# minimax rational functions

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##### 1: 3.11 Approximation Techniques
###### §3.11(iii) MinimaxRational Approximations
Then the minimax (or best uniform) rational approximation … The theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when $p_{n}(x)$ is replaced by a rational function $R_{k,\ell}(x)$. … A collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). …
##### 2: 7.24 Approximations
• 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 $\mathrm{g}\left(x\right)$.

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

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

• ##### 3: 5.23 Approximations
###### §5.23(i) Rational 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. … See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\Gamma\left(z\right)$. …
##### 4: Bibliography J
• 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.
• ##### 5: 9.19 Approximations
• Moshier (1989, §6.14) provides minimax rational approximations for calculating $\mathrm{Ai}\left(x\right)$, $\mathrm{Ai}'\left(x\right)$, $\mathrm{Bi}\left(x\right)$, $\mathrm{Bi}'\left(x\right)$. 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. The precision in the coefficients is 21S.

• ##### 6: 6.20 Approximations
• 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).

• ##### 7: 25.20 Approximations
• 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 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 $10^{-4},10^{-8},10^{-12}$.