row.val[j] = u.row.val[m];
            u.row.idx[j] = u.row.idx[m];

            /* Eliminate zero from column c
             */
            --( temp.s_cact[c] );
            k = --( u.col.len[c] ) + u.col.start[c];

            for ( i = k; u.col.idx[i] != r; --i )
               ;

            u.col.idx[i] = u.col.idx[k];
         }
      }
   }


   /*  create space for fill in row file
    */
   ll = u.row.len[r];

   if ( ll + fill > u.row.max[r] )
      remaxRow( r, ll + fill );

   ll += u.row.start[r];

   /*  fill creating update loop (II)
    */
   for ( j = u.row.start[prow], m = j + u.row.len[prow]; j < m; ++j )
   {
      c = u.row.idx[j];

      if ( temp.s_mark[c] )
      {
         x = - work[c] * lx;

         if ( x != 0 )
         {
            /* produce fill element in row r
             */
            u.row.val[ll] = x;
            u.row.idx[ll] = c;
            ll++;
            u.row.len[r]++;

            /* produce fill element in column c
             */

            if ( u.col.len[c] >= u.col.max[c] )
               remaxCol( c, u.col.len[c] + 1 );

            u.col.idx[u.col.start[c] + ( u.col.len[c] )++] = r;

            temp.s_cact[c]++;
         }
      }
      else
         temp.s_mark[c] = 1;
   }

   /*  move row to appropriate list.
    */
   removeDR( temp.pivot_row[r] );

   init2DR( temp.pivot_row[r], temp.pivot_rowNZ[u.row.len[r]] );

   assert( row.perm[r] < 0 );

   temp.s_max[r] = -1;

   return lv;
}

/*
 *      Eliminate pivot element
 */
void CLUFactorRational::eliminatePivot( int prow, int pos )
{
   int i, j, k, m = -1;
   int lv = -1;  // This value should never be used.
   int pcol;
   Rational pval;
   int pbeg = u.row.start[prow];
   int plen = --( u.row.len[prow] );
   int pend = pbeg + plen;


   /*  extract pivot element   */
   i = pbeg + pos;
   pcol = u.row.idx[i];
   pval = u.row.val[i];
   removeDR( temp.pivot_col[pcol] );
   initDR( temp.pivot_col[pcol] );

   /*  remove pivot from pivot row     */
   u.row.idx[i] = u.row.idx[pend];
   u.row.val[i] = u.row.val[pend];

   /*  set pivot element and construct L vector */
   setPivot( temp.stage++, pcol, prow, pval );

   /**@todo If this test failes, lv has no value. I suppose that in this
    *       case none of the loops below that uses lv is executed.
    *       But this is unproven.
    */

   if ( temp.s_cact[pcol] - 1 > 0 )
      lv = makeLvec( temp.s_cact[pcol] - 1, prow );

   /*  init working vector,
    *  remove pivot row from working matrix
    *  and remove columns from list.
    */
   for ( i = pbeg; i < pend; ++i )
   {
      j = u.row.idx[i];
      temp.s_mark[j] = 1;
      work[j] = u.row.val[i];
      removeDR( temp.pivot_col[j] );
      m = u.col.start[j] + u.col.len[j] - temp.s_cact[j];

      for ( k = m; u.col.idx[k] != prow; ++k )
         ;

      u.col.idx[k] = u.col.idx[m];

      u.col.idx[m] = prow;

      temp.s_cact[j]--;
   }

   /*  perform L and update loop
    */
   for ( i = u.col.len[pcol] - temp.s_cact[pcol];
         ( m = u.col.idx[u.col.start[pcol] + i] ) != prow;
         ++i )
   {
      assert( row.perm[m] < 0 );
      assert( lv >= 0 );
      updateRow( m, lv++, prow, pcol, pval );
   }

   /*  skip pivot row  */

   m = u.col.len[pcol];

   for ( ++i; i < m; ++i )
   {
      assert( lv >= 0 );
      updateRow( u.col.idx[u.col.start[pcol] + i], lv++, prow, pcol, pval );
   }

   /*  remove pivot column from column file.
    */
   u.col.len[pcol] -= temp.s_cact[pcol];

   /*  clear working vector and reinsert columns to lists
    */
   for ( i = u.row.start[prow], pend = i + plen; i < pend; ++i )
   {
      j = u.row.idx[i];
      work[j] = 0;
      temp.s_mark[j] = 0;
      init2DR( temp.pivot_col[j], temp.pivot_colNZ[temp.s_cact[j]] );
      assert( col.perm[j] < 0 );
   }
}


/*
 *      Factorize nucleus.
 */
void CLUFactorRational::eliminateNucleus( const Rational& threshold )
{
   int r, c;
   CLUFactorRational::Pring *pivot;

   if ( stat == SLinSolverRational::SINGULAR )
      return;

   temp.pivots.mkwtz = -1;

   temp.pivots.idx = -1;

   temp.pivots.pos = -1;

   while ( temp.stage < thedim - 1 )
   {
#ifndef NDEBUG
      int i;
      // CLUFactorRationalIsConsistent(fac);

      for ( i = 0; i < thedim; ++i )
         if ( col.perm[i] < 0 )
            assert( temp.s_mark[i] == 0 );

#endif

      if ( temp.pivot_rowNZ[1].next != &( temp.pivot_rowNZ[1] ) )
      {
         if( timeLimitReached() )
            return;

         /* row singleton available */
         eliminateRowSingletons();
      }
      else
         if ( temp.pivot_colNZ[1].next != &( temp.pivot_colNZ[1] ) )
         {
            if( timeLimitReached() )
               return;

            /* column singleton available */
            eliminateColSingletons();
         }
         else
         {
            initDR( temp.pivots );
            selectPivots( threshold );

            assert( temp.pivots.next != &temp.pivots &&
                    "ERROR: no pivot element selected" );

            for ( pivot = temp.pivots.next; pivot != &temp.pivots;
                  pivot = pivot->next )
            {
               if( timeLimitReached() )
                  return;

               eliminatePivot( pivot->idx, pivot->pos );
            }
         }

      if ( temp.pivot_rowNZ->next != temp.pivot_rowNZ ||
            temp.pivot_colNZ->next != temp.pivot_colNZ )
      {
         stat = SLinSolverRational::SINGULAR;
         return;
      }
   }

   if ( temp.stage < thedim )
   {
      /*      Eliminate remaining element.
       *      Note, that this must be both, column and row singleton.
       */
      assert( temp.pivot_rowNZ[1].next != &( temp.pivot_rowNZ[1] ) &&
              "ERROR: one row must be left" );
      assert( temp.pivot_colNZ[1].next != &( temp.pivot_colNZ[1] ) &&
              "ERROR: one col must be left" );
      r = temp.pivot_rowNZ[1].next->idx;
      c = temp.pivot_colNZ[1].next->idx;
      u.row.len[r] = 0;
      u.col.len[c]--;
      setPivot( temp.stage, c, r, u.row.val[u.row.start[r]] );
   }
}

/*****************************************************************************/

int CLUFactorRational::setupColVals()
{
   int i;
   int n = thedim;

   u.col.val.reDim(u.col.size);

   for ( i = 0; i < thedim; i++ )
      u.col.len[i] = 0;

   maxabs = 0;

   for ( i = 0; i < thedim; i++ )
   {
      int     k   = u.row.start[i];
      int*    idx = &u.row.idx[k];
      Rational*   val = &u.row.val[k];
      int     len = u.row.len[i];

      n += len;

      while ( len-- > 0 )
      {
         assert(( *idx >= 0 ) && ( *idx < thedim ) );

         k = u.col.start[*idx] + u.col.len[*idx];

         assert(( k >= 0 ) && ( k < u.col.size ) );

         u.col.len[*idx]++;

         assert( u.col.len[*idx] <= u.col.max[*idx] );

         u.col.idx[k] = i;
         u.col.val[k] = *val;

         if ( spxAbs( *val ) > maxabs )
            maxabs = spxAbs( *val );

         idx++;

         val++;
      }
   }

   return n;
}

/*****************************************************************************/

#ifdef WITH_L_ROWS
void CLUFactorRational::setupRowVals()
{
   int   i, j, k, m;
   int   vecs, mem;
   int*  l_row;
   int*  idx;
   int*  beg;
   int*  l_ridx;
   int*  l_rbeg;
   int*  rorig;
   int*  rrorig;
   int*  rperm;
   int*  rrperm;

   vecs  = l.firstUpdate;
   l_row = l.row;
   idx   = l.idx;
   DVectorRational& val = l.val;
   int validx = 0;
   beg   = l.start;
   mem   = beg[vecs];

   if ( l.ridx )
      spx_free( l.ridx );

   if ( l.rbeg )
      spx_free( l.rbeg );

   if ( l.rorig )
      spx_free( l.rorig );

   if ( l.rperm )
      spx_free( l.rperm );

   l.rval.reDim(mem);

   spx_alloc( l.ridx, mem );

   spx_alloc( l.rbeg, thedim + 1 );

   spx_alloc( l.rorig, thedim );

   spx_alloc( l.rperm, thedim );

   l_ridx = l.ridx;

   DVectorRational& l_rval = l.rval;

   l_rbeg = l.rbeg;

   rorig  = l.rorig;

   rrorig = row.orig;

   rperm  = l.rperm;

   rrperm = row.perm;

   for ( i = thedim; i--; *l_rbeg++ = 0 )
   {
      *rorig++ = *rrorig++;
      *rperm++ = *rrperm++;
   }

   *l_rbeg = 0;

   l_rbeg = l.rbeg + 1;

   for ( i = mem; i--; )
      l_rbeg[*idx++]++;

   idx = l.idx;

   for ( m = 0, i = thedim; i--; l_rbeg++ )
   {
      j = *l_rbeg;
      *l_rbeg = m;
      m += j;
   }

   assert( m == mem );

   l_rbeg = l.rbeg + 1;

   for ( i = j = 0; i < vecs; ++i )
   {
      m = l_row[i];
      assert( idx == &l.idx[l.start[i]] );

      for ( ; j < beg[i + 1]; j++ )
      {
         k = l_rbeg[*idx++]++;
         assert( k < mem );
         l_ridx[k] = m;
         l_rval[k] = val[validx];
         validx++; // was l_rval[k] = *val++; with Rational* val
      }
   }

   assert( l.rbeg[thedim] == mem );

   assert( l.rbeg[0] == 0 );
}

#endif

/*****************************************************************************/

void CLUFactorRational::factor(
   const SVectorRational** vec,         ///< Array of column vector pointers
   const Rational& threshold            ///< pivoting threshold
   )
{
   MSG_DEBUG( std::cout << "CLUFactorRational::factor()\n" );

   factorTime->start();

   stat = SLinSolverRational::OK;

   l.start[0]    = 0;
   l.firstUpdate = 0;
   l.firstUnused = 0;

   temp.init( thedim );
   initPerm();

   initFactorMatrix( vec );

   if ( stat )
      goto TERMINATE;

   if( timeLimitReached() )
      goto TERMINATE;

