In mathematics, the arithmeticgeometric 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:
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:
These two sequences converge to the same number, which is the arithmeticgeometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y).
This can be used for algorithmic purposes as in the AGM method.
Contents
Example
To find the arithmeticgeometric mean of a_{0} = 24 and g_{0} = 6, first calculate their arithmetic mean and geometric mean, thus:
and then iterate as follows:
The first four iterations give the following values:
The arithmeticgeometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.45817148173.
Properties
The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmetic and geometric means); as a consequence, (g_{n}) is an increasing sequence, (a_{n}) is a decreasing sequence, and g_{n} ≤ M(x, y) ≤ a_{n}. These are strict inequalities if x ≠ y.
M(x, y) is thus 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 an integralform expression for M(x, y):
where K(m) is the complete elliptic integral of the first kind:
Indeed, since the arithmeticgeometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula.
The reciprocal of the arithmeticgeometric mean of 1 and the square root of 2 is called Gauss's constant.
named after Carl Friedrich Gauss.
The geometricharmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. The arithmeticharmonic mean can be similarly defined, but takes the same value as the geometric mean.
Proof of existence
From inequality of arithmetic and geometric means we can conclude that:
and thus
that is, the sequence g_{i} is nondecreasing. Furthermore, it is easy to see that it is also bounded above by the larger of x and y (which follows from the fact that both arithmetic and geometric means of two numbers both lie between them). Thus by the monotone convergence theorem the sequence is convergent, so there exists a g such that:
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