Computation of multiple eigenvalues of infinite tridiagonal matrices
HTML articles powered by AMS MathViewer
- 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 $zJβ_{\nu }(z)+HJ_{\nu }(z)$; (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)β(C) for the computation of their βdouble pairsβ.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- Fayez A. Alhargan, A complete method for the computations of Mathieu characteristic numbers of integer orders, SIAM Rev. 38 (1996), no.Β 2, 239β255. MR 1391228, DOI 10.1137/1038040
- Tom M. Apostol, Mathematical analysis, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974. MR 0344384
- N. Asai, Y. Miyazaki, D. Cai, K. Hirasawa, and Y. Ikebe, Numerical Methods for $zJβ_{\nu }(z)+HJ_{\nu }(z)=0$ by Eigenvalue Problem, The Transactions of the Institute of Electronics, Information and Communication Engineers A, Vol. J79-A, No. 7 (1996), 1256-1265. (Later translated into English and appeared in Electronics and Communications in Japan, Part 3, Vol. 80, No. 7 (1997), 44-54.)
- Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24β82. MR 213062, DOI 10.1137/1009002
- Yasuhiko Ikebe, Nobuyoshi Asai, Yoshinori Miyazaki, and DongSheng Cai, The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994), 1996, pp.Β 599β618. MR 1400455, DOI 10.1016/0024-3795(95)00699-0
- Yasuhiko Ikebe, Yasushi Kikuchi, Issei Fujishiro, Nobuyoshi Asai, Kouichi Takanashi, and Minoru Harada, 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 $m$, Linear Algebra Appl. 194 (1993), 35β70. MR 1243819, DOI 10.1016/0024-3795(93)90112-2
- Y. Miyazaki, N. Asai, D. Cai, and Y. Ikebe, A Numerical Computation of the Inverse Characteristic Values of Mathieuβs Equation, Transactions of the Japan Society for Industrial and Applied Mathematics, 8(2), (1998), 199-222 (in Japanese).
- Y. Miyazaki, N. Asai, D. Cai, and Y. Ikebe, The Computation of Eigenvalues of Spheroidal Differential Equations by Matrix Method, JSIAM Annual Meeting, (1997), 224-225 (in Japanese).
- Yoshinori Miyazaki, Yasushi Kikuchi, DongSheng Cai, and Yasuhiko Ikebe, Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative, Math. Comp. 70 (2001), no.Β 235, 1195β1204. MR 1710636, DOI 10.1090/S0025-5718-00-01241-2
- Y. Miyazaki, Y. Kikuchi, D. Cai, and Y. Ikebe, The Computation of Double Eigenvalues for Infinite Matrices of a Certain Class with Newtonβs Method, Abstracts of Plenary and Invited Lectures Delivered at the Second Congress ISAAC 1999 (1999), 148β149.
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425β428. MR 6, DOI 10.2307/2303037
- Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 727118
Additional Information
- Yoshinori Miyazaki
- Affiliation: Faculty of Communications and Informatics, Shizuoka Sangyo University, Shizuoka 426-8668, Japan
- Email: yoshi@ssu.ac.jp
- Nobuyoshi Asai
- Affiliation: School of Computer Science and Engineering, University of Aizu, Fukushima-ken 965-8580, Japan
- Yasushi Kikuchi
- Affiliation: Faculty of Science, Division II, Tokyo University of Science, Tokyo, 162-8601, Japan
- DongSheng Cai
- Affiliation: Institute of Information Sciences and Electronics, University of Tsukuba, Ibaraki 305-8573, Japan
- Yasuhiko Ikebe
- Affiliation: Research Center for Information Science, Meisei University, Tokyo, 191-8506, Japan
- Received by editor(s): March 2, 2002
- Received by editor(s) in revised form: August 12, 2002
- Published electronically: June 19, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 719-730
- MSC (2000): Primary 34L16
- DOI: https://doi.org/10.1090/S0025-5718-03-01555-2
- MathSciNet review: 2031402