   //   initMaxabs = initMaxabs;

   colSingletons();

   if ( stat != SLinSolverRational::OK )
      goto TERMINATE;

   if( timeLimitReached() )
      goto TERMINATE;

   rowSingletons();

   if ( stat != SLinSolverRational::OK )
      goto TERMINATE;

   if ( temp.stage < thedim )
   {
      if( timeLimitReached() )
         goto TERMINATE;

      initFactorRings();
      eliminateNucleus( threshold );
      freeFactorRings();
   }

TERMINATE:

   l.firstUpdate = l.firstUnused;

   if ( stat == SLinSolverRational::OK )
   {
#ifdef WITH_L_ROWS
      setupRowVals();
#endif
      nzCnt = setupColVals();
   }

   factorTime->stop();

   factorCount++;
}

void CLUFactorRational::dump() const
{
   int i, j, k;

   // Dump regardless of the verbosity level if this method is called;

   /*  Dump U:
    */

   for ( i = 0; i < thedim; ++i )
   {
      if ( row.perm[i] >= 0 )
         std::cout << "DCLUFA01 diag[" << i << "]: [" << col.orig[row.perm[i]]
         << "] = " << diag[i] << std::endl;

      for ( j = 0; j < u.row.len[i]; ++j )
         std::cout << "DCLUFA02   u[" << i << "]: ["
         << u.row.idx[u.row.start[i] + j] << "] = "
         << u.row.val[u.row.start[i] + j] << std::endl;
   }

   /*  Dump L:
    */
   for ( i = 0; i < thedim; ++i )
   {
      for ( j = 0; j < l.firstUnused; ++j )
         if ( col.orig[row.perm[l.row[j]]] == i )
         {
            std::cout << "DCLUFA03 l[" << i << "]" << std::endl;

            for ( k = l.start[j]; k < l.start[j + 1]; ++k )
               std::cout << "DCLUFA04   l[" << k - l.start[j]
               << "]:  [" << l.idx[k]
               << "] = "  << l.val[k] << std::endl;

            break;
         }
   }

   return;
}

/*****************************************************************************/
/*
 *      Ensure that row memory is at least size.
 */
void CLUFactorRational::minRowMem( int size )
{
   if ( u.row.val.dim() < size )
   {
      u.row.val.reDim(size);
      spx_realloc( u.row.idx, size );
   }
   assert(u.row.val.dim() >= size);
}

/*****************************************************************************/
/*
 *      Ensure that column memory is at least size.
 */
void CLUFactorRational::minColMem( int size )
{

   if ( u.col.size < size )
   {
      u.col.size = size;
      spx_realloc( u.col.idx, size );
   }
}

void CLUFactorRational::forestMinColMem( int size )
{

   if ( u.col.size < size )
   {
      u.col.size = size;
      spx_realloc( u.col.idx, size );
      u.col.val.reDim(size);
   }
}

void CLUFactorRational::minLMem( int size )
{

   if ( size > l.val.dim() )
   {
      int newsize = int( 0.2 * l.val.dim() + size );
      l.val.reDim(newsize);
      spx_realloc( l.idx, l.val.dim() );
   }
}


int CLUFactorRational::makeLvec( int p_len, int p_row )
{

   if ( l.firstUnused >= l.startSize )
   {
      l.startSize += 100;
      spx_realloc( l.start, l.startSize );
   }

   int* p_lrow = l.row;

   int* p_lbeg = l.start;
   int first   = p_lbeg[l.firstUnused];

   assert( p_len > 0 && "ERROR: no empty columns allowed in L vectors" );

   minLMem( first + p_len );
   p_lrow[l.firstUnused] = p_row;
   l.start[++( l.firstUnused )] = first + p_len;

   assert( l.start[l.firstUnused] <= l.val.dim() );
   assert( l.firstUnused <= l.startSize );
   return first;
}


/*****************************************************************************/

bool CLUFactorRational::isConsistent() const
{
#ifdef ENABLE_CONSISTENCY_CHECKS
   int              i, j, k, ll;
   Dring            *ring;
   CLUFactorRational::Pring *pring;

   /*  Consistency only relevant for real factorizations
    */

   if ( stat )
      return true;

   /*  Test column ring list consistency.
    */
   i = 0;

   for ( ring = u.col.list.next; ring != &( u.col.list ); ring = ring->next )
   {
      assert( ring->idx >= 0 );
      assert( ring->idx < thedim );
      assert( ring->prev->next == ring );

      if ( ring != u.col.list.next )
      {
         assert( u.col.start[ring->prev->idx] + u.col.max[ring->prev->idx]
                 == u.col.start[ring->idx] );
      }

      ++i;
   }

   assert( i == thedim );

   assert( u.col.start[ring->prev->idx] + u.col.max[ring->prev->idx]
           == u.col.used );


   /*  Test row ring list consistency.
    */
   i = 0;

   for ( ring = u.row.list.next; ring != &( u.row.list ); ring = ring->next )
   {
      assert( ring->idx >= 0 );
      assert( ring->idx < thedim );
      assert( ring->prev->next == ring );
      assert( u.row.start[ring->prev->idx] + u.row.max[ring->prev->idx]
              == u.row.start[ring->idx] );
      ++i;
   }

   assert( i == thedim );

   assert( u.row.start[ring->prev->idx] + u.row.max[ring->prev->idx]
           == u.row.used );


   /*  Test consistency of individual svectors.
    */

   for ( i = 0; i < thedim; ++i )
   {
      assert( u.row.max[i] >= u.row.len[i] );
      assert( u.col.max[i] >= u.col.len[i] );
   }


   /*  Test consistency of column file to row file of U
    */
   for ( i = 0; i < thedim; ++i )
   {
      for ( j = u.row.start[i] + u.row.len[i] - 1; j >= u.row.start[i]; j-- )
      {
         k = u.row.idx[j];

         for ( ll = u.col.start[k] + u.col.len[k] - 1; ll >= u.col.start[k]; ll-- )
         {
            if ( u.col.idx[ll] == i )
               break;
         }

         assert( u.col.idx[ll] == i );

         if ( row.perm[i] < 0 )
         {
            assert( col.perm[k] < 0 );
         }
         else
         {
            assert( col.perm[k] < 0 || col.perm[k] > row.perm[i] );
         }
      }
   }

   /*  Test consistency of row file to column file of U
    */
   for ( i = 0; i < thedim; ++i )
   {
      for ( j = u.col.start[i] + u.col.len[i] - 1; j >= u.col.start[i]; j-- )
      {
         k = u.col.idx[j];

         for ( ll = u.row.start[k] + u.row.len[k] - 1; ll >= u.row.start[k]; ll-- )
         {
            if ( u.row.idx[ll] == i )
               break;
         }

         assert( u.row.idx[ll] == i );

         assert( col.perm[i] < 0 || row.perm[k] < col.perm[i] );
      }
   }

   /*  Test consistency of nonzero count lists
    */
   if ( temp.pivot_colNZ && temp.pivot_rowNZ )
   {
      for ( i = 0; i < thedim - temp.stage; ++i )
      {
         for ( pring = temp.pivot_rowNZ[i].next; pring != &( temp.pivot_rowNZ[i] ); pring = pring->next )
         {
            assert( row.perm[pring->idx] < 0 );
         }

         for ( pring = temp.pivot_colNZ[i].next; pring != &( temp.pivot_colNZ[i] ); pring = pring->next )
         {
            assert( col.perm[pring->idx] < 0 );
         }
      }
   }

#endif // CONSISTENCY_CHECKS

   return true;
}

void CLUFactorRational::solveUright( Rational* wrk, Rational* vec )
{

   for ( int i = thedim - 1; i >= 0; i-- )
   {
      int  r = row.orig[i];
      int  c = col.orig[i];
      Rational x = wrk[c] = diag[r] * vec[r];

      vec[r] = 0;

      if ( x != 0 )
         //if (isNotZero(x))
      {
         if( timeLimitReached() )
            return;

         for ( int j = u.col.start[c]; j < u.col.start[c] + u.col.len[c]; j++ )
            vec[u.col.idx[j]] -= x * u.col.val[j];
      }
   }
}

int CLUFactorRational::solveUrightEps( Rational* vec, int* nonz, Rational* rhs )
{
   int i, j, r, c, n;
   int *rorig, *corig;
   int *cidx, *clen, *cbeg;
   Rational x;

   int *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;

   cidx = u.col.idx;
   DVectorRational& cval = u.col.val;
   clen = u.col.len;
   cbeg = u.col.start;

   n = 0;

   for ( i = thedim - 1; i >= 0; --i )
   {
      r = rorig[i];
      x = diag[r] * rhs[r];

      if ( x != 0 )
      {
         c = corig[i];
         vec[c] = x;
         nonz[n++] = c;
         val = &cval[cbeg[c]];
         idx = &cidx[cbeg[c]];
         j = clen[c];

         while ( j-- > 0 )
            rhs[*idx++] -= x * ( *val++ );
      }
   }

   return n;
}

void CLUFactorRational::solveUright2( Rational* p_work1, Rational* vec1, Rational* p_work2, Rational* vec2 )
{
   int i, j, r, c;
   int *rorig, *corig;
   int *cidx, *clen, *cbeg;
   Rational x1, x2;

   int* idx;
   Rational* val;

   rorig = row.orig;
   corig = col.orig;

   cidx = u.col.idx;
   DVectorRational& cval = u.col.val;
   clen = u.col.len;
   cbeg = u.col.start;

   for ( i = thedim - 1; i >= 0; --i )
   {
      r = rorig[i];
      c = corig[i];
      p_work1[c] = x1 = diag[r] * vec1[r];
      p_work2[c] = x2 = diag[r] * vec2[r];
      vec1[r] = vec2[r] = 0;

      if ( x1 != 0 && x2 != 0 )
      {
         val = &cval[cbeg[c]];
         idx = &cidx[cbeg[c]];
         j = clen[c];

         while ( j-- > 0 )
         {
            vec1[*idx] -= x1 * ( *val );
            vec2[*idx++] -= x2 * ( *val++ );
         }
      }
      else
         if ( x1 != 0 )
         {
            val = &cval[cbeg[c]];
            idx = &cidx[cbeg[c]];
            j = clen[c];

            while ( j-- > 0 )
               vec1[*idx++] -= x1 * ( *val++ );
         }
         else
            if ( x2 != 0 )
            {
               val = &cval[cbeg[c]];
               idx = &cidx[cbeg[c]];
               j = clen[c];

               while ( j-- > 0 )
                  vec2[*idx++] -= x2 * ( *val++ );
            }
   }
}

int CLUFactorRational::solveUright2eps( Rational* p_work1, Rational* vec1, Rational* p_work2, Rational* vec2, int* nonz )
{
   int i, j, r, c, n;
   int *rorig, *corig;
   int *cidx, *clen, *cbeg;
   bool notzero1, notzero2;
   Rational x1, x2;

