Computation of multiple eigenvalues of infinite tridiagonal matrices

Miyazaki, Yoshinori; Asai, Nobuyoshi; Kikuchi, Yasushi; Cai, DongSheng; Ikebe, Yasuhiko

doi:10.1090/S0025-5718-03-01555-2

Computation of multiple eigenvalues of infinite tridiagonal matrices
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by Yoshinori Miyazaki, Nobuyoshi Asai, Yasushi Kikuchi, DongSheng Cai and Yasuhiko Ikebe PDF

Math. Comp. 73 (2004), 719-730 Request permission

Abstract:

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of $J_{\nu }(z)$; (B) the zeros of $zJ’_{\nu }(z)+HJ_{\nu }(z)$; (C) the EVP of the Mathieu differential equation; and (D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)–(C) for the computation of their “double pairs”.

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