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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 Clenshaw algorithm ： ウィキペディア英語版 Clenshaw algorithm In numerical analysis, the Clenshaw algorithm,〔 Note that this paper is written in terms of the ''Shifted'' Chebyshev polynomials of the first kind $T^$ *_n(x) = T_n(2x-1).〕 also called Clenshaw summation,〔 is a recursive method to evaluate a linear combination of Chebyshev polynomials. It is a generalization of Horner's method for evaluating a linear combination of monomials. It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term recurrence relation. ==Clenshaw algorithm== In full generality, the Clenshaw algorithm computes the weighted sum of a finite series of functions $\backslash phi\_k(x)$: :$S(x)\; =\; \backslash sum\_^n\; a\_k\; \backslash phi\_k(x)$ where $\backslash phi\_k,\backslash ;\; k=0,\; 1,\; \backslash ldots$ is a sequence of functions that satisfy the linear recurrence relation :$\backslash phi\_(x)\; =\; \backslash alpha\_k(x)\backslash ,\backslash phi\_k(x)\; +\; \backslash beta\_k(x)\backslash ,\backslash phi\_(x),$ where the coefficients $\backslash alpha\_k(x)$ and $\backslash beta\_k(x)$ are known in advance. The algorithm is most useful when $\backslash phi\_k(x)$ are functions that are complicated to compute directly, but $\backslash alpha\_k(x)$ and $\backslash beta\_k(x)$ are particularly simple. In the most common applications, $\backslash alpha(x)$ does not depend on $k$, and $\backslash beta$ is a constant that depends on neither $x$ nor $k$. To perform the summation for given series of coefficients $a\_0,\; \backslash ldots,\; a\_n$, compute the values $b\_k(x)$ by the "reverse" recurrence formula: :$$ \begin b_(x) &= b_(x) = 0, \\ b_k(x) &= a_k + \alpha_k(x)\,b_(x) + \beta_(x)\,b_(x). \end Note that this computation makes no direct reference to the functions $\backslash phi\_k(x)$. After computing $b\_2(x)$ and $b\_1(x)$, the desired sum can be expressed in terms of them and the simplest functions $\backslash phi\_0(x)$ and $\backslash phi\_1(x)$: :$S(x)\; =\; \backslash phi\_0(x)\backslash ,a\_0\; +\; \backslash phi\_1(x)\backslash ,b\_1(x)\; +\; \backslash beta\_1(x)\backslash ,\backslash phi\_0(x)\backslash ,b\_2(x).$ See Fox and Parker for more information and stability analyses. 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「Clenshaw 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.