   int* idx;
   Rational* val;

   rorig = row.orig;
   corig = col.orig;

   cidx = u.col.idx;
   DVectorRational& cval = u.col.val;
   clen = u.col.len;
   cbeg = u.col.start;

   n = 0;

   for ( i = thedim - 1; i >= 0; --i )
   {
      c = corig[i];
      r = rorig[i];
      p_work1[c] = x1 = diag[r] * vec1[r];
      p_work2[c] = x2 = diag[r] * vec2[r];
      vec1[r] = vec2[r] = 0;
      notzero1 = x1 != 0 ? 1 : 0;
      notzero2 = x2 != 0 ? 1 : 0;

      if ( notzero1 && notzero2 )
      {
         *nonz++ = c;
         n++;
         val = &cval[cbeg[c]];
         idx = &cidx[cbeg[c]];
         j = clen[c];

         while ( j-- > 0 )
         {
            vec1[*idx] -= x1 * ( *val );
            vec2[*idx++] -= x2 * ( *val++ );
         }
      }
      else
         if ( notzero1 )
         {
            p_work2[c] = 0;
            *nonz++ = c;
            n++;
            val = &cval[cbeg[c]];
            idx = &cidx[cbeg[c]];
            j = clen[c];

            while ( j-- > 0 )
               vec1[*idx++] -= x1 * ( *val++ );
         }
         else
            if ( notzero2 )
            {
               p_work1[c] = 0;
               val = &cval[cbeg[c]];
               idx = &cidx[cbeg[c]];
               j = clen[c];

               while ( j-- > 0 )
                  vec2[*idx++] -= x2 * ( *val++ );
            }
            else
            {
               p_work1[c] = 0;
               p_work2[c] = 0;
            }
   }

   return n;
}

void CLUFactorRational::solveLright( Rational* vec )
{
   int i, j, k;
   int end;
   Rational x;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = 0; i < end; ++i )
   {
      if (( x = vec[lrow[i]] ) != 0 )
      {
         if( timeLimitReached() )
            return;

         MSG_DEBUG( std::cout << "y" << lrow[i] << "=" << vec[lrow[i]] << std::endl; )

         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            MSG_DEBUG( std::cout << "                         -> y" << *idx << " -= " << x << " * " << *val << " = " << x * ( *val ) << "    -> " << vec[*idx] - x * ( *val ) << std::endl; )
            vec[*idx++] -= x * ( *val++ );
         }
      }
   }

   if ( l.updateType )                   /* Forest-Tomlin Updates */
   {
      MSG_DEBUG( std::cout << "performing FT updates..." << std::endl; )

      end = l.firstUnused;

      for ( ; i < end; ++i )
      {
         x = 0;
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
            x += vec[*idx++] * ( *val++ );

         vec[lrow[i]] -= x;

         MSG_DEBUG( std::cout << "y" << lrow[i] << "=" << vec[lrow[i]] << std::endl; )
      }

      MSG_DEBUG( std::cout << "finished FT updates." << std::endl; )
   }
}

void CLUFactorRational::solveLright2( Rational* vec1, Rational* vec2 )
{
   int i, j, k;
   int end;
   Rational x2;
   Rational x1;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = 0; i < end; ++i )
   {
      x1 = vec1[lrow[i]];
      x2 = vec2[lrow[i]];

      if ( x1 != 0 && x2 != 0 )
      {
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            vec1[*idx] -= x1 * ( *val );
            vec2[*idx++] -= x2 * ( *val++ );
         }
      }
      else
         if ( x1 != 0 )
         {
            k = lbeg[i];
            idx = &( lidx[k] );
            val = &( lval[k] );

            for ( j = lbeg[i + 1]; j > k; --j )
               vec1[*idx++] -= x1 * ( *val++ );
         }
         else
            if ( x2 != 0 )
            {
               k = lbeg[i];
               idx = &( lidx[k] );
               val = &( lval[k] );

               for ( j = lbeg[i + 1]; j > k; --j )
                  vec2[*idx++] -= x2 * ( *val++ );
            }
   }

   if ( l.updateType )                   /* Forest-Tomlin Updates */
   {
      end = l.firstUnused;

      for ( ; i < end; ++i )
      {
         x1 = 0;
         x2 = 0;
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            x1 += vec1[*idx] * ( *val );
            x2 += vec2[*idx++] * ( *val++ );
         }

         vec1[lrow[i]] -= x1;

         vec2[lrow[i]] -= x2;
      }
   }
}

void CLUFactorRational::solveUpdateRight( Rational* vec )
{
   int i, j, k;
   int end;
   Rational x;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   assert( !l.updateType );             /* no Forest-Tomlin Updates */

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUnused;

   for ( i = l.firstUpdate; i < end; ++i )
   {
      if (( x = vec[lrow[i]] ) != 0 )
      {
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
            vec[*idx++] -= x * ( *val++ );
      }
   }
}

void CLUFactorRational::solveUpdateRight2( Rational* vec1, Rational* vec2 )
{
   int i, j, k;
   int end;
   Rational x1, x2;
   int *lrow, *lidx;
   int *lbeg;

   int* idx;
   Rational* val;

   assert( !l.updateType );             /* no Forest-Tomlin Updates */

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUnused;

   for ( i = l.firstUpdate; i < end; ++i )
   {
      x1 = vec1[lrow[i]];
      x2 = vec2[lrow[i]];

      if ( x1 != 0 && x2 != 0 )
      {
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            vec1[*idx] -= x1 * ( *val );
            vec2[*idx++] -= x2 * ( *val++ );
         }
      }
      else
         if ( x1 != 0 )
         {
            k = lbeg[i];
            idx = &( lidx[k] );
            val = &( lval[k] );

            for ( j = lbeg[i + 1]; j > k; --j )
               vec1[*idx++] -= x1 * ( *val++ );
         }
         else
            if ( x2 != 0 )
            {
               k = lbeg[i];
               idx = &( lidx[k] );
               val = &( lval[k] );

               for ( j = lbeg[i + 1]; j > k; --j )
                  vec2[*idx++] -= x2 * ( *val++ );
            }
   }
}

int CLUFactorRational::solveRight4update( Rational* vec, int* nonz,
                                           Rational* rhs, Rational* forest, int* forestNum, int* forestIdx )
{
   solveLright( rhs );

   if ( forest )
   {
      int n = 0;

      for ( int i = 0; i < thedim; i++ )
      {
         forestIdx[n] = i;
         forest[i]    = rhs[i];
         n           += rhs[i] != 0 ? 1 : 0;
      }

      *forestNum = n;
   }

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      solveUright( vec, rhs );
      solveUpdateRight( vec );
      return 0;
   }
   else
      return solveUrightEps( vec, nonz, rhs );
}

void CLUFactorRational::solveRight( Rational* vec, Rational* rhs )
{
   solveLright( rhs );
   solveUright( vec, rhs );

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
      solveUpdateRight( vec );
}

int CLUFactorRational::solveRight2update( Rational* vec1,
                                  Rational* vec2,
                                  Rational* rhs1,
                                  Rational* rhs2,
                                  int* nonz,
                                  Rational* forest,
                                  int* forestNum,
                                  int* forestIdx )
{
   solveLright2( rhs1, rhs2 );

   if ( forest )
   {
      int n = 0;

      for ( int i = 0; i < thedim; i++ )
      {
         forestIdx[n] = i;
         forest[i]    = rhs1[i];
         n           += rhs1[i] != 0 ? 1 : 0;
      }

      *forestNum = n;
   }

   if ( !l.updateType )         /* no Forest-Tomlin Updates */
   {
      solveUright2( vec1, rhs1, vec2, rhs2 );
      solveUpdateRight2( vec1, vec2 );
      return 0;
   }
   else
      return solveUright2eps( vec1, rhs1, vec2, rhs2, nonz );
}

void CLUFactorRational::solveRight2(
   Rational* vec1,
   Rational* vec2,
   Rational* rhs1,
   Rational* rhs2 )
{
   solveLright2( rhs1, rhs2 );

   if ( l.updateType )           /* Forest-Tomlin Updates */
      solveUright2( vec1, rhs1, vec2, rhs2 );
   else
   {
      solveUright2( vec1, rhs1, vec2, rhs2 );
      solveUpdateRight2( vec1, vec2 );
   }
}

/*****************************************************************************/
void CLUFactorRational::solveUleft( Rational* p_work, Rational* vec )
{
   for ( int i = 0; i < thedim; ++i )
   {
      int  c  = col.orig[i];
      int  r  = row.orig[i];

      assert( c >= 0 );  // Inna/Tobi: otherwise, vec[c] would be strange...
      assert( r >= 0 );  // Inna/Tobi: otherwise, diag[r] would be strange...

      Rational x  = vec[c];

      //      ASSERT_WARN( "WSOLVE01", spxAbs( x ) < 1e40 );
      //      ASSERT_WARN( "WSOLVE02", spxAbs( vec[c] ) < 1e40 );

#if defined(WITH_WARNINGS) || !defined(NDEBUG)
      Rational y = vec[c];
#endif

      vec[c]  = 0;

      if ( x != 0 )
      {
         if( timeLimitReached() )
            return;

         //         ASSERT_WARN( "WSOLVE03", spxAbs( diag[r] ) < 1e40 );

         x        *= diag[r];
         p_work[r] = x;

         //         ASSERT_WARN( "WSOLVE04", spxAbs( x ) < 1e40 );

         int end = u.row.start[r] + u.row.len[r];

         for ( int m = u.row.start[r]; m < end; m++ )
         {
            //            ASSERT_WARN( "WSOLVE05", spxAbs( u.row.val[m] ) < 1e40 );
            //            ASSERT_WARN( "WSOLVE06", spxAbs( vec[u.row.idx[m]] ) < infinity );
            vec[u.row.idx[m]] -= x * u.row.val[m];
            //            ASSERT_WARN( "WSOLVE07", spxAbs( vec[u.row.idx[m]] ) < 1e40 );
         }
      }

