### 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 |
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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 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

## Fingerprint Dive into the research topics of 'The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J<sub>0</sub>(z) - iJ<sub>1</sub>(z) and of Bessel functions J<sub>m</sub>(z) of any real order m'. Together they form a unique fingerprint.

## 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