  Balancing a matrix is a preprocessing step while solving the nonsymmetric
eigenvalue problem. Balancing a matrix reduces the norm of the matrix and
hopefully this will improve the accuracy of the computation. Experiments have
shown that balancing can improve the accuracy of the computed eigenval- ues.
However, there exists examples where balancing increases the eigenvalue
condition number (potential loss in accuracy), deteriorates eigenvector
accuracy, and deteriorates the backward error of the eigenvalue decomposition.
In this paper we propose a change to the stopping criteria of the LAPACK
balancing al- gorithm, GEBAL. The new stopping criteria is better at
determining when a matrix is nearly balanced. Our experiments show that the new
algorithm is able to maintain good backward error, while improving the
eigenvalue accuracy when possible. We present stability analysis, numerical
experiments, and a case study to demonstrate the benefit of the new stopping
criteria.
