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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 Lanczos approximation ： ウィキペディア英語版 Lanczos approximation In mathematics, the Lanczos approximation is a method for computing the Gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical alternative to the more popular Stirling's approximation for calculating the Gamma function with fixed precision. ==Introduction== The Lanczos approximation consists of the formula :$\backslash Gamma(z+1)\; =\; \backslash sqrt\; \backslash right)\}^\; \}\; e^\backslash right)\}\; A\_g(z)$ for the Gamma function, with :$A\_g(z)\; =\; \backslash fracp\_0(g)\; +\; p\_1(g)\; \backslash frac\; +\; p\_2(g)\; \backslash frac\; +\; \backslash cdots.$ Here ''g'' is a constant that may be chosen arbitrarily subject to the restriction that Re(''z'') > 1/2.〔Pugh thesis〕 The coefficients ''p'', which depend on ''g'', are slightly more difficult to calculate (see below). Although the formula as stated here is only valid for arguments in the right complex half-plane, it can be extended to the entire complex plane by the reflection formula, :$\backslash Gamma(1-z)\; \backslash ;\; \backslash Gamma(z)\; =\; .$ The series ''A'' is convergent, and may be truncated to obtain an approximation with the desired precision. By choosing an appropriate ''g'' (typically a small integer), only some 5-10 terms of the series are needed to compute the Gamma function with typical single or double floating-point precision. If a fixed ''g'' is chosen, the coefficients can be calculated in advance and the sum is recast into the following form: :$A\_g(z)\; =\; c\_0\; +\; \backslash sum\_^\; \backslash frac$ Thus computing the Gamma function becomes a matter of evaluating only a small number of elementary functions and multiplying by stored constants. The Lanczos approximation was popularized by ''Numerical Recipes'', according to which computing the Gamma function becomes "not much more difficult than other built-in functions that we take for granted, such as sin ''x'' or ''e''''x''". The method is also implemented in the GNU Scientific Library. 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「Lanczos approximation」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.