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7 Error Functions, Dawson’s and Fresnel IntegralsComputation

§7.24 Approximations

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Permalink:
http://dlmf.nist.gov/7.24
See also:
Annotations for Ch.7
Contents
  1. §7.24(i) Approximations in Terms of Elementary Functions
  2. §7.24(ii) Expansions in Chebyshev Series
  3. §7.24(iii) Padé-Type Expansions

§7.24(i) Approximations in Terms of Elementary Functions

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    Hastings (1955) gives several minimax polynomial and rational approximations for \operatorname{erf}x, \operatorname{erfc}x and the auxiliary functions \mathrm{f}\left(x\right) and \mathrm{g}\left(x\right).

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    Cody (1969) provides minimax rational approximations for \operatorname{erf}x and \operatorname{erfc}x. The maximum relative precision is about 20S.

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    Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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    Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F\left(x\right) (maximum relative precision 20S–22S).

§7.24(ii) Expansions in Chebyshev Series

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    Luke (1969b, pp. 323–324) covers \frac{1}{2}\sqrt{\pi}\operatorname{erf}x and e^{x^{2}}F\left(x\right) for -3\leq x\leq 3 (the Chebyshev coefficients are given to 20D); \sqrt{\pi}xe^{x^{2}}\operatorname{erfc}x and 2xF\left(x\right) for x\geq 3 (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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    Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions \mathrm{f}\left(x\right) and \mathrm{g}\left(x\right) for x\geq 3 (15D).

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    Schonfelder (1978) gives coefficients of Chebyshev expansions for x^{-1}\operatorname{erf}x on 0\leq x\leq 2, for xe^{x^{2}}\operatorname{erfc}x on [2,\infty), and for e^{x^{2}}\operatorname{erfc}x on [0,\infty) (30D).

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    Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for (1+2x)e^{x^{2}}\operatorname{erfc}x on (0,\infty) (22D).

§7.24(iii) Padé-Type Expansions

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Permalink:
http://dlmf.nist.gov/7.24.iii
See also:
Annotations for §7.24 and Ch.7
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    Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for F\left(z\right), \operatorname{erf}z, \operatorname{erfc}z, C\left(z\right), and S\left(z\right); approximate errors are given for a selection of z-values.

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