Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems Ax=λBx with singular B

In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem Ax = ABx are needed. If exact linear solves with A - σB are available, implicitly restarted Arnoldi with purification is a common approach for problems where B is positive semidefinite. In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of the problems due to the singularity of B. Secondly, if exact solves are not available, Jacobi-Davidson QZ will be presented as a robust method to compute a few specific eigenvalues. Results are illustrated by numerical experiments.

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