# iterative refinement

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##### 1: Gloria Wiersma
Then she began working with the staff of the Physics Laboratory Office of Electronic Commerce in Scientific and Engineering Data, developing and refining the Laboratory website until her retirement in 2007. …
##### 3: 3.8 Nonlinear Equations
###### Bisection Method
The convergence of iterative methods …
19.22.8 $\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)},$
19.22.9 $\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{% M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}% \right),$
19.22.10 $R_{D}\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_{0}\right)% a_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
19.22.12 $R_{J}\left(0,g_{0}^{2},a_{0}^{2},p_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_% {0}\right)p_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
19.22.15 $p_{0}^{2}=a_{0}^{2}(q_{0}^{2}+g_{0}^{2})/(q_{0}^{2}+a_{0}^{2}).$
When $a_{0}$ and $g_{0}$ are positive numbers, define …
19.8.2 $c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}.$
19.8.3 $c_{n+1}=\frac{a_{n}-g_{n}}{2}=\frac{c_{n}^{2}}{4a_{n+1}},$
19.8.4 $\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\,% \mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos}^{2}\theta+g_{0}^{2}{\sin}^{2}\theta}}=% \frac{1}{\pi}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0% }^{2})}}.$
19.8.6 $E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n% =0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}-\sum_{n=2}^% {\infty}2^{n-1}c_{n}^{2}\right),$ $-\infty, $a_{0}=1$, $g_{0}=k^{\prime}$,
##### 6: 29.20 Methods of Computation
A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …
##### 7: 19.24 Inequalities
19.24.5 $\frac{1}{a_{n}}\leq\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)\leq% \frac{1}{g_{n}},$ $n=0,1,2,\dots$,
##### 8: 3.9 Acceleration of Convergence
When applied repeatedly, Aitken’s process is known as the iterated $\Delta^{2}$-process. … …
##### 9: 22.20 Methods of Computation
Four iterations of (22.20.1) lead to $c_{4}=\Sci{6.5}{-12}$. … If $k=k^{\prime}=1/\sqrt{2}$, then three iterations of (22.20.1) give $M=0.84721\;30848$, and from (22.20.6) $K=\pi/(2M)=1.85407\;46773$ — in agreement with the value of $\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{2}/\left(4\sqrt{\pi}\right)$; compare (23.17.3) and (23.22.2). … If $k^{\prime}=1-i$, then four iterations of (22.20.1) give $K=1.23969\;74481+i0.56499\;30988$. …
##### 10: 17.12 Bailey Pairs
When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …