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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 Lebedev quadrature ： ウィキペディア英語版 Lebedev quadrature In numerical analysis, Lebedev quadrature, named after Vyacheslav Ivanovich Lebedev, is an approximation to the surface integral of a function over a three-dimensional sphere. The grid is constructed so to have octahedral rotation and inversion symmetry. The number and location of the grid points together with a corresponding set of integration weights are determined by enforcing the exact integration of polynomials (or equivalently, spherical harmonics) up to a given order, leading to a sequence of increasingly dense grids analogous to the one-dimensional Gauss-Legendre scheme. The Lebedev grid is often employed in the numerical evaluation of volume integrals in the spherical coordinate system, where it is combined with a one-dimensional integration scheme for the radial coordinate. Applications of the grid are found in fields such as computational chemistry and neutron transport. ==Angular integrals== The surface integral of a function over the unit sphere, :$I()\; =\; \backslash frac\backslash int\; \backslash mathrm\backslash Omega\backslash \; f(\backslash Omega)\; =\; \backslash frac\backslash int\_0^\backslash pi\; \backslash sin(\backslash theta)\backslash mathrm\backslash theta\backslash int\_0^\backslash mathrm\backslash varphi\backslash \; f(\backslash theta,\backslash varphi),$ is approximated in the Lebedev scheme as :$\backslash tilde()\; =\; \backslash sum\_^\backslash \; w\_i\; f(\backslash theta\_i,\backslash varphi\_i),$ where the particular grid points and grid weights are to be determined. The use of a single sum, rather than two one dimensional schemes from discretizing the ''θ'' and ''φ'' integrals individually, leads to more efficient procedure: fewer total grid points are required to obtain similar accuracy. A competing factor is the computational speedup available when using the direct product of two one-dimensional grids. Despite this, the Lebedev grid still outperforms product grids. However, the use of two one-dimensional integration better allows for fine tuning of the grids, and simplifies the use of any symmetry of the integrand to remove symmetry equivalent grid points. 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「Lebedev 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.