Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some mathematical constants, in particular, computing .

The AGM is defined as the limit of the interdependent sequences $a_i$ and $g_i$. Assuming $x \geq y \geq 0$, we write:$\begin a_0 &= x,\\ g_0 &= y\\ a_ &= \tfrac12(a_n + g_n),\\ g_ &= \sqrt\, . \end$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 can be extended to complex numbers and, when the branches of the square root are allowed to be taken inconsistently, it is a multivalued function.

Example

To find the arithmetic–geometric mean of and , iterate as follows:$\begin a_1 & = & \tfrac12(24 + 6) & = & 15\\ g_1 & = & \sqrt & = & 12\\ a_2 & = & \tfrac12(15 + 12) & = & 13.5\\ g_2 & = & \sqrt & = & 13.416\ 407\ 8649\dots\\ & & \vdots & & \end$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.

Properties

Both the geometric mean and arithmetic mean of two positive numbers and are between the two numbers. (They are ''strictly'' between when .) The geometric mean of two positive numbers is never greater than the arithmetic mean. So the geometric means are an increasing sequence ; the arithmetic means are a decreasing sequence ; and for any . These are strict inequalities if .

is thus a number between and ; it is also between the geometric and arithmetic mean of and .

If then .

There is an integral-form expression for :$\begin M(x,y) &= \frac \left( \int_0^\frac\frac \right)^\\ &=\pi\left(\int_0^\infty \frac\right)^\\ &= \frac \cdot \frac \end$where is the complete elliptic integral of the first kind:$K(k) = \int_0^\frac\frac$Since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in elliptic filter design.

The arithmetic–geometric mean is connected to the Jacobi theta function $\theta_3$ by$M(1,x)=\theta_3^\left(\exp \left(-\pi \frac\right)\right)=\left(\sum_\exp \left(-n^2 \pi \frac\right)\right)^,$which upon setting $x=1/\sqrt$ gives$M(1,1/\sqrt)=\left(\sum_e^\right)^.$

Related concepts

The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is Gauss's constant.$\frac = G = 0.8346268\dots$In 1799, Gauss provedBy 1799, Gauss had two proofs of the theorem, but neither of them was rigorous from the modern point of view. that$M(1,\sqrt)=\frac$where $\varpi$ is the lemniscate constant.

In 1941, $M(1,\sqrt)$ (and hence $G$) was proved transcendental by Theodor Schneider.In particular, he proved that the beta function $\Beta (a,b)$ is transcendental for all $a,b\in\mathbb\setminus\mathbb$ such that $a+b\notin \mathbb_0^-$. The fact that $M(1,\sqrt)$ is transcendental follows from $M(1,\sqrt)=\tfrac\Beta \left(\tfrac,\tfrac\right).$ The set $\$ is algebraically independent over $\mathbb$, but the set $\$ (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over $\mathbb$. In fact,$\pi=2\sqrt\frac.$The geometric–harmonic mean GH can be calculated using analogous sequences of geometric and harmonic means, and in fact .
The arithmetic–harmonic mean is equivalent to 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

The inequality of arithmetic and geometric means implies that$g_n \leq a_n$and thus$g_ = \sqrt \geq \sqrt = g_n$that is, the sequence is nondecreasing and bounded above by the larger of and . By the monotone convergence theorem, the sequence is convergent, so there exists a such that:$\lim_g_n = g$However, we can also see that:$a_n = \frac$
and so:
$\lim_a_n = \lim_\frac = \frac = g$

Q.E.D.

Proof of the integral-form expression

This proof is given by Gauss.
Let

$I(x,y) = \int_0^\frac ,$

Changing the variable of integration to $\theta'$, where

$\sin\theta = \frac \Rightarrow d(\sin\theta)=d\left(\frac\right)\Rightarrow \cos\theta\ d\theta =2x \frac\ \cos\theta' d\theta'$

$\cos\theta = \frac=\frac=\frac ,$

$\Rightarrow \cos\theta\ d\theta =\frac\ d\theta =2x \frac\ \cos\theta' d\theta' ,$

$\Rightarrow d\theta = \frac \frac \ ,$
$\sqrt = \frac= \frac$

This yields
$\frac = \frac = \frac ,$

gives

$\begin I(x,y) &= \int_0^\frac\\ &= I\bigl(\tfrac,\sqrt\bigr) . \end$

Thus, we have

$\begin I(x,y) &= I(a_1, g_1) = I(a_2, g_2) = \cdots\\ &= I\bigl(M(x,y),M(x,y)\bigr) = \pi/\bigr(2M(x,y)\bigl) . \end$
The last equality comes from observing that $I(z,z) = \pi/(2z)$.

Finally, we obtain the desired result

$M(x,y) = \pi/\bigl(2 I(x,y) \bigr) .$

Applications

The number ''π''

According to the Gauss–Legendre algorithm,

$\pi = \frac ,$

where

$c_j = \frac\left(a_-g_\right) ,$

with $a_0=1$ and $g_0=1/\sqrt$, which can be computed without loss of precision using

$c_j = \frac .$

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

Taking $a_0 = 1$ and $g_0 = \cos\alpha$ yields the AGM

$M(1,\cos\alpha) = \frac ,$

where is a complete elliptic integral of the first kind:

$K(k) = \int_0^(1 - k^2 \sin^2\theta)^ \, d\theta.$

That is to say that this quarter period may be efficiently computed through the AGM,
$K(k) = \frac .$

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

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{{DEFAULTSORT:Arithmetic-Geometric Mean
Means
Special functions
Elliptic functions
Articles containing proofs