# iterated

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##### 1: 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, $a_{0}=1$, $g_{0}=k^{\prime}$,
##### 4: 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. …
##### 5: 19.24 Inequalities
19.24.5 $\frac{1}{a_{n}}\leq\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)\leq% \frac{1}{g_{n}},$ $n=0,1,2,\dots$,
##### 6: 3.9 Acceleration of Convergence
When applied repeatedly, Aitken’s process is known as the iterated $\Delta^{2}$-process. … …
##### 7: 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 $K=1.23969\;74481+i0.56499\;30988$. …
##### 8: 17.12 Bailey Pairs
When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …
##### 10: Bibliography J
• G. Julia (1918) Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).