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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 = \tfrac{1}{2}(x + y)
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} = \tfrac{1}{2}(a_n + g_n)
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).

This can be used for algorithmic purposes as in the AGM method.

Contents

[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=\tfrac12(24+6)=15,
g_1=\sqrt{24 \times 6}=12,

and then iterate as follows:

a_2=\tfrac12(15+12)=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

The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmetic and geometric means); as a consequence, (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ≤ M(x,y) ≤ an. 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 a closed form expression for M(x,y):

\Mu(x,y) = \frac{\pi}{4} \frac{x + y}{K\!\left[\left( \frac{x - y}{x + y} \right)^2 \right] }

where K(m)\, is the complete elliptic integral of the first kind:

K(m)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-m\sin^2(\theta)}}

Indeed, since the arithmetic-geometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula.

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] Proof of existence

From inequality of arithmetic and geometric means we can conclude that:

g_i\leqslant a_i

and thus

g_{i+1}=\sqrt{g_i\cdot a_i}\geqslant \sqrt{g_i\cdot g_i}=g_i

that is, the sequence gi 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, from Bolzano-Weierstrass theorem, there exists a convergent subsequence of gi. However, since the sequence is nondecreasing, we can conclude that the sequence itself is convergent, so there exists a g such that:

\lim_{n\to \infty}g_n=g

However, we can also see that:

a_i=\frac{g_{i+1}^2}{g_i}

and so:

\lim_{n\to \infty}a_n=\lim_{n\to \infty}\frac{g_{n+1}^2}{g_{n}}=\frac{g^2}{g}=g

Q.E.D.

[edit] See also

[edit] References

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