Arithmetic-geometric mean: Definition from Answers.com

In mathematics, the arithmetic-geometric mean (AGM) of two positive real numbers x and y is defined as follows:

First compute the arithmetic mean of x and y and call it a₁. Next compute the geometric mean of x and y and call it g₁; this is the square root of the product xy:

$a_1 = \tfrac{1}{2}(x + y)$

$g_1 = \sqrt{xy}.$

Then iterate this operation with a₁ taking the place of x and g₁ taking the place of y. In this way, two sequences (a_n) and (g_n) are defined:

$a_{n+1} = \tfrac{1}{2}(a_n + g_n)$

$g_{n+1} = \sqrt{a_n g_n}.$

These two sequences converge to the same number, which is the arithmetic-geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y).

This can be used for algorithmic purposes as in the AGM method.

1 Example
2 Properties
3 Proof of existence
4 See also
5 References

Example

To find the arithmetic-geometric mean of a₀ = 24 and g₀ = 6, first calculate their arithmetic mean and geometric mean, thus:

$a_1=\tfrac12(24+6)=15,$

$g_1=\sqrt{24 \times 6}=12,$

and then iterate as follows:

$a_2=\tfrac12(15+12)=13.5,$

$g_2=\sqrt{15 \times 12}=13.41640786500\dots$ etc.

The first four iterations give the following values:

n	a_n	g_n
0	24	6
1	15	12
2	13.5	13.41640786500...
3	13.45820393250...	13.45813903099...
4	13.45817148175...	13.45817148171...

The arithmetic-geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.45817148173.

Properties

The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmetic and geometric means); as a consequence, (g_n) is an increasing sequence, (a_n) is a decreasing sequence, and g_n ≤ M(x,y) ≤ a_n. These are strict inequalities if x≠y.

M(x, y) is thus a number between the geometric and arithmetic mean of x and y; in particular it is between x and y.

If r > 0, then M(rx, ry) = r M(x, y).

There is a closed form expression for M(x,y):

$\Mu(x,y) = \frac{\pi}{4} \cdot \frac{x + y}{K[\left( \frac{x - y}{x + y} \right)^2 ] }$

where $K(m)\,$ is the complete elliptic integral of the first kind:

$K(m)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-m\sin^2(\theta)}}$

Indeed, since the arithmetic-geometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula.

The reciprocal of the arithmetic-geometric mean of 1 and the square root of 2 is called Gauss's constant.

$\frac{1}{\Mu(1, \sqrt{2})} = G = 0.8346268\dots$

named after Carl Friedrich Gauss.

The geometric-harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. The arithmetic-harmonic mean can be similarly defined, but takes the same value as the geometric mean.

Proof of existence

From inequality of arithmetic and geometric means we can conclude that:

$g_i\leqslant a_i$

and thus

$g_{i+1}=\sqrt{g_i\cdot a_i}\geqslant \sqrt{g_i\cdot g_i}=g_i$

that is, the sequence g_i is nondecreasing. Furthermore, it is easy to see that it is also bounded above by the larger of x and y (which follows from the fact that both arithmetic and geometric means of two numbers both lie between them). Thus, from Bolzano-Weierstrass theorem, there exists a convergent subsequence of g_i. However, since the sequence is nondecreasing, we can conclude that the sequence itself is convergent, so there exists a g such that:

$\lim_{n\to \infty}g_n=g$

However, we can also see that:

$a_i=\frac{g_{i+1}^2}{g_i}$

and so:

$\lim_{n\to \infty}a_n=\lim_{n\to \infty}\frac{g_{n+1}^2}{g_{n}}=\frac{g^2}{g}=g$

Q.E.D.

References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (October 2008)

Jonathan Borwein, Peter Borwein, Pi and the AGM. A study in analytic number theory and computational complexity. Reprint of the 1987 original. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. xvi+414 pp. ISBN 0-471-31515-X MR 1641658
M. Hazewinkel (2001), "Arithmetic-geometric mean process", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/a/a130280.htm
Weisstein, Eric W., "Arithmetic-Geometric mean" from MathWorld.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	mean (mathematics)
	Landen's transformation
	Gauss–Legendre algorithm

	When would you use the geometric mean vs arithmetic mean? Read answer...
	What is the geometric mean and the arithmetic mean of 7 x sqrt3 and 21 x sqrt3? Read answer...
	Who proposed the doctrine that population increase in a geometric ratio while the means of subsistence increases in an arithmetic ration? Read answer...

	Why isthe Geometric mean less than or equal to the Arithmetic mean?
	Which circumstances is arithmetic mean superior than geometric mean?
	Prove arithmetic mean greater than geometric mean?

		Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Arithmetic-geometric mean". Read more

Arithmetic-geometric mean

Contents

Example

Properties

Proof of existence

See also

References