Journal
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume 34, Issue 3, Pages 1089-1111Publisher
SIAM PUBLICATIONS
DOI: 10.1137/120879841
Keywords
derivatives of eigenvalues and eigenvectors; multiple eigenvalues; very close eigenvalues; quadratic eigenvalue problems
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Funding
- NSFC [10901024]
- Fundamental Research Funds for the Central Universities [BUPT2013RC0903]
- NUS [R-146-000-140-112]
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We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.
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