Arithmetic–geometric mean
In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers x and y is defined as follows:
First, compute the arithmetic and geometric means of x and y, calling them a1 and g1 respectively (the latter is the principal square root of the product xy):
Then, use iteration, with a1 taking the place of x and g1 taking the place of y. In this way, two sequences (an) and (gn) are defined:
These two sequences converge to the same number, which is the arithmetic–geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y).
This can be used for algorithmic purposes as in the AGM method, which makes it possible to construct fast algorithms for calculating exponential and trigonometric functions, as well as some mathematical constants, in particular, to quickly compute .
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