Arithmetic-geometric mean

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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:

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:

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.

Contents 1 Example 2 Properties 3 See also 4 References

[edit] Example

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

and then iterate as follows:

etc.

The first four iterations give the following values:

n	an	gn
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.

[edit] 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):

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

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.

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.

[edit] See also

• Inequality of arithmetic and geometric means
• Gauss–Legendre algorithm

[edit] 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 MR1641658
• M. Hazewinkel (2001), "Arithmetic-geometric mean process", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Weistein, Eric W., "Arithmetic-Geometric mean" from MathWorld.