Arithmetic–geometric mean


In mathematics, the arithmetic–geometric mean of two positive real numbers and is defined as follows:
Call and and :
Then define the two interdependent sequences and as
These two sequences converge to the same number, the arithmetic–geometric mean of and ; it is denoted by, or sometimes by.
The arithmetic-geometric mean is used in fast algorithms for exponential and trigonometric functions, as well as some mathematical constants, in particular, computing.

Example

To find the arithmetic–geometric mean of and, iterate as follows:
The first five iterations give the following values:
The number of digits in which and agree approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately.

History

The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were further analyzed by Gauss.

Properties

The geometric mean of two positive numbers is never bigger than the arithmetic mean. As a consequence, for, is an increasing sequence, is a decreasing sequence, and. These are strict inequalities if.
is thus a number between the geometric and arithmetic mean of and ; it is also between and.
If, then.
There is an integral-form expression for :
where is the complete elliptic integral of the first kind:
Indeed, since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals via this formula. In engineering, it is used for instance in elliptic filter design.

Related concepts

The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is called Gauss's constant, 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.
The arithmetic–geometric mean can be used to compute – among others – logarithms, complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions.

Proof of existence

From the inequality of arithmetic and geometric means we can conclude that:
and thus
that is, the sequence is nondecreasing.
Furthermore, it is easy to see that it is also bounded above by the larger of and . Thus, by the monotone convergence theorem, the sequence is convergent, so there exists a such that:
However, we can also see that:
and so:
Q.E.D.

Proof of the integral-form expression

This proof is given by Gauss.
Let
Changing the variable of integration to, where
gives
Thus, we have
The last equality comes from observing that.
Finally, we obtain the desired result

The AGM method

noticed that the sequences
as
have the same limit:
the arithmetic–geometric mean, agm.
It is possible to use this fact to construct fast algorithms for calculating elementary transcendental functions and some classical constants, in particular, the constant pi|.

Applications

The number ''π''

For example, according to the Gauss–Salamin formula:
where
which can be computed without loss of precision using

Complete elliptic integral ''K''(sin''α'')

Taking , yields the agm,
where is a complete elliptic integral of the first kind,
That is to say that this quarter period may be efficiently computed through the agm,

Other applications

Using this property of the AGM along with the ascending transformations of Landen, Richard Brent suggested the first AGM algorithms for the fast evaluation of elementary transcendental functions. Subsequently, many authors went on to study the use of the AGM algorithms.

Other