Editing Gauss–Legendre algorithm (section)

{{Short description|Quickly converging computation of π}}
The '''Gauss–Legendre algorithm''' is an [[algorithm]] to compute the digits of [[Pi|{{pi}}]]. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of&nbsp;{{pi}}. However, it has some drawbacks (for example, it is [[Random-access_memory|computer memory]]-intensive) and therefore all record-breaking calculations for many years have used other methods, almost always the [[Chudnovsky algorithm]]. For details, see [[chronology of computation of π|Chronology of computation of {{pi}}]].

The method is based on the individual work of [[Carl Friedrich Gauss]] (1777–1855) and [[Adrien-Marie Legendre]] (1752–1833) combined with modern algorithms for multiplication and [[square root]]s. It repeatedly replaces two numbers by their [[arithmetic mean|arithmetic]] and [[geometric mean]], in order to approximate their [[arithmetic-geometric mean]].

The version presented below is also known as the '''Gauss–Euler''', '''Brent–Salamin''' (or '''Salamin–Brent''') '''algorithm''';<ref>[[Richard Brent (scientist)|Brent, Richard]], ''Old and New Algorithms for pi'', Letters to the Editor, Notices of the AMS 60(1), p. 7</ref> it was independently discovered in 1975 by [[Richard Brent (scientist)|Richard Brent]] and [[Eugene Salamin (mathematician)|Eugene Salamin]]. It was used to compute the first 206,158,430,000 decimal digits of {{pi}} on September 18 to 20, 1999, and the results were checked with [[Borwein's algorithm]].