// Reverse preconditioning step 2: inverse balancing.
  // Done in two steps: inverse scaling, then inverse permutation

  // inverse scaling (diagonal transformation)
  for (octave_idx_type i = 0; i < nc; i++)
    for (octave_idx_type j = 0; j < nc; j++)
       retval(i,j) *= dscale(i) / dscale(j);

  OCTAVE_QUIT;

  // construct balancing permutation vector
  Array<octave_idx_type> iperm (nc);
  for (octave_idx_type i = 0; i < nc; i++)
    iperm(i) = i;  // initialize to identity permutation

  // trailing permutations must be done in reverse order
  for (octave_idx_type i = nc - 1; i >= ihi; i--)
    {
      octave_idx_type swapidx = static_cast<octave_idx_type> (dpermute(i)) - 1;
      octave_idx_type tmp = iperm(i);
      iperm(i) = iperm(swapidx);
      iperm(swapidx) = tmp;
    }

  // leading permutations in forward order
  for (octave_idx_type i = 0; i < (ilo-1); i++)
    {
      octave_idx_type swapidx = static_cast<octave_idx_type> (dpermute(i)) - 1;
      octave_idx_type tmp = iperm(i);
      iperm(i) = iperm(swapidx);
      iperm(swapidx) = tmp;
    }

  // construct inverse balancing permutation vector
  Array<octave_idx_type> invpvec (nc);
  for (octave_idx_type i = 0; i < nc; i++)
    invpvec(iperm(i)) = i;     // Thanks to R. A. Lippert for this method

  OCTAVE_QUIT;

  ComplexMatrix tmpMat = retval;
  for (octave_idx_type i = 0; i < nc; i++)
    for (octave_idx_type j = 0; j < nc; j++)
      retval(i,j) = tmpMat(invpvec(i),invpvec(j));

  // Reverse preconditioning step 1: fix trace normalization.