Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative. (English) Zbl 0971.34075
Summary: In 1975 one of the coauthors, Y. Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as an 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 Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi and M. Harada [Linear Algebra Appl. 194, 35-70 (1993; Zbl 0805.65037)], where an extremely accurate asymptotic error estimate is shown.
Here, the authors apply this error formula to the former case to obtain error formulas in a closed, explicit form.
The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi and M. Harada [Linear Algebra Appl. 194, 35-70 (1993; Zbl 0805.65037)], where an extremely accurate asymptotic error estimate is shown.
Here, the authors apply this error formula to the former case to obtain error formulas in a closed, explicit form.
MSC:
| 34L16 | Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators |
| 65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |
Keywords:
Coulomb wave function; eigenvalue problem for infinite matrices; three-term recurrence relations; error estimateCitations:
Zbl 0805.65037Software:
EISPACKReferences:
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| [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). |
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| [9] | Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. · Zbl 0732.47001 |
| [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). · Zbl 0325.65016 |
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