Rational approximations for the Fresnel integrals
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A class of simple rational polynomial approximations for the Fresnel integrals is given with maximum errors from 1.7 x 10-3 down to 4 x 10-8. The domain [0, oo] is not subdivided. The format is particularly convenient for programmable hand calculators and microcomputer subroutines. The best-known algorithms for computer evaluation of the Fresnel integrals, (1) C(x) + iS(x) -J exp(iP,.t2) dt 0 are those of Hastings [1], [41 and of Boersma [2]. The former is of very limited precision. The latter is relatively cumbersome and is subject to nonobvious underflow (subtraction) errors, as is direct evaluation of the standard Maclaurin and asymptotic series. An extensive set of rational polynomial approximations has been given by Cody [3]. These are efficient for very high-precision work, but tedious for intermediate levels of precision because the domain is subdivided into five intervals. Our approach follows Hastings but recasts the formulas into the polar form: (2) C(x) = 2RI(x) sinF1Pr(AJk(x)-x2)I, (3) S(x) = 2 + Rim(x) cos[ 7T(Ajk(x) -X2)] where the RIm and Ajk functions are rational approximations of the form I M2 (4) R,01 = L c1xi/ E dix% i=o i=0 I k (S) Ajk = E a X'/ E b,x. ,=o i=o The domain of x is positive real numbers from zero to infinity. We consider error functions of the form (6) SR=RIM -Ro, (7) SA = '7TRo(Ak AO) where R0 and Ao are exact values. These functions represent orthogonal errors in the plane of the Cornu spiral, with the coefficient inserted in (7) to make the magnitudes Received May 2, 1983; revised January 9, 1984. 1980 Mathematics Subject Classification. Primary 65D20; Secondary 33A20, 41A50, 78A50.
[1] W. J. Cody,et al. Chebyshev approximations for the Fresnel integrals , 1968 .
[2] Irene A. Stegun,et al. Handbook of Mathematical Functions. , 1966 .