level-index arithmetic

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1: 19.8 Quadratic Transformations

… ►

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►As

n\to\infty

a_{n}

and

g_{n}

converge to a common limit

M\left(a_{0},g_{0}\right)

called the AGM (Arithmetic-Geometric Mean) of

a_{0}

and

g_{0}

. …showing that the convergence of

c_{n}

to 0 and of

a_{n}

and

g_{n}

M\left(a_{0},g_{0}\right)

is quadratic in each case. … ►

19.8.5

K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)},

-\infty<k^{2}<1

ⓘ

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, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

… ►Again,

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …

2: 3.1 Arithmetics and Error Measures

… ►

§3.1(ii) Interval Arithmetic

… ►

§3.1(iii) Rational Arithmetics

… ►

§3.1(iv) Level-Index Arithmetic

… ►In level-index arithmetic

x

is represented by

\ell+a

(or

-(\ell+a)

for negative numbers). … ►For further references on level-index arithmetic (and also other arithmetics) see Anuta et al. (1996). …

3: Bibliography C

… ►

C. W. Clenshaw, F. W. J. Olver, and P. R. Turner (1989) Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

ⓘ

External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.1(iv).
Encodings:: BibTeX

… ►

D. A. Cox (1984) The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

ⓘ

External Links:: ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.8(i).
Encodings:: BibTeX

►

D. A. Cox (1985) Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

ⓘ

External Links:: ISSN 0002-9920, MathReview (Willard Parker)
Referenced by:: §19.8(i).
Encodings:: BibTeX

…

4: 4.44 Other Applications

§4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …

5: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

6: 4.48 Software

… ►All scientific programming languages, libraries, and systems support computation of at least some of the elementary functions in standard floating-point arithmetic (§3.1(i)). … ►Here we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter when the arithmetic is nonstandard. … ►A more complete list of available software for computing these functions is found in the Software Index; again, software that uses only standard floating-point arithmetic is excluded. … ►

§4.48(ii) Interval Arithmetic

…

7: 15.17 Mathematical Applications

… ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … …

8: 22.20 Methods of Computation

… ►

§22.20(ii) Arithmetic-Geometric Mean

… ►Then as

n\to\infty

sequences

\{a_{n}\}

\{b_{n}\}

converge to a common limit

M=M\left(a_{0},b_{0}\right)

, the arithmetic-geometric mean of

a_{0},b_{0}

. … ►The rate of convergence is similar to that for the arithmetic-geometric mean. … ►using the arithmetic-geometric mean. … ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute

\operatorname{am}\left(x,k\right)

. …

9: Annie A. M. Cuyt

… ►As a consequence her expertise spans a wide range of activities from pure abstract mathematics to computer arithmetic and different engineering applications. …

10: 19.22 Quadratic Transformations

… ►

§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

►The AGM,

M\left(a_{0},g_{0}\right)

, of two positive numbers

a_{0}

and

g_{0}

is defined in §19.8(i). … ►

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

… ►As

n\to\infty

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …