minimax rational approximation

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1: 7.24 Approximations

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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).

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2: 3.11 Approximation Techniques

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§3.11(iii) Minimax Rational Approximations

… ►Then the minimax (or best uniform) rational approximation … ►A collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). … ► ►

Example

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3: 5.23 Approximations

… ►Cody and Hillstrom (1967) gives minimax rational approximations for

\ln\Gamma\left(x\right)

for the ranges

0.5\leq x\leq 1.5

1.5\leq x\leq 4

4\leq x\leq 12

; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to

\Gamma\left(x\right)

and

\ln\Gamma\left(x\right)

in the intervals

0\leq x\leq 1

8\leq x\leq 1000

12\leq x\leq 1000

; precision is variable. Cody et al. (1973) gives minimax rational approximations for

\psi\left(x\right)

for the ranges

0.5\leq x\leq 3

and

3\leq x<\infty

; precision is variable. …

4: 6.20 Approximations

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Hastings (1955) gives several minimax polynomial and rational approximations for $E_{1}\left(x\right)+\ln x$ , $xe^{x}E_{1}\left(x\right)$ , and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ . These are included in Abramowitz and Stegun (1964, Ch. 5).

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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5: 25.20 Approximations

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Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ and $2\leq x<\infty$ , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ . For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$ .

6: 9.19 Approximations

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Moshier (1989, §6.14) provides minimax rational approximations for calculating $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ . They are in terms of the variable $\zeta$ , where $\zeta=\tfrac{2}{3}x^{3/2}$ when $x$ is positive, $\zeta=\tfrac{2}{3}(-x)^{3/2}$ when $x$ is negative, and $\zeta=0$ when $x=0$ . The approximations apply when $2\leq\zeta<\infty$ , that is, when $3^{2/3}\leq x<\infty$ or $-\infty<x\leq-3^{2/3}$ . The precision in the coefficients is 21S.

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7: Bibliography J

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J. H. Johnson and J. M. Blair (1973) REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

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Referenced by:: §3.11(iii).
Encodings:: BibTeX

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