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Rational Chebyshev approximations for the modified Bessel functions $I_{0}(x)$ and $I_{1}(x)$


Author: J. M. Blair
Journal: Math. Comp. 28 (1974), 581-583
MSC: Primary 65D20
DOI: https://doi.org/10.1090/S0025-5718-1974-0341810-1
MathSciNet review: 0341810
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Abstract: This note presents nearly-best rational approximations for the functions ${I_0}(x)$ and ${I_1}(x)$, with relative errors ranging down to ${10^{ - 23}}$.


References [Enhancements On Off] (What's this?)

    Y. L. Luke, The Special Functions and Their Approximations. Vol. 2, Math. in Science and Engineering, vol. 53, Academic Press, New York, 1969. MR 40 #2909.
  • 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
  • Jet Wimp, Polynomial expansions of Bessel functions and some associated functions, Math. Comp. 16 (1962), 446–458. MR 148956, DOI https://doi.org/10.1090/S0025-5718-1962-0148956-3
  • I. Gargantini, "On the application of the process of equalization of maxima to obtain rational approximations to certain modified Bessel functions," Comm. ACM, v. 9, 1966, pp. 859-863. A. E. Russon & J. M. Blair, Rational Function Minimax Approximations for the Bessel Functions ${K_0}(x)$ and ${K_1}(x)$, Report AECL-3461, Atomic Energy of Canada Limited, Chalk River, Ontario, 1969. J. F. Hart et al., Computer Approximations, Wiley, New York, 1968. J. H. Johnson & J. M. Blair, REMES 2—A FORTRAN Program to Calculate Rational Minimax Approximations to a Given Function, Report AECL-4210, Atomic Energy of Canada Limited, Chalk River, Ontario, 1973. B. S. Berger & H. McAllister, "A table of the modified Bessel functions ${K_n}(x)$ and ${I_n}(x)$ to at least 60S for $n = 0,1$ and $x = 1,2, \cdots ,40,$" Math. Comp., v. 24, 1970, p. 488, RMT 34.

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Keywords: Rational Chebyshev approximations, Bessel functions
Article copyright: © Copyright 1974 American Mathematical Society