Calculation of derivatives of multiple eigenpairs of unsymmetrical quadratic eigenvalue problems

Abstract

New methods are presented for computing the derivatives of multiple eigenvalues and the corresponding eigenvectors of unsymmetrical quadratic eigenvalue problems. The expressions of eigenpair derivatives are derived in terms of the eigenvalues and eigenvectors of quadratic eigenvalue problems, and the use of rather undesirable state-space representation is avoided. Hence the cost of computation is greatly reduced. The proposed methods are valid for both the case of distinct eigenvalue derivatives and the case of equal eigenvalue derivatives. Numerical results show that the proposed methods are efficient.

Keywords:

• eigenvalue derivatives
• eigenvector derivatives
• sensitivity analysis
• quadratic eigenvalue problems
• multiple eigenvalues

2000 AMS Subject Classification :

• 65F15
• 15A18

Acknowledgements

This research was supported by the Mathematical Tianyuan Foundation of China (No.10626019) We thank the referees for comments which helped improve the presentation of our results.