      //      ASSERT_WARN( "WSOLVE08", spxAbs( y ) < 1e40 );
   }
}



void CLUFactorRational::solveUleft2( Rational* p_work1, Rational* vec1, Rational* p_work2, Rational* vec2 )
{
   Rational x1;
   Rational x2;
   int i, k, r, c;
   int *rorig, *corig;
   int *ridx, *rlen, *rbeg, *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;

   ridx = u.row.idx;
   DVectorRational& rval = u.row.val;
   rlen = u.row.len;
   rbeg = u.row.start;

   for ( i = 0; i < thedim; ++i )
   {
      c = corig[i];
      r = rorig[i];
      x1 = vec1[c];
      x2 = vec2[c];

      if (( x1 != 0 ) && ( x2 != 0 ) )
      {
         x1 *= diag[r];
         x2 *= diag[r];
         p_work1[r] = x1;
         p_work2[r] = x2;
         k = rbeg[r];
         idx = &ridx[k];
         val = &rval[k];

         for ( int m = rlen[r]; m != 0; --m )
         {
            vec1[*idx] -= x1 * ( *val );
            vec2[*idx] -= x2 * ( *val++ );
            idx++;
         }
      }
      else
         if ( x1 != 0 )
         {
            x1 *= diag[r];
            p_work1[r] = x1;
            k = rbeg[r];
            idx = &ridx[k];
            val = &rval[k];

            for ( int m = rlen[r]; m != 0; --m )
               vec1[*idx++] -= x1 * ( *val++ );
         }
         else
            if ( x2 != 0 )
            {
               x2 *= diag[r];
               p_work2[r] = x2;
               k = rbeg[r];
               idx = &ridx[k];
               val = &rval[k];

               for ( int m = rlen[r]; m != 0; --m )
                  vec2[*idx++] -= x2 * ( *val++ );
            }
   }
}

int CLUFactorRational::solveLleft2forest( Rational* vec1, int* /* nonz */, Rational* vec2 )
{
   int i;
   int j;
   int k;
   int end;
   Rational x1, x2;
   Rational *val;
   int *lidx, *idx, *lrow;
   int *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      j = lrow[i];
      x1 = vec1[j];
      x2 = vec2[j];

      if ( x1 != 0 )
      {
         if ( x2 != 0 )
         {
            k = lbeg[i];
            val = &lval[k];
            idx = &lidx[k];

            for ( j = lbeg[i + 1]; j > k; --j )
            {
               vec1[*idx] -= x1 * ( *val );
               vec2[*idx++] -= x2 * ( *val++ );
            }
         }
         else
         {
            k = lbeg[i];
            val = &lval[k];
            idx = &lidx[k];

            for ( j = lbeg[i + 1]; j > k; --j )
               vec1[*idx++] -= x1 * ( *val++ );
         }
      }
      else
         if ( x2 != 0 )
         {
            k = lbeg[i];
            val = &lval[k];
            idx = &lidx[k];

            for ( j = lbeg[i + 1]; j > k; --j )
               vec2[*idx++] -= x2 * ( *val++ );
         }
   }

   return 0;
}

void CLUFactorRational::solveLleft2( Rational* vec1, int* /* nonz */, Rational* vec2 )
{
   int i, j, k, r;
   int x1not0, x2not0;
   Rational x1, x2;

   Rational *val;
   int *ridx, *idx;
   int *rbeg;
   int *rorig;

   ridx  = l.ridx;
   DVectorRational& rval = l.rval;
   rbeg  = l.rbeg;
   rorig = l.rorig;

#ifndef WITH_L_ROWS
   DVectorRational& lval  = l.val;
   int*    lidx  = l.idx;
   int*    lrow  = l.row;
   int*    lbeg  = l.start;

   i = l.firstUpdate - 1;

   for ( ; i >= 0; --i )
   {
      k = lbeg[i];
      val = &lval[k];
      idx = &lidx[k];
      x1 = 0;
      x2 = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
      {
         x1 += vec1[*idx] * ( *val );
         x2 += vec2[*idx++] * ( *val++ );
      }

      vec1[lrow[i]] -= x1;

      vec2[lrow[i]] -= x2;
   }

#else
   for ( i = thedim; i--; )
   {
      r = rorig[i];
      x1 = vec1[r];
      x2 = vec2[r];
      x1not0 = ( x1 != 0 );
      x2not0 = ( x2 != 0 );

      if ( x1not0 && x2not0 )
      {
         k = rbeg[r];
         j = rbeg[r + 1] - k;
         val = &rval[k];
         idx = &ridx[k];

         while ( j-- > 0 )
         {
            assert( row.perm[*idx] < i );
            vec1[*idx] -= x1 * *val;
            vec2[*idx++] -= x2 * *val++;
         }
      }
      else
         if ( x1not0 )
         {
            k = rbeg[r];
            j = rbeg[r + 1] - k;
            val = &rval[k];
            idx = &ridx[k];

            while ( j-- > 0 )
            {
               assert( row.perm[*idx] < i );
               vec1[*idx++] -= x1 * *val++;
            }
         }
         else
            if ( x2not0 )
            {
               k = rbeg[r];
               j = rbeg[r + 1] - k;
               val = &rval[k];
               idx = &ridx[k];

               while ( j-- > 0 )
               {
                  assert( row.perm[*idx] < i );
                  vec2[*idx++] -= x2 * *val++;
               }
            }
   }

#endif
}

int CLUFactorRational::solveLleftForest( Rational* vec, int* /* nonz */ )
{
   int i, j, k, end;
   Rational x;
   Rational *val;
   int *idx, *lidx, *lrow, *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      if (( x = vec[lrow[i]] ) != 0 )
      {
         if( timeLimitReached() )
            return 0;

         k = lbeg[i];
         val = &lval[k];
         idx = &lidx[k];

         for ( j = lbeg[i + 1]; j > k; --j )
            vec[*idx++] -= x * ( *val++ );
      }
   }

   return 0;
}

void CLUFactorRational::solveLleft( Rational* vec )
{

#ifndef WITH_L_ROWS
   int*  idx;
   Rational* val;
   DVectorRational& lval  = l.val;
   int*  lidx  = l.idx;
   int*  lrow  = l.row;
   int*  lbeg  = l.start;

   for ( int i = l.firstUpdate - 1; i >= 0; --i )
   {
      if( timeLimitReached() )
         return;

      int k = lbeg[i];
      val = &lval[k];
      idx = &lidx[k];
      x = 0;

      for ( int j = lbeg[i + 1]; j > k; --j )
         x += vec[*idx++] * ( *val++ );

      vec[lrow[i]] -= x;
   }

#else
   for ( int i = thedim - 1; i >= 0; --i )
   {
      int  r = l.rorig[i];
      Rational x = vec[r];

      if ( x != 0 )
      {
         if( timeLimitReached() )
            return;

         for ( int k = l.rbeg[r]; k < l.rbeg[r + 1]; k++ )
         {
            int j = l.ridx[k];

            assert( l.rperm[j] < i );

            vec[j] -= x * l.rval[k];
         }
      }
   }

#endif // WITH_L_ROWS
}

int CLUFactorRational::solveLleftEps( Rational* vec, int* nonz )
{
   int i, j, k, n;
   int r;
   Rational x;
   Rational *val;
   int *ridx, *idx;
   int *rbeg;
   int* rorig;

   ridx = l.ridx;
   DVectorRational& rval = l.rval;
   rbeg = l.rbeg;
   rorig = l.rorig;
   n = 0;
#ifndef WITH_L_ROWS
   DVectorRational& lval = l.val;
   int*  lidx = l.idx;
   int*  lrow = l.row;
   int*  lbeg = l.start;

   for ( i = l.firstUpdate - 1; i >= 0; --i )
   {
      k = lbeg[i];
      val = &lval[k];
      idx = &lidx[k];
      x = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
         x += vec[*idx++] * ( *val++ );

      vec[lrow[i]] -= x;
   }

#else
   for ( i = thedim; i--; )
   {
      r = rorig[i];
      x = vec[r];

      if ( x != 0 )
      {
         *nonz++ = r;
         n++;
         k = rbeg[r];
         j = rbeg[r + 1] - k;
         val = &rval[k];
         idx = &ridx[k];

         while ( j-- > 0 )
         {
            assert( row.perm[*idx] < i );
            vec[*idx++] -= x * *val++;
         }
      }
      else
         vec[r] = 0;
   }

#endif

   return n;
}

void CLUFactorRational::solveUpdateLeft( Rational* vec )
{
   int i, j, k, end;
   Rational x;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   assert( !l.updateType );             /* Forest-Tomlin Updates */

   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      k = lbeg[i];
      val = &lval[k];
      idx = &lidx[k];
      x = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
         x += vec[*idx++] * ( *val++ );

      vec[lrow[i]] -= x;
   }
}

void CLUFactorRational::solveUpdateLeft2( Rational* vec1, Rational* vec2 )
{
   int i, j, k, end;
   Rational x1, x2;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   assert( !l.updateType );             /* Forest-Tomlin Updates */

   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      k = lbeg[i];
      val = &lval[k];
      idx = &lidx[k];
      x1 = 0;
      x2 = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
      {
         x1 += vec1[*idx] * ( *val );
         x2 += vec2[*idx++] * ( *val++ );
      }

      vec1[lrow[i]] -= x1;

      vec2[lrow[i]] -= x2;
   }
}

int CLUFactorRational::solveUpdateLeft( Rational* vec, int* nonz, int n )
{
   int i, j, k, end;
   Rational x, y;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   assert( !l.updateType );             /* no Forest-Tomlin Updates! */

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      k = lbeg[i];
      assert( k >= 0 && k < l.val.dim() );
      val = &lval[k];
      idx = &lidx[k];
      x = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
      {
         assert( *idx >= 0 && *idx < thedim );
         x += vec[*idx++] * ( *val++ );
      }

      k = lrow[i];

      y = vec[k];

      if ( y == 0 )
      {
         y = -x;

         if ( y != 0 )
         {
            nonz[n++] = k;
            vec[k] = y;
         }
      }
      else
      {
         y -= x;
         //vec[k] = ( y != 0 ) ? y : MARKER;
         vec[k] = y;
      }
   }

   return n;
}

void CLUFactorRational::solveLeft( Rational* vec, Rational* rhs )
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      solveUpdateLeft( rhs );
      solveUleft( vec, rhs );
      solveLleft( vec );
   }
   else
   {
      solveUleft( vec, rhs );
      solveLleftForest( vec, 0 );
      solveLleft( vec );
   }
}

int CLUFactorRational::solveLeftEps( Rational* vec, Rational* rhs, int* nonz )
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      solveUpdateLeft( rhs );
      solveUleft( vec, rhs );
      return solveLleftEps( vec, nonz );
   }
   else
   {
      solveUleft( vec, rhs );
      solveLleftForest( vec, nonz );
      return solveLleftEps( vec, nonz );
   }
}

int CLUFactorRational::solveLeft2(
   Rational* vec1,
   int* nonz,
   Rational* vec2,
   Rational* rhs1,
   Rational* rhs2 )
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      solveUpdateLeft2( rhs1, rhs2 );
      solveUleft2( vec1, rhs1, vec2, rhs2 );
      solveLleft2( vec1, nonz, vec2 );
      return 0;
   }
   else
   {
      solveUleft2( vec1, rhs1, vec2, rhs2 );
      solveLleft2forest( vec1, nonz, vec2 );
      solveLleft2( vec1, nonz, vec2 );
      return 0;
   }
}

int CLUFactorRational::solveUleft( Rational* vec, int* vecidx,
                                    Rational* rhs, int* rhsidx, int rhsn )
{
   Rational x, y;
   int i, j, k, n, r, c;
   int *rorig, *corig, *cperm;
   int *ridx, *rlen, *rbeg, *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;
   cperm = col.perm;

