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On Chebyshev–Davidson Method for Symmetric Generalized Eigenvalue Problems

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Abstract

As we know, polynomial filtering technique is efficient for accelerating convergence of standard eigenvalue problems, which, however, has not appeared for solving generalized eigenvalue problems. In this paper, by integrating the effectiveness and robustness of the Chebyshev polynomial filters, we propose the Chebyshev–Davidson method for computing some extreme eigenvalues and corresponding eigenvectors of generalized matrix pencils. In this method, both matrix factorizations and solving systems of linear equations are all avoided. Convergence analysis indicates that the Chebyshev–Davidson method achieves quadratic convergence locally in an ideal situation. Furthermore, numerical experiments are carried out to demonstrate the convergence properties and to show great superiority and robustness over some state-of-the art iteration methods.

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Data Availability

The data used to support the findings of this study are included within the article.

Notes

  1. Matlab Code available at http://www.math.uu.nl/people/sleijpen/.

  2. Matlab Code available at http://www-math.cudenver.edu/~aknyazev/software/CG/toward/lobpcg.m.

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Acknowledgements

The author is very much indebted to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.

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  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, People’s Republic of China

    Cun-Qiang Miao

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Supported by The National Natural Science Foundation of China (No. 11901361), P. R. China.

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Miao, CQ. On Chebyshev–Davidson Method for Symmetric Generalized Eigenvalue Problems. J Sci Comput 85, 53 (2020). https://doi.org/10.1007/s10915-020-01360-4

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  • DOI: https://doi.org/10.1007/s10915-020-01360-4

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