Arithmetic-geometric mean

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:

$a_1 = \frac{x+y}{2}$

$g_1 = \sqrt{xy}.$

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:

$a_{n+1} = \frac{a_n + g_n}{2}$

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

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).

1 Example
2 Properties
3 Implementation in Python
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:

$a_1=\frac{24+6}{2}=15,$

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

and then iterate as follows:

$a_2=\frac{15+12}{2}=13.5,$

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

M(x, y) is 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):

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

where K(x) is the complete elliptic integral of the first kind.

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.

Implementation in Python

The following example code in Python computes the arithmetic-geometric mean of two positive real numbers:

from math import sqrt

def avg(a, b, delta=None):
        if None==delta:
                delta=(a+b)/2*1E-10
        if(abs(b-a)>delta):
                return avg((a+b)/2.0, sqrt(a*b), delta)
        else:
                return (a+b)/2.0

References

This article or section includes a list of references or external links, but its sources remain unclear because it lacks inline citations.
You can 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
Eric W. Weisstein, Arithmetic-Geometric mean at MathWorld.

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Arithmetic-geometric mean

Contents

Example

Properties

Implementation in Python

See also

References