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:
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.
To find the arithmetic–geometric mean of a0 = 24 and g0 = 6, first calculate their arithmetic mean and geometric mean, thus:
and then iterate as follows:
The first five iterations give the following values:
n an gn 0 24 6 1 Template:Underline5 Template:Underline2 2 Template:Underline.5 Template:Underline.416407864998738178455042… 3 Template:Underline203932499369089227521… Template:Underline139030990984877207090… 4 Template:Underline45176983217305… Template:Underline06053858316334… 5 Template:Underline20… Template:Underline06…
As can be seen, the number of digits in agreement (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.
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 an integral-form expression for M(x,y):
where K(k) is the complete elliptic integral of the first kind:
Indeed, since the arithmetic–geometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula. In engineering, it is used for instance in elliptic filter design.
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.
The arithmetic–geometric mean can be used to compute logarithms and complete elliptic integrals of the first kind. A modified arithmetic–geometric mean can be used to efficiently compute complete elliptic integrals of the second kind.
Proof of existence
From inequality of arithmetic and geometric means we can conclude that:
that is, the sequence gn 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:
However, we can also see that:
Proof of the integral-form expression
This proof is given by Gauss. Let
Thus, we have
Finally, we obtain the desired result
- 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 Template:MR
- Zoltán Daróczy, Zsolt Páles, Gauss-composition of means and the solution of the Matkowski–Suto problem. Publ. Math. Debrecen 61/1-2 (2002), 157–218.