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Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear eigenproblems. I. Classical algorithms

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Abstract

We study the local convergence rates of several most widely used single-vector Newton-like methods for the solution of a degenerate eigenvalue of nonlinear algebraic eigenvalue problems of the form \(T(\lambda )v=0\). This problem has not been completely understood, since the Jacobian associated with Newton’s method is singular at the desired eigenpair, and the standard convergence theory is not applicable. In fact, Newton’s method generally converges only linearly towards singular roots. In this paper, we show that the local convergence of inverse iteration, Rayleigh functional iteration and the Jacobi–Davidson method are at least quadratic for semi-simple eigenvalues. For defective eigenvalues, Newton-like methods converge only linearly in general. The results are illustrated by numerical experiments.

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Acknowledgments

We thank the referees for their comments and suggestions, which helped improve our presentation.

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Authors and Affiliations

  1. Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA, 19122-6094, USA

    Daniel B. Szyld

  2. Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 41010, Lafayette, LA, 70504-1010, USA

    Fei Xue

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  1. Daniel B. Szyld
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  2. Fei Xue
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Correspondence to Fei Xue.

Additional information

This work was supported by the U. S. National Science Foundation under grant DMS-1115520.

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Szyld, D.B., Xue, F. Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear eigenproblems. I. Classical algorithms. Numer. Math. 129, 353–381 (2015). https://doi.org/10.1007/s00211-014-0639-8

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  • DOI: https://doi.org/10.1007/s00211-014-0639-8

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