### Abstract

Consider computing simple eigenvalues of a given compact infinite matrix re- garded as operating in the complex Hilbert space l^{2} 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 J_{0}(z) - iJ_{1}(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 J_{m}(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 l^{2} 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 language | English |
---|---|

Pages (from-to) | 35-70 |

Number of pages | 36 |

Journal | Linear Algebra and Its Applications |

Volume | 194 |

Issue number | C |

DOIs | |

Publication status | Published - 1993 Nov 15 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

_{0}(z) - iJ

_{1}(z) and of Bessel functions J

_{m}(z) of any real order m.

*Linear Algebra and Its Applications*,

*194*(C), 35-70. https://doi.org/10.1016/0024-3795(93)90112-2

**The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J _{0}(z) - iJ_{1}(z) and of Bessel functions J_{m}(z) of any real order m.** / Ikebe, Yasuhiko; Kikuchi, Yasushi; Fujishiro, Issei; Asai, Nobuyoshi; Takanashi, Kouichi; Harada, Minoru.

Research output: Contribution to journal › Article

_{0}(z) - iJ

_{1}(z) and of Bessel functions J

_{m}(z) of any real order m',

*Linear Algebra and Its Applications*, vol. 194, no. C, pp. 35-70. https://doi.org/10.1016/0024-3795(93)90112-2

_{0}(z) - iJ

_{1}(z) and of Bessel functions J

_{m}(z) of any real order m. Linear Algebra and Its Applications. 1993 Nov 15;194(C):35-70. https://doi.org/10.1016/0024-3795(93)90112-2

}

TY - JOUR

T1 - 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

AU - Ikebe, Yasuhiko

AU - Kikuchi, Yasushi

AU - Fujishiro, Issei

AU - Asai, Nobuyoshi

AU - Takanashi, Kouichi

AU - Harada, Minoru

PY - 1993/11/15

Y1 - 1993/11/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38248998697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38248998697&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(93)90112-2

DO - 10.1016/0024-3795(93)90112-2

M3 - Article

AN - SCOPUS:38248998697

VL - 194

SP - 35

EP - 70

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -