Computation of modified Bessel functions and their ratios

Author:
D. E. Amos

Journal:
Math. Comp. **28** (1974), 239-251

MSC:
Primary 33A40; Secondary 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1974-0333287-7

MathSciNet review:
0333287

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An efficient algorithm for calculating ratios , is presented. This algorithm in conjunction with the recursion relation for gives an alternative to other recursive methods for when approximations for low-order Bessel functions are available. Sharp bounds on and are also established in addition to some monotonicity properties of and .

**[1]**M. Abramowitz & I. A. Stegun (Editors),*Handbook of Mathematical Functions, With Formulas, Graphs and Mathematical Tables*, Nat. Bur. Standards Appl. Math. Ser., 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**29**#4914. MR**0167642 (29:4914)****[2]**D. E. Amos, "Bounds on iterated coerror functions and their ratios,"*Math. Comp.*, v. 27, 1973, pp. 413-427. MR**0331723 (48:10055)****[3]**D. E. Amos,*Evaluation of Some Cumulative Distribution Functions by Numerical Quadrature*, Symposium Proceedings, Sixth Annual Symposium of Interface: Computer Science and Statistics, University of California, Berkeley, October 16-17, 1972.**[4]**C. W. Clenshaw,*Chebyshev Series for Mathematical Functions*, National Physical Laboratory Mathematical Tables, vol. 5, Department of Scientific and Industrial Research, Her Majesty's Stationery Office, London, 1962. MR**26**#362. MR**0142793 (26:362)****[5]**C. W. Clenshaw & S. M. Picken,*Chebyshev Series for Bessel Functions of Fractional Order*, National Physical Laboratory Mathematical Tables, vol. 8, Her Majesty's Stationery Office, London, 1966. MR**34**#2948. MR**0203095 (34:2948)****[6]**W. Gautschi, "Computational aspects of three-term recurrence relations,"*SIAM Rev.*, v. 9, 1967, pp. 24-82. MR**35**#3927. MR**0213062 (35:3927)****[7]**Y. L. Luke,*On Generating Bessel Functions by Use of the Backward Recurrence Formula*, Aerospace Research Laboratories Report ARL 72-0030, Wright-Patterson Air Force Base, Ohio. MR**0326022 (48:4368)****[8]**Y. L. Luke, "Inequalities for generalized hypergeometric functions,"*J. Approximation Theory*, v. 5, 1972, pp. 41-65. MR**0350082 (50:2575)****[9]**Y. L. Luke,*The Special Functions and Their Approximations*. Vol. II, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR**40**#2909. MR**0249668 (40:2909)****[10]**F. W. J. Olver, "Numerical solution of second order linear difference equations,"*J. Res. Nat. Bur. Standards Sect. B*, v. 71B, 1967, pp. 111-129. MR**36**#4841. MR**0221789 (36:4841)****[11]**F. W. J. Olver,*Tables for Bessel Functions of Moderate or Large Orders*, National Physical Laboratory Mathematical Tables, vol. 6, Department of Scientific and Industrial Research, Her Majesty's Stationery Office, London, 1962. MR**26**#5190. MR**0147676 (26:5190)****[12]**F. W. J. Olver, "The asymptotic expansion of Bessel functions of large order,"*Philos. Trans. Roy Soc. London Ser. A*, v. 247. 1954, pp. 328-368. MR**16**, 696. MR**0067250 (16:696a)**

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0333287-7

Keywords:
Modified Bessel functions,
recursive computation,
ratios of Bessel functions,
bounds on Bessel functions

Article copyright:
© Copyright 1974
American Mathematical Society