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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 Geometric–harmonic mean ： ウィキペディア英語版 Geometric–harmonic mean In mathematics, the geometric–harmonic mean M(''x'', ''y'') of two positive real numbers ''x'' and ''y'' is defined as follows: we form the geometric mean of ''g''0 = ''x'' and ''h''0 = ''y'' and call it ''g''1, i.e. ''g''1 is the square root of ''xy''. We also form the harmonic mean of ''x'' and ''y'' and call it ''h''1, i.e. ''h''1 is the reciprocal of the arithmetic mean of the reciprocals of ''x'' and ''y''. These may be done sequentially (in any order) or simultaneously. Now we can iterate this operation with ''g''1 taking the place of ''x'' and ''h''1 taking the place of ''y''. In this way, two sequences (''g''''n'') and (''h''''n'') are defined: :$g\_\; =\; \backslash sqrt$ and :$h\_\; =\; \backslash frac\; +\; \backslash frac\}$ Both of these sequences converge to the same number, which we call the geometric–harmonic mean M(''x'', ''y'') of ''x'' and ''y''. The geometric–harmonic mean is also designated as the harmonic–geometric mean. (cf. Wolfram MathWorld below.) The existence of the limit can be proved by the means of Bolzano–Weierstrass theorem in a manner almost identical to the proof of existence of arithmetic–geometric mean. ==Properties== M(''x'', ''y'') is a number between the geometric and harmonic mean of ''x'' and ''y''; in particular it is between ''x'' and ''y''. M(''x'', ''y'') is also homogeneous, i.e. if ''r'' > 0, then M(''rx'', ''ry'') = ''r'' M(''x'', ''y''). If AG(''x'', ''y'') is the arithmetic–geometric mean, then we also have :$M(x,y)\; =\; \backslash frac,\backslash frac)\}$ 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「Geometric–harmonic 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.