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":

:$a\_1\; =\; frac\{x+y\}\{2\}$

:$g\_1\; =\; sqrt\{xy\}.$

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:

:$a\_\{n+1\}\; =\; frac\{a\_n\; +\; g\_n\}\{2\}$

:$g\_\{n+1\}\; =\; sqrt\{a\_n\; g\_n\}.$

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").

Example

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

:$a\_1=frac\{24+6\}\{2\}=15,$

:$g\_1=sqrt\{24\; imes\; 6\}=12,$

and then iterate as follows:

:$a\_2=frac\{15+12\}\{2\}=13.5,$

:$g\_2=sqrt\{15\; imes\; 12\}=13.41640786500dots$ 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"):

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

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.

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

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