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Rational Chebyshev approximations for the Bessel functions $ J\sb{0}(x)$, $ J\sb{1}(x)$, $ Y\sb{0}(x)$, $ Y\sb{1}(x)$


Authors: C. A. Wills, J. M. Blair and P. L. Ragde
Journal: Math. Comp. 39 (1982), 617-623
MSC: Primary 65D20; Secondary 33A40, 41A50
DOI: https://doi.org/10.1090/S0025-5718-1982-0669653-4
MathSciNet review: 669653
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Abstract: This report presents near-minimax rational approximations for the Bessel functions $ {J_0}(x)$, $ {J_1}(x)$, $ {Y_0}(x)$, and $ {Y_1}(x)$ for the complete range of x, with relative errors ranging down to $ {10^{ - 23}}$. The first thirty zeros of each function are listed to 35D. The tabulated zeros and the McMahon asymptotic formulae may be used to construct an algorithm which retains relative accuracy in the neighborhood of zeros.


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  • [1] M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, Nat. Bur. Standards, Appl. Math. Series No. 55, U. S. Government Printing Office, Washington, D. C., 1965.
  • [2] R. P. Brent, "A FORTRAN Multiple-Precision Arithmetic Package," ACM Trans. Math. Software, v. 4, 1978, pp. 57-70.
  • [3] C W. Clenshaw & Susan M. Picken, Chebyshev Series for Bessel Functions of Fractional Order, Nat. Phys. Lab. Math. Tables, v. 8, Her Majesty's Stationery Office, London, 1956. MR 0203095 (34:2948)
  • [4] W. J. Cody, "The FUNPACK Package of Special Function Subroutines," ACM Trans. Math. Software, v. 1, 1975, pp. 13-25.
  • [5] W. J. Cody, R. M. Motley & L. W. Fullerton, "Coefficients for the approximation of $ {Y_v}(x)$," AMD Technical Memorandum #284, Argonne National Laboratory. (In preparation.)
  • [6] D. Döring, "Über die McMahon-Entwicklungen," Z. Angew. Math. Phys., v. 18, 1967, pp. 461-473. MR 0214272 (35:5123)
  • [7] J. F. Hart, et al., Computer Approximations, Wiley, New York, 1968.
  • [8] K. Hayashi, Tafeln der Besselschen, Theta, Kugel--und anderer Funktionen, Springer-Verlag, Berlin, 1930.
  • [9] J. H. Johnson & J. M. Blair, REMES2: A FORTRAN Program to Calculate Rational Minimax Approximations to a Given Function, Atomic Energy of Canada Limited, Report AECL-4210, Chalk River, Ontario, 1973.
  • [10] Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, 1975. MR 0501762 (58:19039)
  • [11] S. Makinouchi, Zeros of Bessel Functions $ {J_v}(x)$ and $ {Y_v}(x)$ Accurate to Twenty-Nine Significant Digits, Osaka University Technology Report No. 685, 1965.
  • [12] C. Mesztenyi & C. Witzgall, "Stable evaluation of polynomials," J. Res. Nat. Bur. Standards Sect. B, v. 71B, 1967, pp. 11-17. MR 0212994 (35:3859)
  • [13] F. W. J. Olver (Ed.), Bessel Functions, Part III. Zeros and Associated Values, Royal Society Mathematical Tables, v. 7, Cambridge Univ. Press, Cambridge, 1960. MR 0119441 (22:10202)
  • [14] J. Wimp, "Polynomial expansions of Bessel functions and some associated functions," Math. Comp., v. 16, 1962, pp. 446-458. MR 0148956 (26:6452)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0669653-4
Keywords: Rational Chebyshev approximations, Bessel functions, McMahon expansions
Article copyright: © Copyright 1982 American Mathematical Society

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