# error control

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5 matching pages ♦

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## 5 matching pages

##### 1: 2.7 Differential Equations

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2.7.23
$$|{\u03f5}_{j}(x)|,\frac{1}{2}{f}^{-1/2}(x)|{\u03f5}_{j}^{\prime}(x)|\le \mathrm{exp}\left(\frac{1}{2}{\mathcal{V}}_{{a}_{j},x}\left(F\right)\right)-1,$$
$j=1,2$,

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►Here $F(x)$ is the *error-control function*►
2.7.24
$$F(x)=\int \left(\frac{1}{{f}^{1/4}}\frac{{d}^{2}}{{dx}^{2}}\left(\frac{1}{{f}^{1/4}}\right)-\frac{g}{{f}^{1/2}}\right)dx,$$

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

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##### 2: Bibliography K

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Algorithm 421. Complex gamma function with error control.
Comm. ACM 15 (4), pp. 271–272.
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##### 3: 15.19 Methods of Computation

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►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

##### 4: 13.29 Methods of Computation

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►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

##### 5: 19.38 Approximations

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►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.
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►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi $ near $\pi /2$ with the improvements made in the 1970 reference.
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