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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–Kronrod quadrature formula ： ウィキペディア英語版 Gauss–Kronrod quadrature formula In numerical mathematics, the Gauss–Kronrod quadrature formula is a method for numerical integration (calculating approximate values of integrals). Gauss–Kronrod quadrature is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order (the latter called an ''embedded'' rule). The difference between these two approximations is used to estimate the calculational error of the integration. These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss. Gauss–Kronrod quadrature is used in the QUADPACK library, the GNU Scientific Library, the NAG Numerical Libraries and R.〔(http://stat.ethz.ch/R-manual/R-patched/library/stats/html/integrate.html )〕 == Description == The problem in numerical integration is to approximate definite integrals of the form :$\backslash int\_a^b\; f(x)\backslash ,dx.$ Such integrals can be approximated, for example, by ''n''-point Gaussian quadrature :$\backslash int\_a^b\; f(x)\backslash ,dx\; \backslash approx\; \backslash sum\_^n\; w\_i\; f(x\_i).$ where ''w''''i'', ''x''''i'' are the weights and points at which to evaluate the function ''f''(''x''). If the interval (''b'' ) is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at the midpoint for odd numbers of evaluation points), and thus the integrand must be evaluated at every point. Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding $n+1$ points to an $n$-point rule in such a way that the resulting rule is of order $2n+1$. These extra points are the zeros of Stieltjes polynomials. This allows for computing higher-order estimates while reusing the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error. 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–Kronrod quadrature formula」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.