# minimax polynomials

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##### 1: 19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^{2}$ with $0\leq m<1$ can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. …
##### 2: 3.11 Approximation Techniques
###### §3.11(i) MinimaxPolynomial Approximations
If we have a sufficiently close approximation … For examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … 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 $R_{k,\ell}(x)$. …
##### 3: 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 $\mathrm{g}\left(x\right)$.

• ##### 4: 5.23 Approximations
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. …
##### 5: 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).

• ##### 6: 18.38 Mathematical Applications
The scaled Chebyshev polynomial $2^{1-n}T_{n}\left(x\right)$, $n\geq 1$, enjoys the “minimax” property on the interval $[-1,1]$, that is, $|2^{1-n}T_{n}\left(x\right)|$ has the least maximum value among all monic polynomials of degree $n$. …
##### 7: Bibliography J
• L. Jager (1997) Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).
• X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.
• X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.
• 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.
• ##### 8: 25.20 Approximations
• Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$, $5\leq s\leq 11$, $11\leq s\leq 25$, $25\leq s\leq 55$. Precision is varied, with a maximum of 20S.

• 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 $10^{-4},10^{-8},10^{-12}$.