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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. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. …

2: 3.1 Arithmetics and Error Measures

… ►Let $E_{\min }\le E\le E_{\max }$ with and $E_{\max }>0$. …The integers $p$, $E_{\min }$, and $E_{\max }$ are characteristics of the machine. … ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{\min }=-126$, $E_{\max }=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{\min }=-1022$, $E_{\max }=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{\min }=-16382$, $E_{\max }=16383$) are as in Figure 3.1.1. … ► $N_{\min }\le x\le N_{\max }$, and …Then rounding by chopping or rounding down of $x$ gives $x_{-}$, with maximum relative error ${ϵ}_M$. …

3: 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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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 ${10}^{-4},{10}^{-8},{10}^{-12}$.

4: 33.25 Approximations

… ►Maximum relative errors range from 1. …

5: 7.24 Approximations

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Cody (1969) provides minimax rational approximations for $\mathrm{erf}x$ and $\mathrm{erfc}x$. 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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6: 14.16 Zeros

… ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)$ if any of the following sets of conditions hold: … ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)+1$ if either of the following sets of conditions holds: … ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. …

7: 15.15 Sums

… ►Here $z_0$ ($\ne 0$) is an arbitrary complex constant and the expansion converges when $|z-z_0|>\max (|z_0|,|z_0-1|)$. …

8: 29.20 Methods of Computation

… ►Alternatively, the zeros can be found by locating the maximum of function $g$ in (29.12.11).

9: 1.10 Functions of a Complex Variable

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§1.10(v) Maximum-Modulus Principle

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Analytic Functions

… ►If $f(z)$ is continuous on $\overline{D}$ and analytic in $D$, then $\left|f(z)\right|$ attains its maximum on $\partial D$. ►

Harmonic Functions

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Schwarz’s Lemma

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10: Bibliography P

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Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.

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Notes:

Includes Fortran code, maximum accuracy 20D. For certification and remarks see same journal v. 13 (1970), pp. 448–449 and p. 750; v. 15 (1972), p. 1074.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii).

Encodings:

BibTeX

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

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Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.

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Notes:

Single-precision Fortran, maximum accuracy 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 680; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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