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1: 3.8 Nonlinear Equations

… ► … ► … ►

Bisection Method

… ►The convergence of iterative methods … ►

2: 19.22 Quadratic Transformations

… ►

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)},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

►

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),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

… ►

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},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.22(ii)
Permalink:: http://dlmf.nist.gov/19.22.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

… ►

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},

ⓘ

Symbols:: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $p_{j}$ : coefficient, $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.8(i)
Permalink:: http://dlmf.nist.gov/19.22.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

… ►

19.22.15

p_{0}^{2}=a_{0}^{2}(q_{0}^{2}+g_{0}^{2})/(q_{0}^{2}+a_{0}^{2}).

ⓘ

Symbols:: $p_{j}$ : coefficient, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

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3: 19.8 Quadratic Transformations

… ►When

a_{0}

and

g_{0}

are positive numbers, define … ►

19.8.2

c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}.

ⓘ

Symbols:: $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

… ►

19.8.3

c_{n+1}=\frac{a_{n}-g_{n}}{2}=\frac{c_{n}^{2}}{4a_{n+1}},

ⓘ

Symbols:: $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

… ►

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})}}.

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.8(i)
Permalink:: http://dlmf.nist.gov/19.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

… ►

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<k^{2}<1

a_{0}=1

g_{0}=k^{\prime}

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $n$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

…

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

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.24.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.24(i), §19.24 and Ch.19

… ►

19.24.9

\frac{1}{2}\,g_{1}^{2}\leq\frac{R_{G}\left(a_{0}^{2},g_{0}^{2},0\right)}{R_{F}% \left(a_{0}^{2},g_{0}^{2},0\right)}\leq\frac{1}{2}\,a_{1}^{2},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.24(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.24.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.24(i), §19.24 and Ch.19

…

6: 3.9 Acceleration of Convergence

… ►When applied repeatedly, Aitken’s process is known as the iterated $\Delta^{2}$ -process. … … ►

§3.9(v) Levin’s and Weniger’s Transformations

…

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: 3.2 Linear Algebra

… ►

Iterative Refinement

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9: 17.12 Bailey Pairs

… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …

10: 18.20 Hahn Class: Explicit Representations

…