   /*  move rhsidx to a heap
    */

   for ( i = 0; i < rhsn; )
      enQueueMin( rhsidx, &i, cperm[rhsidx[i]] );

   ridx = u.row.idx;

   DVectorRational& rval = u.row.val;

   rlen = u.row.len;

   rbeg = u.row.start;

   n = 0;

   while ( rhsn > 0 )
   {
      i = deQueueMin( rhsidx, &rhsn );
      assert( i >= 0 && i < thedim );
      c = corig[i];
      assert( c >= 0 && c < thedim );
      x = rhs[c];
      rhs[c] = 0;

      if ( x != 0 )
      {
         r = rorig[i];
         assert( r >= 0 && r < thedim );
         vecidx[n++] = r;
         x *= diag[r];
         vec[r] = x;
         k = rbeg[r];
         assert( k >= 0 && k < u.row.val.dim() );
         idx = &ridx[k];
         val = &rval[k];

         for ( int m = rlen[r]; m; --m )
         {
            j = *idx++;
            assert( j >= 0 && j < thedim );
            y = rhs[j];

            if ( y == 0 )
            {
               y = -x * ( *val++ );

               if ( y != 0 )
               {
                  rhs[j] = y;
                  enQueueMin( rhsidx, &rhsn, cperm[j] );
               }
            }
            else
            {
               y -= x * ( *val++ );
               //               rhs[j] = ( y != 0 ) ? y : MARKER;
               rhs[j] = y;
            }
         }
      }
   }

   return n;
}


void CLUFactorRational::solveUleftNoNZ( Rational* vec, Rational* rhs, int* rhsidx, int rhsn )
{
   Rational x, y;
   int i, j, k, r, c;
   int *rorig, *corig, *cperm;
   int *ridx, *rlen, *rbeg, *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;
   cperm = col.perm;

   /*  move rhsidx to a heap
    */

   for ( i = 0; i < rhsn; )
      enQueueMin( rhsidx, &i, cperm[rhsidx[i]] );

   ridx = u.row.idx;

   DVectorRational& rval = u.row.val;

   rlen = u.row.len;

   rbeg = u.row.start;

   while ( rhsn > 0 )
   {
      i = deQueueMin( rhsidx, &rhsn );
      assert( i >= 0 && i < thedim );
      c = corig[i];
      assert( c >= 0 && c < thedim );
      x = rhs[c];
      rhs[c] = 0;

      if ( x != 0 )
      {
         r = rorig[i];
         assert( r >= 0 && r < thedim );
         x *= diag[r];
         vec[r] = x;
         k = rbeg[r];
         assert( k >= 0 && k < u.row.val.dim() );
         idx = &ridx[k];
         val = &rval[k];

         for ( int m = rlen[r]; m; --m )
         {
            j = *idx++;
            assert( j >= 0 && j < thedim );
            y = rhs[j];

            if ( y == 0 )
            {
               y = -x * ( *val++ );

               if ( y != 0 )
               {
                  rhs[j] = y;
                  enQueueMin( rhsidx, &rhsn, cperm[j] );
               }
            }
            else
            {
               y -= x * ( *val++ );
               //               rhs[j] = ( y != 0 ) ? y : MARKER;
               rhs[j] = y;
            }
         }
      }
   }
}


int CLUFactorRational::solveLleftForest( Rational* vec, int* nonz, int n )
{
   int i, j, k, end;
   Rational x, y;
   Rational *val;
   int *idx, *lidx, *lrow, *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;
   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      assert( i >= 0 && i < l.val.dim() );

      if (( x = vec[lrow[i]] ) != 0 )
      {
         k = lbeg[i];
         assert( k >= 0 && k < l.val.dim() );
         val = &lval[k];
         idx = &lidx[k];

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            int m = *idx++;
            assert( m >= 0 && m < thedim );
            y = vec[m];

            if ( y == 0 )
            {
               y = -x * ( *val++ );

               if ( y != 0 )
               {
                  vec[m] = y;
                  nonz[n++] = m;
               }
            }
            else
            {
               y -= x * ( *val++ );

            }
         }
      }
   }

   return n;
}


void CLUFactorRational::solveLleftForestNoNZ( Rational* vec )
{
   int i, j, k, end;
   Rational x;
   Rational *val;
   int *idx, *lidx, *lrow, *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;
   end = l.firstUpdate;

   for ( i = l.firstUnused - 1; i >= end; --i )
   {
      if (( x = vec[lrow[i]] ) != 0 )
      {
         assert( i >= 0 && i < l.val.dim() );
         k = lbeg[i];
         assert( k >= 0 && k < l.val.dim() );
         val = &lval[k];
         idx = &lidx[k];

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            assert( *idx >= 0 && *idx < thedim );
            vec[*idx++] -= x * ( *val++ );
         }
      }
   }
}


int CLUFactorRational::solveLleft( Rational* vec, int* nonz, int rn )
{
   int i, j, k, n;
   int r;
   Rational x, y;
   Rational *val;
   int *ridx, *idx;
   int *rbeg;
   int *rorig, *rperm;
   int *last;

   ridx  = l.ridx;
   DVectorRational& rval  = l.rval;
   rbeg  = l.rbeg;
   rorig = l.rorig;
   rperm = l.rperm;
   n     = 0;

   i = l.firstUpdate - 1;
#ifndef WITH_L_ROWS
#pragma warn "Not yet implemented, define WITH_L_ROWS"
   DVectorRational& lval = l.val;
   int*    lidx = l.idx;
   int*    lrow = l.row;
   int*    lbeg = l.start;

   for ( ; i >= 0; --i )
   {
      k   = lbeg[i];
      val = &lval[k];
      idx = &lidx[k];
      x   = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
         x += vec[*idx++] * ( *val++ );

      vec[lrow[i]] -= x;
   }

#else
   /*  move rhsidx to a heap
    */
   for ( i = 0; i < rn; )
      enQueueMax( nonz, &i, rperm[nonz[i]] );

   last = nonz + thedim;

   while ( rn > 0 )
   {
      i = deQueueMax( nonz, &rn );
      r = rorig[i];
      x = vec[r];

      if ( x != 0 )
      {
         *( --last ) = r;
         n++;
         k = rbeg[r];
         j = rbeg[r + 1] - k;
         val = &rval[k];
         idx = &ridx[k];

         while ( j-- > 0 )
         {
            assert( l.rperm[*idx] < i );
            int m = *idx++;
            y = vec[m];

            if ( y == 0 )
            {
               y = -x * *val++;

               if ( y != 0 )
               {
                  vec[m] = y;
                  enQueueMax( nonz, &rn, rperm[m] );
               }
            }
            else
            {
               y -= x * *val++;
               //               vec[m] = ( y != 0 ) ? y : MARKER;
               vec[m] = y;
            }
         }
      }
      else
         vec[r] = 0;
   }

   for ( i = 0; i < n; ++i )
      *nonz++ = *last++;

#endif

   return n;
}


void CLUFactorRational::solveLleftNoNZ( Rational* vec )
{
   int i, j, k;
   int r;
   Rational x;
   Rational *val;
   int *ridx, *idx;
   int *rbeg;
   int* rorig;

   ridx = l.ridx;
   DVectorRational& rval = l.rval;
   rbeg = l.rbeg;
   rorig = l.rorig;

#ifndef WITH_L_ROWS
   DVectorRational& lval = l.val;
   int*    lidx = l.idx;
   int*    lrow = l.row;
   int*    lbeg = l.start;

   i = l.firstUpdate - 1;
   assert( i < thedim );

   for ( ; i >= 0; --i )
   {
      k = lbeg[i];
      assert( k >= 0 && k < l.val.dim() );
      val = &lval[k];
      idx = &lidx[k];
      x = 0;

      for ( j = lbeg[i + 1]; j > k; --j )
      {
         assert( *idx >= 0 && *idx < thedim );
         x += vec[*idx++] * ( *val++ );
      }

      vec[lrow[i]] -= x;
   }

#else
   for ( i = thedim; i--; )
   {
      r = rorig[i];
      x = vec[r];

      if ( x != 0 )
      {
         k = rbeg[r];
         j = rbeg[r + 1] - k;
         val = &rval[k];
         idx = &ridx[k];

         while ( j-- > 0 )
         {
            assert( l.rperm[*idx] < i );
            vec[*idx++] -= x * *val++;
         }
      }
   }

#endif
}

int CLUFactorRational::vSolveLright( Rational* vec, int* ridx, int rn )
{
   int i, j, k, n;
   int end;
   Rational x;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = 0; i < end; ++i )
   {
      x = vec[lrow[i]];

      if ( x != 0 )
      {
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            assert( *idx >= 0 && *idx < thedim );
            ridx[rn] = n = *idx++;
            rn += ( vec[n] == 0 ) ? 1 : 0;
            vec[n] -= x * ( *val++ );
            //            vec[n] += ( vec[n] == 0 ) ? MARKER : 0;
         }
      }
   }

   if ( l.updateType )                   /* Forest-Tomlin Updates */
   {
      end = l.firstUnused;

      for ( ; i < end; ++i )
      {
         x = 0;
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            assert( *idx >= 0 && *idx < thedim );
            x += vec[*idx++] * ( *val++ );
         }

         ridx[rn] = j = lrow[i];

         rn += ( vec[j] == 0 ) ? 1 : 0;
         vec[j] -= x;
         //         vec[j] += ( vec[j] == 0 ) ? MARKER : 0;
      }
   }

   return rn;
}

void CLUFactorRational::vSolveLright2( Rational* vec, int* ridx, int* rnptr,
                                        Rational* vec2, int* ridx2, int* rn2ptr )
{
   int i, j, k, n;
   int end;
   Rational x, y;
   Rational x2, y2;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   int rn = *rnptr;
   int rn2 = *rn2ptr;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = 0; i < end; ++i )
   {
      j = lrow[i];
      x2 = vec2[j];
      x = vec[j];

      if ( x != 0 )
      {
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         if ( x2 != 0 )
         {
            for ( j = lbeg[i + 1]; j > k; --j )
            {
               assert( *idx >= 0 && *idx < thedim );
               ridx[rn] = ridx2[rn2] = n = *idx++;
               y = vec[n];
               y2 = vec2[n];
               rn += ( y == 0 ) ? 1 : 0;
               rn2 += ( y2 == 0 ) ? 1 : 0;
               y -= x * ( *val );
               y2 -= x2 * ( *val++ );
               //               vec[n] = y + ( y == 0 ? MARKER : 0 );
               //               vec2[n] = y2 + ( y2 == 0 ? MARKER : 0 );
               vec[n] = y;
               vec2[n] = y2;
            }
         }
         else
         {
            for ( j = lbeg[i + 1]; j > k; --j )
            {
               assert( *idx >= 0 && *idx < thedim );
               ridx[rn] = n = *idx++;
               y = vec[n];
               rn += ( y == 0 ) ? 1 : 0;
               y -= x * ( *val++ );
               //               vec[n] = y + ( y == 0 ? MARKER : 0 );
               vec[n] = y;
            }
         }
      }
      else
         if ( x2 != 0 )
         {
            k = lbeg[i];
            idx = &( lidx[k] );
            val = &( lval[k] );

            for ( j = lbeg[i + 1]; j > k; --j )
            {
               assert( *idx >= 0 && *idx < thedim );
               ridx2[rn2] = n = *idx++;
               y2 = vec2[n];
               rn2 += ( y2 == 0 ) ? 1 : 0;
               y2 -= x2 * ( *val++ );
               //               vec2[n] = y2 + ( y2 == 0 ? MARKER : 0 );
               vec2[n] = y2;
            }
         }
   }

   if ( l.updateType )                   /* Forest-Tomlin Updates */
   {
      end = l.firstUnused;

      for ( ; i < end; ++i )
      {
         x = x2 = 0;
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            assert( *idx >= 0 && *idx < thedim );
            x += vec[*idx] * ( *val );
            x2 += vec2[*idx++] * ( *val++ );
         }

         ridx[rn] = ridx2[rn2] = j = lrow[i];

         rn += ( vec[j] == 0 ) ? 1 : 0;
         rn2 += ( vec2[j] == 0 ) ? 1 : 0;
         vec[j] -= x;
         vec2[j] -= x2;
         //         vec[j] += ( vec[j] == 0 ) ? MARKER : 0;
         //         vec2[j] += ( vec2[j] == 0 ) ? MARKER : 0;
      }
   }

