Arithmetic–geometric mean

Arithmetic–geometric mean
In mathematics, the arithmetic–geometric mean of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means:
Begin the sequences with x and y:
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 or .
The arithmetic–geometric mean is used in fast algorithms for exponential and trigonometric functions, as well as some mathematical constants, in particular, computing .
The arithmetic–geometric mean can be extended to complex numbers and when the branches of the square root are allowed to be taken inconsistently, it is, in general, a multivalued function.
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 (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 .
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 (see inequality of arithmetic and geometric means). 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.
The arithmetic–geometric mean is connected to the Jacobi theta function by
which upon setting gives
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.
In 1799, Gauss proved that
where is the lemniscate constant.
In 1941, (and hence ) was proven transcendental by Theodor Schneider. The set is algebraically independent over , but the set (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over . In fact,
The geometric–harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. One finds that .
The arithmetic–harmonic mean can be similarly defined, but takes the same value as the geometric mean (see section "Calculation" there).
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 (which follows from the fact that both the arithmetic and geometric means of two numbers lie between them). 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
Applications
The number π
For example, according to the Gauss–Legendre algorithm:
where
with and , which can be computed without loss of precision using
Complete elliptic integral K(sinα)
Taking and 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 John Landen, Richard P. 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.
See also
Landen's transformation
Gauss–Legendre algorithm
Generalized mean
References
Notes
Citations
Sources
Means
Special functions
Elliptic functions
Articles containing proofs