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:

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

$g_1 = \sqrt{xy}.$

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

[edit] 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.

[edit] 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.

[edit] 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

[edit] See also

Inequality of arithmetic and geometric means

[edit] References

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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