[Paper Review] On matrix balancing and eigenvector computation

Rodney James, Julien Langou+1 more|arXiv (Cornell University)|Jan 22, 2014

Matrix Theory and Algorithms9 references3 citations

TL;DR

This paper proposes a modified stopping criterion for the LAPACK GEBAL algorithm that incorporates diagonal elements in the balancing norm computation, using the 2-norm to prevent over-balancing. The new criterion maintains backward error stability while improving eigenvalue accuracy in cases where traditional balancing degrades accuracy, particularly for dense, poorly scaled matrices.

ABSTRACT

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.

Motivation & Objective

• To address the issue that standard matrix balancing can deteriorate eigenvalue and eigenvector accuracy and increase backward error in certain cases.
• To improve the stopping criterion of the LAPACK GEBAL algorithm by including diagonal elements in the norm computation.
• To ensure that balancing does not over-apply when the matrix is already nearly balanced, thus preserving backward error.
• To demonstrate through analysis and experiments that the new criterion maintains or improves eigenvalue accuracy without degrading numerical stability.
• To provide a practical fix that enhances reliability of eigensolvers for dense, ill-conditioned matrices.

Proposed method

• Modify the GEBAL stopping criterion to use the 2-norm of the entire row and column, including diagonal elements, rather than excluding them.
• Use the 2-norm to compute the balancing factor f = sqrt(r/c), where r and c are the 2-norms of the off-diagonal row and column, respectively.
• Apply the diagonal similarity transformation D^{-1}AD iteratively until the norm reduction falls below a threshold based on the full row and column norms.
• Introduce a new convergence test that prevents over-balancing by detecting when further iterations would not meaningfully improve balance.
• Use backward error analysis to show that the new criterion avoids increasing the backward error of the eigenvalue decomposition.
• Validate the method on test matrices, including a detailed case study with a Hessenberg-like matrix where traditional balancing fails.

Experimental results

Research questions

• RQ1 Does including diagonal elements in the balancing norm improve backward error stability?
• RQ2 Can a modified stopping criterion prevent over-balancing and preserve numerical accuracy in eigenvalue computations?
• RQ3 How does the new algorithm compare to the original GEBAL in terms of eigenvalue and eigenvector accuracy?
• RQ4 In what cases does the new criterion prevent deterioration of the eigenvalue condition number?
• RQ5 Is the 2-norm-based balancing criterion more robust than the current norm-ignoring approach for ill-scaled matrices?

Key findings

• The new stopping criterion prevents over-balancing by including diagonal elements in the norm computation, reducing the risk of deteriorating backward error.
• Numerical experiments show that the modified algorithm maintains good backward error while improving eigenvalue accuracy in cases where the original GEBAL fails.
• For the case study matrix with small ε, the new algorithm avoids increasing the eigenvalue condition number, unlike the original GEBAL.
• The 2-norm-based balancing prevents balancing of Hessenberg matrices in a way that degrades accuracy, which was a known failure case of the original algorithm.
• The new method improves eigenvalue accuracy when the matrix is poorly scaled, especially when accurate eigenvectors are not required.
• The backward error analysis confirms that the new criterion avoids increasing the backward error of the eigenvalue decomposition, even when the norm reduction is small.