AGM Method - Method

Method

Gauss noticed that the sequences

$\begin{align} a_0 & & b_0 \\ a_1 & = \frac{a_0+b_0}{2}, & b_1 & = \sqrt{a_0 b_0} \\ a_2 & = \frac{a_1+b_1}{2}, & b_2 & = \sqrt{a_1 b_1} \\ & {}\ \ \vdots & & {}\ \ \vdots \\ a_{N+1} & = \frac{a_N + b_N}{2}, & b_{N+1} & = \sqrt{a_N b_N} \end{align}$

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.

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