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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gauss–Legendre algorithm ： ウィキペディア英語版 Gauss–Legendre algorithm The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π. However, the drawback is that it is memory intensive and it is therefore sometimes not used over Machin-like formulas. 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 roots. It repeatedly replaces two numbers by their 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;〔Brent, Richard, ''Old and New Algorithms for pi'', Letters to the Editor, Notices of the AMS 60(1), p. 7〕 it was independently discovered in 1975 by Richard Brent and Eugene Salamin. It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with Borwein's algorithm. == Algorithm == # Initial value setting: $a\_0\; =\; 1\backslash qquad\; b\_0\; =\; \backslash frac\backslash qquad\; p\_0\; =\; 1.\backslash !$ # Repeat the following instructions until the difference of $a\_n\backslash !$ and $b\_n\backslash !$ is within the desired accuracy: b_ & = \sqrt, \\ t_ & = t_n - p_n(a_-a_)^2, \\ p_ & = 2p_n. \end # π is then approximated as: $\backslash pi\; \backslash approx\; \backslash frac)^2\}\{4t\_\{n+1\}\}.\backslash !$ The first three iterations give (approximations given up to and including the first incorrect digit): :$3.140\backslash dots\backslash !$ :$3.14159264\backslash dots\backslash !$ :$3.1415926535897932382\backslash dots\backslash !$ The algorithm has second-order convergent nature, which essentially means that the number of correct digits doubles with each step of the algorithm. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gauss–Legendre algorithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.