New backward recurrences for Bessel functions
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- by Henry C. Thacher PDF
- Math. Comp. 33 (1979), 744-764 Request permission
Abstract:
The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate ${J_v}(x)\;(|x|\;{\text {small}})$, ${e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})$, and the modulus and phase of $H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})$. The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for ${K_{v + 1}}(x)/{K_v}(x)$ the convergence of which increases with $|x|$. The procedures were tested numerically both for integer and fractional values of v, and for real and complex x.References
- W. G. Bickley, L. J. Comrie, J. C. P. Miller, D. H. Sadler, and A. J. Thompson, Bessel functions. Part II. Functions of positive integer order, British Association for the Advancement of Science, Mathematical Tables, vol. 10, University Press, Cambridge, 1952. MR 0050973
- Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002
- W. Gautschi, Zur Numerik rekurrenter Relationen, Computing (Arch. Elektron. Rechnen) 9 (1972), 107–126 (German, with English summary). MR 312714, DOI 10.1007/bf02236961
- Walter Gautschi, Computational methods in special functions—a survey, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 1–98. MR 0391476
- Henry C. Thacher Jr., Series solutions to differential equations by backward recurrence, Information processing 71 (Proc. IFIP Congress, Ljubljana, 1971) North-Holland, Amsterdam, 1972, pp. 1287–1291. MR 0461923
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
- Sin Hitotumatu, On the numerical computation of Bessel functions through continued fraction, Comment. Math. Univ. St. Paul. 16 (1967/68), 89–113. MR 233492
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
- Boro Döring, Complex zeros of cylinder functions, Math. Comp. 20 (1966), 215–222. MR 192632, DOI 10.1090/S0025-5718-1966-0192632-1
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 744-764
- MSC: Primary 65D20; Secondary 33A40
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521289-1
- MathSciNet review: 521289