Abstract
The computation of a defective eigenpair of nonlinear algebraic eigenproblems of the form \(T(\lambda )v=0\) is challenging due to its ill-posedness and the linear convergence of classical single-vector Newton-like methods. In this paper, we propose and study new accelerated Newton-like methods for defective eigenvalues which exhibit quadratic local convergence at the cost of solving two linear systems per iteration. To the best of our knowledge, the accelerated algorithms are the most efficient methods for solving defective eigenpairs. The analyses are illustrated by numerical experiments.
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Notes
Note that \(mirror\) is a special problem for which classical Rayleigh functional iteration (RFI) converges quadratically [9, Section 4.4], and thus there is actually no need to use accelerated algorithms.
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We thank the referees for their comments and suggestions, which helped improve our presentation.
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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: II. Accelerated algorithms. Numer. Math. 129, 383–403 (2015). https://doi.org/10.1007/s00211-014-0640-2
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DOI: https://doi.org/10.1007/s00211-014-0640-2