New backward recurrences for Bessel functions

Author:
Henry C. Thacher

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate , , and the modulus and phase of . 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 the convergence of which increases with . The procedures were tested numerically both for integer and fractional values of *v*, and for real and complex *x*.

**[1]**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. X, University Press, Cambridge, 1952. MR**0050973****[2]**Walter Gautschi,*Computational aspects of three-term recurrence relations*, SIAM Rev.**9**(1967), 24–82. MR**0213062**, https://doi.org/10.1137/1009002**[3]**W. Gautschi,*Zur Numerik rekurrenter Relationen*, Computing (Arch. Elektron. Rechnen)**9**(1972), 107–126 (German, with English summary). MR**0312714****[4]**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) Academic Press, New York, 1975, pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35. MR**0391476****[5]**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****[6]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[7]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[8]**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697****[9]**Sin Hitotumatu,*On the numerical computation of Bessel functions through continued fraction*, Comment. Math. Univ. St. Paul.**16**(1967/1968), 89–113. MR**0233492****[10]**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****[11]**Boro Döring,*Complex zeros of cylinder functions*, Math. Comp.**20**(1966), 215–222. MR**0192632**, https://doi.org/10.1090/S0025-5718-1966-0192632-1

Retrieve articles in *Mathematics of Computation*
with MSC:
65D20,
33A40

Retrieve articles in all journals with MSC: 65D20, 33A40

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521289-1

Keywords:
Bessel functions,
modified Bessel functions,
Airy function,
Hankel function,
Kelvin function,
power series,
continued fraction,
differential equation,
recurrence,
Miller algorithm

Article copyright:
© Copyright 1979
American Mathematical Society