Computation of modified Bessel functions and their ratios

Amos, D. E.

doi:10.1090/S0025-5718-1974-0333287-7

Computation of modified Bessel functions and their ratios
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by D. E. Amos PDF

Math. Comp. 28 (1974), 239-251 Request permission

Abstract:

An efficient algorithm for calculating ratios ${r_v}(x) = {I_{v + 1}}(x)/{I_v}(x),v \geqq 0,x \geqq 0$, is presented. This algorithm in conjunction with the recursion relation for ${r_v}(x)$ gives an alternative to other recursive methods for ${I_v}(x)$ when approximations for low-order Bessel functions are available. Sharp bounds on ${r_v}(x)$ and ${I_v}(x)$ are also established in addition to some monotonicity properties of ${r_v}(x)$ and $r’_{v}(x)$.

References

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
D. E. Amos, Bounds on iterated coerror functions and their ratios, Math. Comp. 27 (1973), 413–427. MR 331723, DOI 10.1090/S0025-5718-1973-0331723-2

Evaluation of Some Cumulative Distribution Functions by Numerical Quadrature

C. W. Clenshaw, Chebyshev series for mathematical functions, National Physical Laboratory Mathematical Tables, Vol. 5, Her Majesty’s Stationery Office, London, 1962. Department of Scientific and Industrial Research. MR 0142793
C. W. Clenshaw and Susan 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 0203095
Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002
Yudell L. Luke, On generating Bessel functions by use of the backward recurrence formula, Aerospace Research Laboratories, Air Force Systems Command, United States Air Force, Wright-Patterson Air Force Base, Ohio, 1972. ARL 72-0030. MR 0326022
Yudell L. Luke, Inequalities for generalized hypergeometric functions, J. Approximation Theory 5 (1972), 41–65. MR 350082, DOI 10.1016/0021-9045(72)90028-7
Yudell L. Luke, The special functions and their approximations. Vol. II, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0249668
F. W. J. Olver, Numerical solution of second-order linear difference equations, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 111–129. MR 221789
F. W. J. Olver, Tables for Bessel functions of moderate or large orders, National Physical Laboratory Mathematical Tables, Vol. 6, Her Majesty’s Stationery Office, London, 1962. Department of Scientific and Industrial Research. MR 0147676
F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A 247 (1954), 328–368. MR 67250, DOI 10.1098/rsta.1954.0021