   *rnptr = rn;

   *rn2ptr = rn2;
}

void CLUFactorRational::vSolveLright3( Rational* vec, int* ridx, int* rnptr,
                                        Rational* vec2, int* ridx2, int* rn2ptr,
                                        Rational* vec3, int* ridx3, int* rn3ptr)
{
   int i, j, k, n;
   int end;
   Rational x, y;
   Rational x2, y2;
   Rational x3, y3;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   int rn = *rnptr;
   int rn2 = *rn2ptr;
   int rn3 = *rn3ptr;

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;

   end = l.firstUpdate;

   for ( i = 0; i < end; ++i )
   {
      j = lrow[i];
      x3 = vec3[j];
      x2 = vec2[j];
      x = vec[j];

      if ( x != 0 )
      {
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         if ( x2 != 0 )
         {
            if ( x3 != 0 )
            {
               // case 1: all three vectors are nonzero at j
               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx[rn] = ridx2[rn2] = ridx3[rn3] = n = *idx++;
                  y = vec[n];
                  y2 = vec2[n];
                  y3 = vec3[n];
                  rn += ( y == 0 ) ? 1 : 0;
                  rn2 += ( y2 == 0 ) ? 1 : 0;
                  rn3 += ( y3 == 0 ) ? 1 : 0;
                  y -= x * ( *val );
                  y2 -= x2 * ( *val );
                  y3 -= x3 * ( *val++ );
                  //                  vec[n] = y + ( y == 0 ? MARKER : 0 );
                  //                  vec2[n] = y2 + ( y2 == 0 ? MARKER : 0 );
                  //                  vec3[n] = y3 + ( y3 == 0 ? MARKER : 0 );
                  vec[n] = y;
                  vec2[n] = y2;
                  vec3[n] = y3;
               }
            }
            else
            {
               // case 2: 1 and 2 are nonzero at j
               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx[rn] = ridx2[rn2] = n = *idx++;
                  y = vec[n];
                  y2 = vec2[n];
                  rn += ( y == 0 ) ? 1 : 0;
                  rn2 += ( y2 == 0 ) ? 1 : 0;
                  y -= x * ( *val );
                  y2 -= x2 * ( *val++ );
                  //                  vec[n] = y + ( y == 0 ? MARKER : 0 );
                  //vec2[n] = y2 + ( y2 == 0 ? MARKER : 0 );
                  vec[n] = y;
                  vec2[n] = y2;
               }
            }
         }
         else
            if ( x3 != 0 )
            {
               // case 3: 1 and 3 are nonzero at j
               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx[rn] = ridx3[rn3] = n = *idx++;
                  y = vec[n];
                  y3 = vec3[n];
                  rn += ( y == 0 ) ? 1 : 0;
                  rn3 += ( y3 == 0 ) ? 1 : 0;
                  y -= x * ( *val );
                  y3 -= x3 * ( *val++ );
                  //                  vec[n] = y + ( y == 0 ? MARKER : 0 );
                  //                  vec3[n] = y3 + ( y3 == 0 ? MARKER : 0 );
                  vec[n] = y;
                  vec3[n] = y3;
               }
            }
            else
            {
               // case 4: only 1 is nonzero at j
               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx[rn] = n = *idx++;
                  y = vec[n];
                  rn += ( y == 0 ) ? 1 : 0;
                  y -= x * ( *val++ );
                  //                  vec[n] = y + ( y == 0 ? MARKER : 0 );
                  vec[n] = y;
               }
            }
      }
      else
         if ( x2 != 0 )
         {
            k = lbeg[i];
            idx = &( lidx[k] );
            val = &( lval[k] );

            if ( x3 != 0 )
            {
               // case 5: 2 and 3 are nonzero at j
               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx2[rn2] = ridx3[rn3] = n = *idx++;
                  y2 = vec2[n];
                  y3 = vec3[n];
                  rn2 += ( y2 == 0 ) ? 1 : 0;
                  rn3 += ( y3 == 0 ) ? 1 : 0;
                  y2 -= x2 * ( *val );
                  y3 -= x3 * ( *val++ );
                  //                  vec2[n] = y2 + ( y2 == 0 ? MARKER : 0 );
                  //                  vec3[n] = y3 + ( y3 == 0 ? MARKER : 0 );
                  vec2[n] = y2;
                  vec3[n] = y3;
               }
            }
            else
            {
               // case 6: only 2 is nonzero at j
               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx2[rn2] = n = *idx++;
                  y2 = vec2[n];
                  rn2 += ( y2 == 0 ) ? 1 : 0;
                  y2 -= x2 * ( *val++ );
                  //                  vec2[n] = y2 + ( y2 == 0 ? MARKER : 0 );
                  vec2[n] = y2;
               }
            }
         }
         else
            if ( x3 != 0 )
            {
               // case 7: only 3 is nonzero at j
               k = lbeg[i];
               idx = &( lidx[k] );
               val = &( lval[k] );

               for ( j = lbeg[i + 1]; j > k; --j )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  ridx3[rn3] = n = *idx++;
                  y3 = vec3[n];
                  rn3 += ( y3 == 0 ) ? 1 : 0;
                  y3 -= x3 * ( *val++ );
                  //                  vec3[n] = y3 + ( y3 == 0 ? MARKER : 0 );
                  vec3[n] = y3;
               }
            }
   }

   if ( l.updateType )                   /* Forest-Tomlin Updates */
   {
      end = l.firstUnused;

      for ( ; i < end; ++i )
      {
         x = x2 = x3 = 0;
         k = lbeg[i];
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            assert( *idx >= 0 && *idx < thedim );
            x += vec[*idx] * ( *val );
            x2 += vec2[*idx] * ( *val );
            x3 += vec3[*idx++] * ( *val++ );
         }

         ridx[rn] = ridx2[rn2] = ridx3[rn3] = j = lrow[i];

         rn += ( vec[j] == 0 ) ? 1 : 0;
         rn2 += ( vec2[j] == 0 ) ? 1 : 0;
         rn3 += ( vec3[j] == 0 ) ? 1 : 0;
         vec[j] -= x;
         vec2[j] -= x2;
         vec3[j] -= x3;
         //         vec[j] += ( vec[j] == 0 ) ? MARKER : 0;
         //         vec2[j] += ( vec2[j] == 0 ) ? MARKER : 0;
         //         vec3[j] += ( vec3[j] == 0 ) ? MARKER : 0;
      }
   }

   *rnptr = rn;

   *rn2ptr = rn2;
   *rn3ptr = rn3;
}

int CLUFactorRational::vSolveUright( Rational* vec, int* vidx,
                                      Rational* rhs, int* ridx, int rn )
{
   int i, j, k, r, c, n;
   int *rorig, *corig;
   int *rperm;
   int *cidx, *clen, *cbeg;
   Rational x, y;

   int *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;
   rperm = row.perm;

   cidx = u.col.idx;
   DVectorRational& cval = u.col.val;
   clen = u.col.len;
   cbeg = u.col.start;

   n = 0;

   while ( rn > 0 )
   {
      /*      Find nonzero with highest permuted row index and setup i and r
       */
      i = deQueueMax( ridx, &rn );
      assert( i >= 0 && i < thedim );
      r = rorig[i];
      assert( r >= 0 && r < thedim );

      x = diag[r] * rhs[r];
      rhs[r] = 0;

      if ( x != 0 )
      {
         c = corig[i];
         assert( c >= 0 && c < thedim );
         vidx[n++] = c;
         vec[c] = x;
         val = &cval[cbeg[c]];
         idx = &cidx[cbeg[c]];
         j = clen[c];

         while ( j-- > 0 )
         {
            assert( *idx >= 0 && *idx < thedim );
            k = *idx++;
            assert( k >= 0 && k < thedim );
            y = rhs[k];

            if ( y == 0 )
            {
               y = -x * ( *val++ );

               if ( y != 0 )
               {
                  rhs[k] = y;
                  enQueueMax( ridx, &rn, rperm[k] );
               }
            }
            else
            {
               y -= x * ( *val++ );
               //               y += ( y == 0 ) ? MARKER : 0;
               rhs[k] = y;
            }
         }

         if ( rn > i*verySparseFactor4right )
         {                                   /* continue with dense case */
            for ( i = *ridx; i >= 0; --i )
            {
               r = rorig[i];
               assert( r >= 0 && r < thedim );
               x = diag[r] * rhs[r];
               rhs[r] = 0;

               if ( x != 0 )
               {
                  c = corig[i];
                  assert( c >= 0 && c < thedim );
                  vidx[n++] = c;
                  vec[c] = x;
                  val = &cval[cbeg[c]];
                  idx = &cidx[cbeg[c]];
                  j = clen[c];

                  while ( j-- > 0 )
                  {
                     assert( *idx >= 0 && *idx < thedim );
                     rhs[*idx++] -= x * ( *val++ );
                  }
               }
            }

            break;
         }
      }
   }

   return n;
}


void CLUFactorRational::vSolveUrightNoNZ( Rational* vec, Rational* rhs, int* ridx, int rn )
{
   int i, j, k, r, c;
   int *rorig, *corig;
   int *rperm;
   int *cidx, *clen, *cbeg;
   Rational x, y;

   int *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;
   rperm = row.perm;

   cidx = u.col.idx;
   DVectorRational& cval = u.col.val;
   clen = u.col.len;
   cbeg = u.col.start;

