Optimal Scaling of Generalized and Polynomial Eigenvalue Problems

Scaling is a commonly used technique for standard eigenvalue problems to improve the sensitivity of the eigenvalues. In this paper we investigate scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal scaling of a PEP with respect to an eigenvalue can be described by the ratio of its normwise and componentwise condition number. Furthermore, the effect of linearization on optimally scaled polynomials is investigated. We introduce a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available and also discuss variable transformations of the type $\lambda=\alpha\mu$ for PEPs of arbitrary degree.

[1]  E. E. Osborne.  On Pre-Conditioning of Matrices , 1960, JACM.

[2]  J. Demmel,et al.  Balancing sparse matrices for computing eigenvalues , 2000 .

[3]  Robert C. Ward,et al.  Balancing the Generalized Eigenvalue Problem , 1981 .

[4]  Nicholas J. Higham,et al.  Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications , 2001, SIAM J. Matrix Anal. Appl..

[5]  Paul Van Dooren,et al.  Normwise Scaling of Second Order Polynomial Matrices , 2004, SIAM J. Matrix Anal. Appl..

[6]  Paul Van Dooren,et al.  Balancing Regular Matrix Pencils , 2006, SIAM J. Matrix Anal. Appl..

[7]  Daniel Kressner,et al.  Numerical Methods for General and Structured Eigenvalue Problems , 2005, Lecture Notes in Computational Science and Engineering.

[8]  Volker Mehrmann,et al.  Vector Spaces of Linearizations for Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[9]  N. Higham.  A Survey of Componentwise Perturbation Theory in Numerical Linear Algebra , 1994 .

[10]  Nicholas J. Higham,et al.  Structured Backward Error and Condition of Generalized Eigenvalue Problems , 1999, SIAM J. Matrix Anal. Appl..

[11]  Nicholas J. Higham,et al.  The Conditioning of Linearizations of Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[12]  D. S. Watkins.  A CASE WHERE BALANCING IS HARMFUL (cid:3) , 2005 .

[13]  C. Reinsch,et al.  Balancing a matrix for calculation of eigenvalues and eigenvectors , 1969 .

[14]  Nicholas J. Higham,et al.  Backward Error of Polynomial Eigenproblems Solved by Linearization , 2007, SIAM J. Matrix Anal. Appl..

[15]  Frann Coise Tisseur.  Backward Error and Condition of Polynomial Eigenvalue Problems , 1999 .