iterative refinement

(0.002 seconds)

1—10 of 23 matching pages

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. …

2: 3.2 Linear Algebra

… ►

Iterative Refinement

…

3: 3.8 Nonlinear Equations

… ► … ► … ►

Bisection Method

… ►The convergence of iterative methods … ►

4: 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

…

5: 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

…

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

ⓘ

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

…

8: 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

…

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. …