   while ( rn > 0 )
   {
      if ( rn > *ridx * verySparseFactor4right )
      {                                       /* continue with dense case */
         for ( i = *ridx; i >= 0; --i )
         {
            assert( i >= 0 && i < thedim );
            r = rorig[i];
            assert( r >= 0 && r < thedim );
            x = diag[r] * rhs[r];
            rhs[r] = 0;

            if ( x != 0 )
            {
               c = corig[i];
               vec[c] = x;
               val = &cval[cbeg[c]];
               idx = &cidx[cbeg[c]];
               j = clen[c];

               while ( j-- > 0 )
               {
                  assert( *idx >= 0 && *idx < thedim );
                  rhs[*idx++] -= x * ( *val++ );
               }
            }
         }

         break;
      }

      /*      Find nonzero with highest permuted row index and setup i and r
       */
      i = deQueueMax( ridx, &rn );

      assert( i >= 0 && i < thedim );

      r = rorig[i];

      assert( r >= 0 && r < thedim );

      x = diag[r] * rhs[r];

      rhs[r] = 0;

      if ( x != 0 )
      {
         c = corig[i];
         vec[c] = x;
         val = &cval[cbeg[c]];
         idx = &cidx[cbeg[c]];
         j = clen[c];

         while ( j-- > 0 )
         {
            k = *idx++;
            assert( k >= 0 && k < thedim );
            y = rhs[k];

            if ( y == 0 )
            {
               y = -x * ( *val++ );

               if ( y != 0 )
               {
                  rhs[k] = y;
                  enQueueMax( ridx, &rn, rperm[k] );
               }
            }
            else
            {
               y -= x * ( *val++ );
               //               y += ( y == 0 ) ? MARKER : 0;
               rhs[k] = y;
            }
         }
      }
   }
}


int CLUFactorRational::vSolveUright2( Rational* vec, int* vidx, Rational* rhs, int* ridx, int rn,
                                       Rational* vec2, Rational* rhs2, int* ridx2, int rn2 )
{
   int i, j, k, r, c, n;
   int *rorig, *corig;
   int *rperm;
   int *cidx, *clen, *cbeg;
   Rational x, y;
   Rational x2, y2;

   int *idx;
   Rational *val;

   rorig = row.orig;
   corig = col.orig;
   rperm = row.perm;

   cidx = u.col.idx;
   DVectorRational& cval = u.col.val;
   clen = u.col.len;
   cbeg = u.col.start;

   n = 0;

   while ( rn + rn2 > 0 )
   {
      /*      Find nonzero with highest permuted row index and setup i and r
       */
      if ( rn <= 0 )
         i = deQueueMax( ridx2, &rn2 );
      else
         if ( rn2 <= 0 )
            i = deQueueMax( ridx, &rn );
         else
            if ( *ridx2 > *ridx )
               i = deQueueMax( ridx2, &rn2 );
            else
               if ( *ridx2 < *ridx )
                  i = deQueueMax( ridx, &rn );
               else
               {
                  i = deQueueMax( ridx, &rn );
                  i = deQueueMax( ridx2, &rn2 );
               }

      assert( i >= 0 && i < thedim );

      r = rorig[i];
      assert( r >= 0 && r < thedim );

      x = diag[r] * rhs[r];
      x2 = diag[r] * rhs2[r];
      rhs[r] = 0;
      rhs2[r] = 0;

      if ( x != 0 )
      {
         c = corig[i];
         vidx[n++] = c;
         vec[c] = x;
         vec2[c] = x2;
         val = &cval[cbeg[c]];
         idx = &cidx[cbeg[c]];
         j = clen[c];

         if ( x2 != 0 )
         {
            while ( j-- > 0 )
            {
               k = *idx++;
               assert( k >= 0 && k < thedim );
               y2 = rhs2[k];

               if ( y2 == 0 )
               {
                  y2 = -x2 * ( *val );

                  if ( y2 != 0 )
                  {
                     rhs2[k] = y2;
                     enQueueMax( ridx2, &rn2, rperm[k] );
                  }
               }
               else
               {
                  y2 -= x2 * ( *val );
                  //                  rhs2[k] = ( y2 != 0 ) ? y2 : MARKER;
                  rhs2[k] = y2;
               }

               y = rhs[k];

               if ( y == 0 )
               {
                  y = -x * ( *val++ );

                  if ( y != 0 )
                  {
                     rhs[k] = y;
                     enQueueMax( ridx, &rn, rperm[k] );
                  }
               }
               else
               {
                  y -= x * ( *val++ );
                  //                  y += ( y == 0 ) ? MARKER : 0;
                  rhs[k] = y;
               }
            }
         }
         else
         {
            while ( j-- > 0 )
            {
               k = *idx++;
               assert( k >= 0 && k < thedim );
               y = rhs[k];

               if ( y == 0 )
               {
                  y = -x * ( *val++ );

                  if ( y != 0 )
                  {
                     rhs[k] = y;
                     enQueueMax( ridx, &rn, rperm[k] );
                  }
               }
               else
               {
                  y -= x * ( *val++ );
                  //                  y += ( y == 0 ) ? MARKER : 0;
                  rhs[k] = y;
               }
            }
         }
      }
      else
         if ( x2 != 0 )
         {
            c = corig[i];
            assert( c >= 0 && c < thedim );
            vec2[c] = x2;
            val = &cval[cbeg[c]];
            idx = &cidx[cbeg[c]];
            j = clen[c];

            while ( j-- > 0 )
            {
               k = *idx++;
               assert( k >= 0 && k < thedim );
               y2 = rhs2[k];

               if ( y2 == 0 )
               {
                  y2 = -x2 * ( *val++ );

                  if ( y2 != 0 )
                  {
                     rhs2[k] = y2;
                     enQueueMax( ridx2, &rn2, rperm[k] );
                  }
               }
               else
               {
                  y2 -= x2 * ( *val++ );
                  //                  rhs2[k] = ( y2 != 0 ) ? y2 : MARKER;
                  rhs2[k] = y2;
               }
            }
         }

      if ( rn + rn2 > i*verySparseFactor4right )
      {                                       /* continue with dense case */
         if ( *ridx > *ridx2 )
            i = *ridx;
         else
            i = *ridx2;

         for ( ; i >= 0; --i )
         {
            assert( i < thedim );
            r = rorig[i];
            assert( r >= 0 && r < thedim );
            x = diag[r] * rhs[r];
            x2 = diag[r] * rhs2[r];
            rhs[r] = 0;
            rhs2[r] = 0;

            if ( x2 != 0 )
            {
               c = corig[i];
               assert( c >= 0 && c < thedim );
               vec2[c] = x2;
               val = &cval[cbeg[c]];
               idx = &cidx[cbeg[c]];
               j = clen[c];

               if ( x != 0 )
               {
                  vidx[n++] = c;
                  vec[c] = x;

                  while ( j-- > 0 )
                  {
                     assert( *idx >= 0 && *idx < thedim );
                     rhs[*idx] -= x * ( *val );
                     rhs2[*idx++] -= x2 * ( *val++ );
                  }
               }
               else
               {
                  while ( j-- > 0 )
                  {
                     assert( *idx >= 0 && *idx < thedim );
                     rhs2[*idx++] -= x2 * ( *val++ );
                  }
               }
            }
            else
               if ( x != 0 )
               {
                  c = corig[i];
                  assert( c >= 0 && c < thedim );
                  vidx[n++] = c;
                  vec[c] = x;
                  val = &cval[cbeg[c]];
                  idx = &cidx[cbeg[c]];
                  j = clen[c];

                  while ( j-- > 0 )
                  {
                     assert( *idx >= 0 && *idx < thedim );
                     rhs[*idx++] -= x * ( *val++ );
                  }
               }
         }

         break;
      }
   }

   return n;
}

int CLUFactorRational::vSolveUpdateRight( Rational* vec, int* ridx, int n )
{
   int i, j, k;
   int end;
   Rational x, y;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   assert( !l.updateType );             /* no Forest-Tomlin Updates */

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;
   end = l.firstUnused;

   for ( i = l.firstUpdate; i < end; ++i )
   {
      assert( i >= 0 && i < thedim );
      x = vec[lrow[i]];

      if ( x != 0 )
      {
         k = lbeg[i];
         assert( k >= 0 && k < l.val.dim() );
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            int m = ridx[n] = *idx++;
            assert( m >= 0 && m < thedim );
            y = vec[m];
            n += ( y == 0 ) ? 1 : 0;
            y = y - x * ( *val++ );
            //            vec[m] = ( y != 0 ) ? y : MARKER;
            vec[m] = y;
         }
      }
   }

   return n;
}

void CLUFactorRational::vSolveUpdateRightNoNZ( Rational* vec )
{
   int i, j, k;
   int end;
   Rational x;
   Rational *val;
   int *lrow, *lidx, *idx;
   int *lbeg;

   assert( !l.updateType );             /* no Forest-Tomlin Updates */

   DVectorRational& lval = l.val;
   lidx = l.idx;
   lrow = l.row;
   lbeg = l.start;
   end = l.firstUnused;

   for ( i = l.firstUpdate; i < end; ++i )
   {
      assert( i >= 0 && i < thedim );

      if (( x = vec[lrow[i]] ) != 0 )
      {
         k = lbeg[i];
         assert( k >= 0 && k < l.val.dim() );
         idx = &( lidx[k] );
         val = &( lval[k] );

         for ( j = lbeg[i + 1]; j > k; --j )
         {
            assert( *idx >= 0 && *idx < thedim );
            vec[*idx++] -= x * ( *val++ );
         }
      }
   }
}


int CLUFactorRational::vSolveRight4update( Rational* vec, int* idx,                             /* result */
                                            Rational* rhs, int* ridx, int rn,                    /* rhs    */
                                            Rational* forest, int* forestNum, int* forestIdx )
{
   rn = vSolveLright( rhs, ridx, rn );

   /*  turn index list into a heap
    */

   if ( forest )
   {
      Rational x;
      int i, j, k;
      int* rperm;
      int* it = forestIdx;

      rperm = row.perm;

      for ( i = j = 0; i < rn; ++i )
      {
         k = ridx[i];
         assert( k >= 0 && k < thedim );
         x = rhs[k];

         if ( x != 0 )
         {
            enQueueMax( ridx, &j, rperm[*it++ = k] );
            forest[k] = x;
         }
         else
            rhs[k] = 0;
      }

      *forestNum = rn = j;
   }
   else
   {
      Rational x;
      int i, j, k;
      int* rperm;

      rperm = row.perm;

      for ( i = j = 0; i < rn; ++i )
      {
         k = ridx[i];
         assert( k >= 0 && k < thedim );
         x = rhs[k];

         if ( x != 0 )
            enQueueMax( ridx, &j, rperm[k] );
         else
            rhs[k] = 0;
      }

      rn = j;
   }

   rn = vSolveUright( vec, idx, rhs, ridx, rn );

