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A rational approximation method for the nonlinear eigenvalue problem

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arxiv 1901.01188 v2 pith:TCUJ5IKW submitted 2019-01-04 math.NA cs.NA

Yousef Saad , Mohamed El-Guide , Agnieszka Międlar This is my paper · ORCID
classification math.NAcs.NA
keywords eigenvaluemethodproblemeigenvaluesformfunctionslinearizationnonlinear
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This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a form of linearization. Eigenpairs of the expanded form of this linearization are not extracted directly. Instead, its structure is exploited to develop a scheme that allows to extract all eigenvalues in a certain region of the complex plane by solving an eigenvalue problem of much smaller dimension. Because of its simple implementation and the ability to work efficiently in large dimensions, the presented method is appealing when solving challenging engineering problems. A few theoretical results are established to explain why the new approach works and numerical experiments are presented to validate the proposed algorithm.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 6 citations worldwide. Full citation record

  1. Convergence analysis of a nonlinear eigensolver based on rational approximation of the resolvent

    math.NA 2026-07 accept novelty 6.0 of 10

    Block probing and zooming-in make resolvent-polefinding for nonlinear eigenproblems accurate and multiplicity-aware, with proved rates and stable barycentric rootfinding.

  2. Rational Minimax Approximations for Matrix-Valued Functions: Existence, Optimality and Algorithms

    math.OC 2026-06 conditional novelty 6.0 of 10

    Matrix-valued rational minimax approximants with a common denominator are proved to exist on dense point sets, with Kolmogorov/Ruttan optimality certificates and an equivalence to the m-d-Lawson dual conditions.

  3. Rational minimax approximation of matrix-valued functions

    math.NA 2025-08 conditional novelty 6.0 of 10

    The paper develops a duality-based framework and an iterative algorithm (m-d-Lawson) for discrete minimax rational approximation of matrix-valued functions with a common denominator.

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