minimax polynomial approximations

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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 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

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§3.11(i) Minimax Polynomial Approximations

… ►If we have a sufficiently close approximation … ►For examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … ►Since $L_0=1$, $L_n$ is a monotonically increasing function of $n$, and (for example) $L_{1000}=4.07\mathrm{\dots }$, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►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 $R_{k,\mathrm{\ell }}(x)$. …

3: 7.24 Approximations

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

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4: 6.20 Approximations

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Hastings (1955) gives several minimax polynomial and rational approximations for $E_1\left(x\right)+\ln x$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, 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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5: 5.23 Approximations

… ►Hart et al. (1968) gives minimax polynomial and rational approximations to $\mathrm{\Gamma }\left(x\right)$ and $\ln \mathrm{\Gamma }\left(x\right)$ in the intervals $0\le x\le 1$, $8\le x\le 1000$, $12\le x\le 1000$; precision is variable. …

6: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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Approximation Theory

►The monic Chebyshev polynomial $2^{1-n}{T}_n\left(x\right)$, $n\ge 1$, 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 $n$. … ►

Integrable Systems

… ►Ultraspherical polynomials are zonal spherical harmonics. …

7: Bibliography J

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Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).

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Notes:

Translated title: Mathieu functions and Klein-Gordon polynomials.

External Links:

ISSN 0764-4442, Document, MathReview, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

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Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:

ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§18.24, §2.4(vi).

Encodings:

BibTeX

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Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:

ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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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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8: 25.20 Approximations

§25.20 Approximations

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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 $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

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Morris (1979) gives rational approximations for ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $0.5\le x\le 1$. Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , 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 ${10}^{-4},{10}^{-8},{10}^{-12}$.