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
      rn = vSolveUpdateRight( vec, idx, rn );

   return rn;
}

int CLUFactorRational::vSolveRight4update2( Rational* vec, int* idx,                             /* result1 */
                                             Rational* rhs, int* ridx, int rn,                    /* rhs1    */
                                             Rational* vec2,                                      /* result2 */
                                             Rational* rhs2, int* ridx2, int rn2,                 /* rhs2    */
                                             Rational* forest, int* forestNum, int* forestIdx )
{
   vSolveLright2( rhs, ridx, &rn, rhs2, ridx2, &rn2 );

   /*  turn index list into a heap
    */

   if ( forest )
   {
      Rational x;
      int i, j, k;
      int* rperm;
      int* it = forestIdx;

      rperm = row.perm;

      for ( i = j = 0; i < rn; ++i )
      {
         k = ridx[i];
         assert( k >= 0 && k < thedim );
         x = rhs[k];

         if ( x != 0 )
         {
            enQueueMax( ridx, &j, rperm[*it++ = k] );
            forest[k] = x;
         }
         else
            rhs[k] = 0;
      }

      *forestNum = rn = j;
   }
   else
   {
      Rational x;
      int i, j, k;
      int* rperm;

      rperm = row.perm;

      for ( i = j = 0; i < rn; ++i )
      {
         k = ridx[i];
         assert( k >= 0 && k < thedim );
         x = rhs[k];

         if ( x != 0 )
            enQueueMax( ridx, &j, rperm[k] );
         else
            rhs[k] = 0;
      }

      rn = j;
   }

   if ( rn2 > thedim*verySparseFactor4right )
   {
      ridx2[0] = thedim - 1;
      /* ridx2[1] = thedim - 2; */
   }
   else
   {
      Rational x;
      /*      Rational  maxabs; */
      int i, j, k;
      int* rperm;

      /*      maxabs = 1;    */
      rperm = row.perm;

      for ( i = j = 0; i < rn2; ++i )
      {
         k = ridx2[i];
         assert( k >= 0 && k < thedim );
         x = rhs2[k];

         if ( x == 0 )
         {
            /*              maxabs = (maxabs < -x) ? -x : maxabs;  */
            enQueueMax( ridx2, &j, rperm[k] );
         }
         else
            rhs2[k] = 0;
      }

      rn2 = j;

   }

   rn = vSolveUright( vec, idx, rhs, ridx, rn );

   vSolveUrightNoNZ( vec2, rhs2, ridx2, rn2 );

   /*
    *  rn = vSolveUright2(vec, idx, rhs, ridx, rn, vec2, rhs2, ridx2, rn2 );
    */

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      rn = vSolveUpdateRight( vec, idx, rn );
      vSolveUpdateRightNoNZ( vec2 );
   }

   return rn;
}

int CLUFactorRational::vSolveRight4update3( Rational* vec, int* idx,                             /* result1 */
                                             Rational* rhs, int* ridx, int rn,                    /* rhs1    */
                                             Rational* vec2,                                      /* result2 */
                                             Rational* rhs2, int* ridx2, int rn2,                 /* rhs2    */
                                             Rational* vec3,                                      /* result3 */
                                             Rational* rhs3, int* ridx3, int rn3,                 /* rhs3    */
                                             Rational* forest, int* forestNum, int* forestIdx )
{

   vSolveLright3( rhs, ridx, &rn, rhs2, ridx2, &rn2, rhs3, ridx3, &rn3 );
   assert( rn >= 0 && rn <= thedim );
   assert( rn2 >= 0 && rn2 <= thedim );
   assert( rn3 >= 0 && rn3 <= thedim );

   /*  turn index list into a heap
    */

   if ( forest )
   {
      Rational x;
      int i, j, k;
      int* rperm;
      int* it = forestIdx;

      rperm = row.perm;

      for ( i = j = 0; i < rn; ++i )
      {
         k = ridx[i];
         assert( k >= 0 && k < thedim );
         x = rhs[k];

         if ( x != 0 )
         {
            enQueueMax( ridx, &j, rperm[*it++ = k] );
            forest[k] = x;
         }
         else
            rhs[k] = 0;
      }

      *forestNum = rn = j;
   }
   else
   {
      Rational x;
      int i, j, k;
      int* rperm;

      rperm = row.perm;

      for ( i = j = 0; i < rn; ++i )
      {
         k = ridx[i];
         assert( k >= 0 && k < thedim );
         x = rhs[k];

         if ( x != 0 )
            enQueueMax( ridx, &j, rperm[k] );
         else
            rhs[k] = 0;
      }

      rn = j;
   }

   if ( rn2 > thedim*verySparseFactor4right )
   {
      ridx2[0] = thedim - 1;
   }
   else
   {
      Rational x;
      int i, j, k;
      int* rperm;

      rperm = row.perm;

      for ( i = j = 0; i < rn2; ++i )
      {
         k = ridx2[i];
         assert( k >= 0 && k < thedim );
         x = rhs2[k];

         if ( x == 0 )
         {
            enQueueMax( ridx2, &j, rperm[k] );
         }
         else
            rhs2[k] = 0;
      }

      rn2 = j;
   }

   if ( rn3 > thedim*verySparseFactor4right )
   {
      ridx3[0] = thedim - 1;
   }
   else
   {
      Rational x;
      int i, j, k;
      int* rperm;

      rperm = row.perm;

      for ( i = j = 0; i < rn3; ++i )
      {
         k = ridx3[i];
         assert( k >= 0 && k < thedim );
         x = rhs3[k];

         if ( x == 0 )
         {
            enQueueMax( ridx3, &j, rperm[k] );
         }
         else
            rhs3[k] = 0;
      }

      rn3 = j;
   }

   rn = vSolveUright( vec, idx, rhs, ridx, rn );

   vSolveUrightNoNZ( vec2, rhs2, ridx2, rn2 );
   vSolveUrightNoNZ( vec3, rhs3, ridx3, rn3 );

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      rn = vSolveUpdateRight( vec, idx, rn );
      vSolveUpdateRightNoNZ( vec2 );
      vSolveUpdateRightNoNZ( vec3 );
   }

   return rn;
}

void CLUFactorRational::vSolveRightNoNZ( Rational* vec2,                         /* result2 */
                                          Rational* rhs2, int* ridx2, int rn2 )   /* rhs2    */
{
   rn2 = vSolveLright( rhs2, ridx2, rn2 );
   assert( rn2 >= 0 && rn2 <= thedim );

   if ( rn2 > thedim*verySparseFactor4right )
   {
      *ridx2 = thedim - 1;
   }
   else
   {
      Rational x;
      /*      Rational  maxabs; */
      int i, j, k;
      int* rperm;

      /*      maxabs = 1;    */
      rperm = row.perm;

      for ( i = j = 0; i < rn2; ++i )
      {
         k = ridx2[i];
         assert( k >= 0 && k < thedim );
         x = rhs2[k];

         if ( x == 0 )
         {
            /*              maxabs = (maxabs < -x) ? -x : maxabs;  */
            enQueueMax( ridx2, &j, rperm[k] );
         }
         else
            rhs2[k] = 0;
      }

      rn2 = j;
   }

   vSolveUrightNoNZ( vec2, rhs2, ridx2, rn2 );

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
      vSolveUpdateRightNoNZ( vec2 );
}

int CLUFactorRational::vSolveLeft( Rational* vec, int* idx,                     /* result */
                                    Rational* rhs, int* ridx, int rn )           /* rhs    */
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      rn = solveUpdateLeft( rhs, ridx, rn );
      rn = solveUleft( vec, idx, rhs, ridx, rn );
   }
   else
   {
      rn = solveUleft( vec, idx, rhs, ridx, rn );
      rn = solveLleftForest( vec, idx, rn );
   }

   if ( rn + l.firstUpdate > verySparseFactor4left * thedim )
   {
      solveLleftNoNZ( vec );
      return 0;
   }
   else
      return solveLleft( vec, idx, rn );
}

int CLUFactorRational::vSolveLeft2( Rational* vec, int* idx,                      /* result */
                                     Rational* rhs, int* ridx, int rn,             /* rhs    */
                                     Rational* vec2,                               /* result2 */
                                     Rational* rhs2, int* ridx2, int rn2 )         /* rhs2    */
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      rn = solveUpdateLeft( rhs, ridx, rn );
      rn = solveUleft( vec, idx, rhs, ridx, rn );
      rn2 = solveUpdateLeft( rhs2, ridx2, rn2 );
      solveUleftNoNZ( vec2, rhs2, ridx2, rn2 );
   }
   else
   {
      rn = solveUleft( vec, idx, rhs, ridx, rn );
      rn = solveLleftForest( vec, idx, rn );
      solveUleftNoNZ( vec2, rhs2, ridx2, rn2 );
      solveLleftForestNoNZ( vec2 );
   }

   rn = solveLleft( vec, idx, rn );

   solveLleftNoNZ( vec2 );

   return rn;
}

int CLUFactorRational::vSolveLeft3( Rational* vec, int* idx,                      /* result */
                                     Rational* rhs, int* ridx, int rn,             /* rhs    */
                                     Rational* vec2,                               /* result2 */
                                     Rational* rhs2, int* ridx2, int rn2,          /* rhs2    */
                                     Rational* vec3,                               /* result3 */
                                     Rational* rhs3, int* ridx3, int rn3 )         /* rhs3    */
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      rn = solveUpdateLeft( rhs, ridx, rn );
      rn = solveUleft( vec, idx, rhs, ridx, rn );
      rn2 = solveUpdateLeft( rhs2, ridx2, rn2 );
      solveUleftNoNZ( vec2, rhs2, ridx2, rn2 );
      rn3 = solveUpdateLeft( rhs3, ridx3, rn3 );
      solveUleftNoNZ( vec3, rhs3, ridx3, rn3 );
   }
   else
   {
      rn = solveUleft( vec, idx, rhs, ridx, rn );
      rn = solveLleftForest( vec, idx, rn );
      solveUleftNoNZ( vec2, rhs2, ridx2, rn2 );
      solveLleftForestNoNZ( vec2 );
      solveUleftNoNZ( vec3, rhs3, ridx3, rn3 );
      solveLleftForestNoNZ( vec3 );
   }

   rn = solveLleft( vec, idx, rn );

   solveLleftNoNZ( vec2 );
   solveLleftNoNZ( vec3 );

   return rn;
}

void CLUFactorRational::vSolveLeftNoNZ( Rational* vec2,                            /* result2 */
                                         Rational* rhs2, int* ridx2, int rn2 )      /* rhs2    */
{

   if ( !l.updateType )          /* no Forest-Tomlin Updates */
   {
      rn2 = solveUpdateLeft( rhs2, ridx2, rn2 );
      solveUleftNoNZ( vec2, rhs2, ridx2, rn2 );
   }
   else
   {
      solveUleftNoNZ( vec2, rhs2, ridx2, rn2 );
      solveLleftForestNoNZ( vec2 );
   }

   solveLleftNoNZ( vec2 );
}
} // namespace soplex