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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 Pseudo-spectral method ： ウィキペディア英語版 Pseudo-spectral method Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely related to spectral methods, but complement the basis by an additional pseudo-spectral basis, which allows to represent functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the fast Fourier transform. ==Motivation with a concrete example== Take the initial-value problem : In the spectral method, the solution $\backslash psi$ is expanded in a suitable set of basis functions, for example plane waves, :$\backslash psi(x,t)\; =\; \backslash frac\; .$ Insertion and equating identical coefficients yields a set of ordinary differential equations for the coefficients, :$i\backslash frac\; c\_n(t)\; =\; (2\backslash pi\; n)^2\; c\_n\; +\; \backslash sum\_k\; V\_\; c\_k,$ where the elements $V\_$ are calculated through the explicit Fourier-transform :$V\_\; =\; \backslash frac\; \backslash int\_0^\; V(x)\; \backslash \; e^\; dx\; .$ The solution would then be obtained by truncating the expansion to $N$ basis functions, and finding a solution for the $c\_n(t)$. In general, this is done by numerical methods, such as Runge–Kutta methods. For the numerical solutions, the right-hand side of the ordinary differential equation has to be evaluated repeatedly at different time steps. At this point, the spectral method has a major problem with the potential term $V(x)$. In the spectral representation, the multiplication with the function $V(x)$ transforms into a vector-matrix multiplication, which scales as $N^2$. Also, the matrix elements $V\_$ need to be evaluated explicitly before the differential equation for the coefficients can be solved, which requires an additional step. In the pseudo-spectral method, this term is evaluated differently. Given the coefficients $c\_n(t)$, an inverse discrete Fourier transform yields the value of the function $\backslash psi$ at discrete grid points $x\_j\; =\; 2\backslash pi\; j/N$. At these grid points, the function is then multiplied, $\backslash psi\text{'}(x\_i,\; t)\; =\; V(x\_i)\; \backslash psi(x\_i,\; t)$, and the result Fourier-transformed back. This yields a new set of coefficients $c\text{'}\_n(t)$ that are used instead of the matrix product $\backslash sum\_k\; V\_\; c\_k(t)$. It can be shown that both methods have similar accuracy. However, the pseudo-spectral method allows the use of a fast Fourier transform, which scales as $O(N\backslash ln\; N)$, and is therefore significantly more efficient than the matrix multiplication. Also, the function $V(x)$ can be used directly without evaluating any additional integrals. 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「Pseudo-spectral method」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.