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Chebyshev approximations for the Fresnel integrals


Author: W. J. Cody
Journal: Math. Comp. 22 (1968), 450-453
DOI: https://doi.org/10.1090/S0025-5718-68-99871-2
MathSciNet review: 0238469
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Abstract | References | Additional Information

Abstract: Rational Chebyshev approimations have been computed for the Fresnel integrals $ C(x)$ and $ S(x)$ for arguments in the intervals $ [0.,1.2]$ and $ [1.2,1.6]$, and for the related functions $ f(x)$ and $ g(x)$ for the intervals $ [1.6,1.9]$, $ [1.9,2.4]$ and $ [2.4,\infty ]$. Maximal relative errors range down to $ 2 \times {10^{ - 19}}$.


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

  • [1] M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Appl. Math. Series, No. 55, U. S. Government Printing Office, Washington, D. C., 1964. MR 31 #1400. MR 0167642 (29:4914)
  • [2] H. E. Syrett & M. W. Wilson, Computation of Fresnel Integrals to 28 Figures: Approximations to 8 and 20 Figures, Univ. of Western Ontario, Canada. (Unpublished.) See Math. Comp., v. 20, 1966, p. 181, RMT 25.
  • [3] J. Boersma, "Computation of Fresnel integrals," Math. Comp., v. 14, 1960, p. 380. MR 22 #12700. MR 0121973 (22:12700)
  • [4] G. Németh, "Chebyshev expansions for Fresnel integrals," Numer. Math., v. 7, 1965, pp. 310-312. MR 32 #3265. MR 0185805 (32:3265)
  • [5] W. Fraser & J. F. Hart, "On the computation of rational approximations to continuous functions," Comm. ACM, v. 5, 1962, pp. 401-403.
  • [6] W. J. Cody & J. Stoer, "Rational Chebyshev approximations using interpolation," Numer. Math., v. 9, 1966, pp. 177-188.
  • [7] P. Henrici, "The quotient-difference algorithm," Nat. Bur. Standards Appl. Math. Ser., no. 49, 1958, pp. 23-46. MR 20 #1410. MR 0094901 (20:1410)
  • [8] J. R. Rick, "On the conditioning of polynomial and rational forms," Numer. Math., v. 7, 1965, pp. 426-435. MR 32 #6710. MR 0189283 (32:6710)


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-68-99871-2
Article copyright: © Copyright 1968 American Mathematical Society

American Mathematical Society