# error control

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## 5 matching pages

##### 1: 2.7 Differential Equations
2.7.23 $|\epsilon_{j}(x)|,\;\;\tfrac{1}{2}f^{-1/2}(x)|\epsilon_{j}^{\prime}(x)|\leq% \exp\left(\tfrac{1}{2}\mathcal{V}_{a_{j},x}\left(F\right)\right)-1,$ $j=1,2$,
Here $F(x)$ is the error-control function
2.7.24 $F(x)=\int\left(\frac{1}{f^{1/4}}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}% \left(\frac{1}{f^{1/4}}\right)-\frac{g}{f^{1/2}}\right)\,\mathrm{d}x,$
2.7.25 $\mathcal{V}_{a_{j},x}\left(F\right)=\left|\int_{a_{j}}^{x}\left|\frac{1}{f^{1/% 4}(t)}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}t}^{2}}\left(\frac{1}{f^{1/4}(t)}% \right)-\frac{g(t)}{f^{1/2}(t)}\right|\,\mathrm{d}t\right|.$
##### 2: Bibliography K
• H. Kuki (1972) Algorithm 421. Complex gamma function with error control. Comm. ACM 15 (4), pp. 271–272.
• ##### 3: 15.19 Methods of Computation
The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.
##### 4: 13.29 Methods of Computation
The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.
##### 5: 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 $0\leq m<1$ 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 $0 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … 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 $\phi$ near $\pi/2$ with the improvements made in the 1970 reference. …