AGM method

In mathematics, the AGM method (for arithmetic–geometric mean) makes it possible to construct fast algorithms for calculation of exponential and trigonometric functions, and some mathematical constants and in particular, to quickly compute $\pi$.

Method

Gauss noticed that the sequences

${\begin{aligned}a_{0}&&b_{0}\\a_{1}&={\frac {a_{0}+b_{0}}{2}},&b_{1}&={\sqrt {a_{0}b_{0}}}\\a_{2}&={\frac {a_{1}+b_{1}}{2}},&b_{2}&={\sqrt {a_{1}b_{1}}}\\&{}\ \ \vdots &&{}\ \ \vdots \\a_{N+1}&={\frac {a_{N}+b_{N}}{2}},&b_{N+1}&={\sqrt {a_{N}b_{N}}}\end{aligned}}$

as

$N\to +\infty ,\,$

have the same limit:

$\lim _{N\to \infty }a_{N}=\lim _{N\to \infty }b_{N}=M(a,b),\,$

the arithmetic–geometric mean.

It is possible to use this fact to construct fast algorithms for calculating elementary transcendental functions and some classical constants, in particular, the constant π.

Applications

The number π

For example, according to the Gauss–Salamin formula:

$\pi ={\frac {4\left(M(1;{\frac {1}{\sqrt {2}}})\right)^{2}}{\displaystyle 1-\sum _{j=1}^{\infty }2^{j+1}c_{j}^{2}}},$

where

$c_{j}={\frac {1}{2}}\left(a_{j-1}-b_{j-1}\right).$

Complete elliptic integral K(α)

At the same time, if we take

$a_{0}=1,\quad b_{0}=\cos \alpha ,$

then

$\lim _{N\to \infty }a_{N}={\frac {\pi }{2K(\alpha )}},$

where K(α) is a complete elliptic integral

$K(\alpha )=\int _{0}^{\pi /2}(1-\alpha \sin ^{2}\theta )^{-1/2}\,d\theta .$

Other applications

Using this property of the AGM and also the ascending transformations of Landen, Richard Brent suggested the first AGM algorithms for fast evaluation of elementary transcendental functions (e^x, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms, see, for example, the book Pi and the AGM by Jonathan and Peter Borwein.

See also

• Gauss–Legendre algorithm