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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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


Author: J. M. Blair
Journal: Math. Comp. 28 (1974), 581-583
MSC: Primary 65D20
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}}$.


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  • [1] 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.
  • [2] C. W. Clenshaw, Chebyshev Series for Mathematical Functions, Nat. Phys. Lab. Math. Tables, vol. 5, Her Majesty's Stationery Office, London, 1962. MR 26 #362. MR 0142793 (26:362)
  • [3] J. Wimp, "Polynomial expansions of Bessel functions and some associated functions," Math. Comp., v. 16, 1962, pp. 446-458. MR 26 #6452. MR 0148956 (26:6452)
  • [4] 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.
  • [5] 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.
  • [6] J. F. Hart et al., Computer Approximations, Wiley, New York, 1968.
  • [7] 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.
  • [8] 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1974-0341810-1
PII: S 0025-5718(1974)0341810-1
Keywords: Rational Chebyshev approximations, Bessel functions
Article copyright: © Copyright 1974 American Mathematical Society