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Solution of a Nonlinear Eigenvalue Problem Using Signed Singular Values

Part of: Numerical linear algebra Basic linear algebra Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  31 January 2018

Kouhei Ooi ,
Yoshinori Mizuno ,
Tomohiro Sogabe ,
Yusaku Yamamoto  and
Shao-Liang Zhang
Kouhei Ooi
Affiliation:
Nagoya University, 1 Furo-cho, Chikusa, Nagoya, Aichi, 464-8603, Japan
Yoshinori Mizuno
Affiliation:
Meteorological Research Institute, 1-1 Nagamine, Tsukuba, Ibaraki, 305-0052, Japan
Tomohiro Sogabe
Affiliation:
Nagoya University, 1 Furo-cho, Chikusa, Nagoya, Aichi, 464-8603, Japan
Yusaku Yamamoto*
Affiliation:
The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan
Shao-Liang Zhang
Affiliation:
Nagoya University, 1 Furo-cho, Chikusa, Nagoya, Aichi, 464-8603, Japan
*
*Corresponding author. Email address:yusaku.yamamoto@uec.ac.jp (Y. Yamamoto)
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Abstract

We propose a robust numerical algorithm for solving the nonlinear eigenvalue problem A(ƛ)x = 0. Our algorithm is based on the idea of finding the value of ƛ for which A(ƛ) is singular by computing the smallest eigenvalue or singular value of A(ƛ) viewed as a constant matrix. To further enhance computational efficiency, we introduce and use the concept of signed singular value. Our method is applicable when A(ƛ) is large and nonsymmetric and has strong nonlinearity. Numerical experiments on a nonlinear eigenvalue problem arising in the computation of scaling exponent in turbulent flow show robustness and effectiveness of our method.

Keywords

Nonlinear eigenvalue problemsigned singular valuescaling exponentturbulent flow

MSC classification

Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 799 - 809
Copyright
Copyright © Global-Science Press 2017 

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