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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 ''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 = (''x'' + ''y'') / 2
:''g''₁ = √(''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:
:
:$g\_\{n+1\}\; =\; sqrt\{a\_n\; g\_n\}.$
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'').

Contents

Example

Properties

Implementation in Python

See also

References

Example

To find the arithmetic-geometric mean of $a\_0=24$ and $g\_0=6$, first calculate their arithmetic mean and geometric mean, thus:
:
:$g\_1=sqrt\{24\; imes\; 6\}=12,$
and then iterate as follows:
:
:$g\_2=sqrt\{15\; imes\; 12\}=13.41640786500dots$ 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''):
:
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.
:
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 the 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

See 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

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