The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J0(z) - iJ1(z) and of Bessel functions Jm(z) of any real order m

Yasuhiko Ikebe, Yasushi Kikuchi, Issei Fujishiro, Nobuyoshi Asai, Kouichi Takanashi, Minoru Harada

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Consider computing simple eigenvalues of a given compact infinite matrix re- garded as operating in the complex Hilbert space l2 by computing the eigenvalues of the truncated finite matrices and taking an obvious limit. In this paper we deal with a special case where the given matrix is compact, complex, and symmetric (but not necessarily Hermitian). Two examples of application are studied. The first is con- cerned with the equation J0(z) - iJ1(z)=0 appearing in the analysis of the solitary-wave runup on a sloping beach, and the second with the zeros of the Bessel function Jm(z) of any real order m. In each case, the problem is reformulated as an eigenvalue problem for a compact complex symmetric tridiagonal matrix operator in l2 whose eigenvalues are all simple. A complete error analysis for the numerical solution by truncation is given, based on the general theorems proved in this paper, where the usefulness of the seldom used generalized Rayleigh quotient is demonstrated.

Original languageEnglish
Pages (from-to)35-70
Number of pages36
JournalLinear Algebra and Its Applications
Volume194
Issue numberC
DOIs
Publication statusPublished - 1993 Nov 15
Externally publishedYes

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Complex Symmetric Matrices
Bessel functions
Bessel Functions
Numerical Computation
Eigenvalue Problem
Eigenvalue
Zero
Rayleigh quotient
Infinite Matrices
Computing
Tridiagonal matrix
Solitary Waves
Symmetric matrix
Error Analysis
Truncation
Hilbert spaces
Beaches
Hilbert space
Solitons
Numerical Solution

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis

Cite this

The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J0(z) - iJ1(z) and of Bessel functions Jm(z) of any real order m. / Ikebe, Yasuhiko; Kikuchi, Yasushi; Fujishiro, Issei; Asai, Nobuyoshi; Takanashi, Kouichi; Harada, Minoru.

In: Linear Algebra and Its Applications, Vol. 194, No. C, 15.11.1993, p. 35-70.

Research output: Contribution to journalArticle

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