Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative

Authors:
Yoshinori Miyazaki, Yasushi Kikuchi, DongSheng Cai and Yasuhiko Ikebe

Journal:
Math. Comp. **70** (2001), 1195-1204

MSC (2000):
Primary 34L16

DOI:
https://doi.org/10.1090/S0025-5718-00-01241-2

Published electronically:
March 24, 2000

MathSciNet review:
1710636

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Abstract | References | Similar Articles | Additional Information

In 1975 one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices. Approximation by truncation is justified but no error estimates are given there.

The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by Ikebe et al. in 1993, where an extremely accurate asymptotic error estimate is shown.

In this paper, we apply this error formula to the former case to obtain error formulas in a closed, explicit form.

**1.**M. Abramowitz,*Asymptotic Expansions of Coulomb Wave Functions*, MR, Vol. VII, No. 1 (1949), 75-84.MR**10:454a****2.**M. Abramowitz and I. A. Stegun,*Handbook of Mathematical Functions*, Dover, N.Y., (1972).MR**34:8606****3.**N. Asai, Y. Miyazaki, D. Cai, K. Hirasawa, and Y. Ikebe,*Matrix Methods for the Numerical Solution of*, Electronics and Communications in Japan, Part 3, Vol. 80, No. 7 (1997), 44-54.**4.**W. Gautschi,*Computational Aspects of Three-Term Recurrence Relations*, SIAM Rev., 9 (1967), 24-82.MR**35:3927****5.**Y. Ikebe,*The Zeros of Regular Coulomb Wave Functions and of Their Derivatives*, Math. Comp., 29(131), (1975), 878-887.MR**51:14529****6.**Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada,*The Eigenvalue Problem for Infinite Compact Complex Symmetric Matrices with Application to the Numerical Computation of Complex Zeros of and of Bessel Functions of Any Real Order*, Linear Algebra Appl., 194 (1993), 35-70.MR**94g:47025****7.**Y. Miyazaki, N. Asai, D. Cai, and Y. Ikebe,*A Numerical Computation of the Inverse Characteristic Values of Mathieu's Equation*, Transactions of the Japan Society for Industrial and Applied Mathematics, 8(2), (1998), 199-222 (in Japanese).**8.**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).**9.**F. Riesz and B. S. Nagy,*Functional Analysis*, Dover, N.Y., (1990).MR**91g:00002****10.**B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler,*Matrix Eigensystem Routines - EISPACK Guide, Second Edition*, Springer-Verlag, (1976).MR**58:1366a**

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Additional Information

**Yoshinori Miyazaki**

Affiliation:
Faculty of Communications and Informatics, Shizuoka Sangyo University, Surugadai 4-1-1, Fujieda, Shizuoka, 426-8668, Japan

Email:
yoshi@fujieda-ssu.ac.jp

**Yasushi Kikuchi**

Affiliation:
Department of Computer Software, The University of Aizu, Tsuruga, Ikkimachi, Aizuwakamatsu, Fukushima, 965-8580, Japan

Email:
kikuchi@u-aizu.ac.jp

**DongSheng Cai**

Affiliation:
Institute of Information Sciences and Electronics, The University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8573, Japan

Email:
cai@is.tsukuba.ac.jp

**Yasuhiko Ikebe**

Affiliation:
Department of Computer Software, The University of Aizu, Tsuruga, Ikkimachi, Aizuwakamatsu, Fukushima, 965-8580, Japan

Email:
ikebe@u-aizu.ac.jp

DOI:
https://doi.org/10.1090/S0025-5718-00-01241-2

Keywords:
Coulomb wave function,
eigenvalue problem for infinite matrices,
three-term recurrence relations,
error estimate

Received by editor(s):
July 27, 1999

Published electronically:
March 24, 2000

Article copyright:
© Copyright 2000
American Mathematical Society