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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–Jacobi quadrature ： ウィキペディア英語版 Gauss–Jacobi quadrature In numerical analysis, Gauss–Jacobi quadrature is a method of numerical quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form :$\backslash int\_^1\; f(x)\; (1-x)^\backslash alpha\; (1+x)^\backslash beta\; \backslash ,\backslash mathrmx$ where ƒ is a smooth function on (1 ) and ''α'', ''β'' > −1. The interval (1 ) can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with ''α'' = ''β'' = 0. Similarly, Chebyshev–Gauss quadrature arises when one takes ''α'' = ''β'' = ±½. More generally, the special case ''α'' = ''β'' turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature. Gauss–Jacobi quadrature uses ''ω''(''x'') = (1 − ''x'')''α'' (1 + ''x'')''β'' as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on ''n'' points has the form :$\backslash int\_^1\; f(x)\; (1-x)^\backslash alpha\; (1+x)^\backslash beta\; \backslash ,\backslash mathrmx\; \backslash approx\; \backslash lambda\_1\; f(x\_1)\; +\; \backslash lambda\_2\; f(x\_2)\; +\; \backslash cdots\; +\; \backslash lambda\_n\; f(x\_n),$ where ''x''1, …, ''x''''n'' are the roots of the Jacobi polynomial of degree ''n''. The weights ''λ''1, …, ''λ''''n'' are given by the formula :$\backslash lambda\_i\; =\; -\; \backslash frac\; \backslash frac\; \backslash frac(x\_i)\},$ where Γ denotes the Gamma function and ''P''''n'' the Jacobi polynomial of degree ''n''. ==References== * . 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–Jacobi quadrature」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.