Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax=\lambda Bx$ with singular $B$

In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem $Ax=\lambda Bx$ are needed. If exact linear solves with $A-\sigma B$ 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.