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 "a"1. Next compute the geometric mean of "x" and "y" and call it "g"1; this is the square root of the product "xy":

:$a_1 = frac\left\{x+y\right\}\left\{2\right\}$

:$g_1 = sqrt\left\{xy\right\}.$

Then iterate this operation with "a"1 taking the place of "x" and "g"1 taking the place of "y". In this way, two sequences ("a""n") and ("g""n") are defined:

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

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

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").

Example

To find the arithmetic-geometric mean of "a"0 = 24 and "g"0 = 6, first calculate their arithmetic mean and geometric mean, thus:

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

:$g_1=sqrt\left\{24 imes 6\right\}=12,$

and then iterate as follows:

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

:$g_2=sqrt\left\{15 imes 12\right\}=13.41640786500dots$ etc.

The first four iterations give the following values:

:

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"):

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

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.

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

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

ee also

* Inequality of arithmetic and geometric means

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 MathSciNet|id=1641658
*SpringerEOM|author=M. Hazewinkel|title=Arithmetic-geometric mean process|urlname=a/a130280
*mathworld|urlname=Arithmetic-GeometricMean|title=Arithmetic-Geometric mean

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