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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 Gaussian quadrature ： ウィキペディア英語版 Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An ''n''-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the points and weights for . The domain of integration for such a rule is conventionally taken as (1 ), so the rule is stated as :$\backslash int\_^1\; f(x)\backslash ,dx\; =\; \backslash sum\_^n\; w\_i\; f(x\_i).$ Gaussian quadrature as above will only produce accurate results if the function ''f''(''x'') is well approximated by a polynomial function within the range . The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as $f(x)\; =\; \backslash omega(x)\; g(x)\backslash ,$, where is approximately polynomial and is known, then alternative weights $w\_i\text{'}$ and points $x\_i\text{'}$ that depend on the weighting function may give better results, where :$\backslash int\_^1\; f(x)\backslash ,dx\; =\; \backslash int\_^1\; \backslash omega(x)\; g(x)\backslash ,dx\; \backslash approx\; \backslash sum\_^n\; w\_i\text{'}\; g(x\_i\text{'}).$ Common weighting functions include $\backslash omega(x)=1/\backslash sqrt\backslash ,$ (Chebyshev–Gauss) and $\backslash omega(x)=e^$ (Gauss–Hermite). It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials. == Gauss–Legendre quadrature == For the simplest integration problem stated above, i.e. with $\backslash omega(x)=1$, the associated polynomials are Legendre polynomials, ''P''''n''(''x''), and the method is usually known as Gauss–Legendre quadrature. With the -th polynomial normalized to give ''P''''n''(1) = 1, the -th Gauss node, , is the -th root of ; its weight is given by :$w\_i\; =\; \backslash frac.$ Some low-order rules for solving the integration problem are listed below. } || 1 |- align="center" | rowspan="2" | 3 || 0 || $\backslash tfrac$ |- align="center" | $\backslash pm\backslash sqrt\}$ || $\backslash tfrac$ |- align="center" | rowspan="2" | 4 || $\backslash pm\backslash sqrt\; -\; \backslash tfrac\backslash sqrt\}\}$ || $\backslash tfrac$ |- align="center" | $\backslash pm\backslash sqrt\; +\; \backslash tfrac\backslash sqrt\}\}$ || $\backslash tfrac$ |- align="center" | rowspan="3" | 5 || 0 || $\backslash tfrac$ |- align="center" | $\backslash pm\backslash tfrac13\backslash sqrt\}\}$ || $\backslash tfrac$ |- align="center" | $\backslash pm\backslash tfrac13\backslash sqrt\}\}$ || $\backslash tfrac$ |} == Change of interval == An integral over must be changed into an integral over before applying the Gaussian quadrature rule. This change of interval can be done in the following way: :$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; \backslash frac\; \backslash int\_^1\; f\backslash left(\backslash fracx\; +\; \backslash frac\backslash right)\backslash ,dx.$ Applying the Gaussian quadrature rule then results in the following approximation: :$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; \backslash frac\; \backslash sum\_^n\; w\_i\; f\backslash left(\backslash fracx\_i\; +\; \backslash frac\backslash right).$ 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「Gaussian 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.