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 (an) and (gn) 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).

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[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 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

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( \frac{x - y}{x + y} \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