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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 Arithmetic–geometric mean ： ウィキペディア英語版 Arithmetic–geometric mean In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers and is defined as follows: First compute the arithmetic mean of and and call it . Next compute the geometric mean of and and call it ; this is the square root of the product : :$\backslash begin$ a_1 &= \tfrac12(x + y)\\ g_1 &= \sqrt \end Then iterate this operation with taking the place of and taking the place of . In this way, two sequences and are defined: :$\backslash begin$ a_ &= \tfrac12(a_n + g_n)\\ g_ &= \sqrt \end These two sequences converge to the same number, which is the arithmetic–geometric mean of and ; it is denoted by , or sometimes by . This can be used for algorithmic purposes as in the AGM method. ==Example== To find the arithmetic–geometric mean of and , first calculate their arithmetic mean and geometric mean, thus: :$\backslash begin$ a_1 &= \tfrac12(24 + 6) = 15\\ g_1 &= \sqrt = 12 \end and then iterate as follows: :$\backslash begin$ a_2 &= \tfrac12(15 + 12) = 13.5\\ g_2 &= \sqrt = 13.41640786500\dots\\ \dots \end The first five iterations give the following values: : As can be seen, the number of digits in agreement (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.〔(agm(24, 6) at WolframAlpha )〕 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「Arithmetic–geometric mean」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.