absolute error

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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

… ►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. … ►If $x^{\ast }$ is an approximation to a real or complex number $x$, then the absolute error is ►

3.1.8 $${ϵ}_a=\left|x^{\ast }-x\right|.$$

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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

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

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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

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3: 8.27 Approximations

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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 $x\to \mathrm{\infty }$.

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4: 18.40 Methods of Computation

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Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points $x_{i,N}$, $i=1,2,\mathrm{\dots },N$ for $N=40$. … Magnify

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5: 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. …

6: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by …

7: 18.15 Asymptotic Approximations

… ►When $\alpha ,\beta \in (-\frac{1}{2},\frac{1}{2})$, the error term in (18.15.1) is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$. …

8: 10.40 Asymptotic Expansions for Large Argument

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§10.40(ii) Error Bounds for Real Argument and Order

… ►Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that $\mathrm{\ell }\ge \max (|\nu |-\frac{1}{2},1)$. ►For the error term in (10.40.1) see §10.40(iii). ►

§10.40(iii) Error Bounds for Complex Argument and Order

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9: 10.17 Asymptotic Expansions for Large Argument

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§10.17(iii) Error Bounds for Real Argument and Order

… ►If these expansions are terminated when $k=\mathrm{\ell }-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. ►

§10.17(iv) Error Bounds for Complex Argument and Order

… ►Corresponding error bounds for (10.17.3) and (10.17.4) are obtainable by combining (10.17.13) and (10.17.14) with (10.4.4). … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

10: 2.7 Differential Equations

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2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval and $F$: error-control function

Referenced by:

Erratum (V1.1.2) for Equation (2.7.25)

Permalink:

http://dlmf.nist.gov/2.7.E25

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value.

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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