A Successive Approximation Algorithm with Rayleigh Functionals for Solving Nonlinear Eigenvalue Problems

doi:10.1587/transfun.2024EAP1118

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A Successive Approximation Algorithm with Rayleigh Functionals for Solving Nonlinear Eigenvalue Problems

2025-12-01

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Takafumi Miyata *

DOI：10.1587/transfun.2024EAP1118

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Abstract

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This paper presents an iterative algorithm for computing an eigenvalue close to a user-specified value and its corresponding eigenvector of a nonlinear eigenvalue problem. This algorithm iterates two parts alternately. The first part is the existing algorithm called the successive approximation algorithm, where the Taylor expansion of a matrix is used to transform the nonlinear problem to the linear problem. By solving the linear problem, an approximate eigenvalue and an approximate eigenvector of the nonlinear problem are computed. The second part refines the approximate eigenvalue computed by the first part. To this end, we approximately compute the Rayleigh functional, which is the solution of the nonlinear equation defined by the approximate eigenvector, and use it as a new approximate eigenvalue. Experimental results show that a combination of the successive approximation algorithm and the Rayleigh functionals converges within fewer iterations and requires less computational time in comparison with the existing successive approximation algorithms.

Keywords:

nonlinear eigenvalue problem

Taylor expansion of a matrix

successive approximation algorithm

Rayleigh functional