Abstract
We investigate the behavior of the Lanczos process when it is used to find all the eigenvalues of large sparse symmetric matrices. We study the convergence of classical Lanczos (i.e., without re-orthogonalization) to the point where there is a cluster of Ritz values around each eigenvalue of the input matrix A. At that point, convergence to all the eigenvalues can be ascertained if A has no multiple eigenvalues. To eliminate multiple eigenvalues, we disperse them by adding to A a random matrix with a small norm; using high-precision arithmetic, we can perturb the eigenvalues and still produce accurate double-precision results. Our experiments indicate that the speed with which Ritz clusters form depends on the local density of eigenvalues and on the unit roundoff, which implies that we can accelerate convergence by using high-precision arithmetic in computations involving the Lanczos iterates.
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Notes
- 1.
We used the lapack unsymmetric eigensolver dgeev to compute the eigenvalues. Although the symmetric subroutine dsyev is more efficient, we did not use it here.
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Acknowledgments
We thank the referees for their valuable comments. The idea of using first-order corrections that we discuss in Sect. 2 was proposed by one of the referees.
This research was supported in part by grant 1045/09 from the Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities), and by grant 2010231 from the US–Israel Binational Science Foundation.
The first author was at Tel Aviv University while conducting this research.
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Alperovich, A., Druinsky, A., Toledo, S. (2014). Experiences with a Lanczos Eigensolver in High-Precision Arithmetic. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2013. Lecture Notes in Computer Science(), vol 8384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55224-3_4
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DOI: https://doi.org/10.1007/978-3-642-55224-3_4
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