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Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $$Ax=\lambda Bx$$ with singular $$B$$. (English) Zbl 1133.65020
Summary: In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem $$Ax=\lambda Bx$$ are needed. If exact linear solvers 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 is presented that avoids most of the problems due to the singularity of $$B$$. Secondly, if exact solvers are not available, Jacobi-Davidson QZ will be presented as a robust method to compute a few specific eigenvalues. The results are illustrated by numerical experiments.

##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65F50 Computational methods for sparse matrices
##### Software:
JDQR; JDQZ; ARPACK; IRAM
Full Text:
##### References:
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