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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. …

2: 3.1 Arithmetics and Error Measures

… ►The lower and upper bounds for the absolute values of the nonzero machine numbers are given by … ►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. … ►If $x^{*}$ is an approximation to a real or complex number $x$, then the absolute error is ►

3.1.8 $${ϵ}_a=\left|x^{*}-x\right|.$$

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Defines:

${ϵ}_a$: absolute error (locally)

A&S Ref:

3.5.1 (in absolute value)

Permalink:

http://dlmf.nist.gov/3.1.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(v), §3.1 and Ch.3

… ►

3.1.9 $${ϵ}_r=\left|\frac{x^{*}-x}{x}\right|=\frac{{ϵ}_a}{\left|x\right|}.$$

ⓘ

Defines:

${ϵ}_r$: relative error (locally)

Symbols:

${ϵ}_a$: absolute error

A&S Ref:

3.5.2 (in absolute value)

Permalink:

http://dlmf.nist.gov/3.1.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(v), §3.1 and Ch.3

…

3: 6.13 Zeros

… ► $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ each have an infinite number of positive real zeros, which are denoted by $c_k$, $s_k$, respectively, arranged in ascending order of absolute value for $k=0,1,2,\mathrm{\dots }$. …

4: 23.16 Graphics

… ►In Figures 23.16.2 and 23.16.3, height corresponds to the absolute value of the function and color to the phase. …

5: 5.3 Graphics

… ►In the graphics shown in this subsection, both the height and color correspond to the absolute value of the function. …

6: 8.27 Approximations

… ►

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

…

7: 9.3 Graphics

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …

8: 14.22 Graphics

… ►In the graphics shown in this section, height corresponds to the absolute value of the function and color to the phase. …

9: 4.3 Graphics

… ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …

10: 7.13 Zeros

… ► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the sector , $\mathrm{erfc}z$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the first quadrant of $ℂ$ $C\left(z\right)$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. …