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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 Clenshaw–Curtis quadrature ： ウィキペディア英語版 Clenshaw–Curtis quadrature Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables $x\; =\; \backslash cos\; \backslash theta$ and use a discrete cosine transform (DCT) approximation for the cosine series. Besides having fast-converging accuracy comparable to Gaussian quadrature rules, Clenshaw–Curtis quadrature naturally leads to nested quadrature rules (where different accuracy orders share points), which is important for both adaptive quadrature and multidimensional quadrature (cubature). Briefly, the function $f(x)$ to be integrated is evaluated at the $N$ extrema or roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function. This polynomial is then integrated exactly. In practice, the integration weights for the value of the function at each node are precomputed, and this computation can be performed in $O(N\; \backslash log\; N)$ time by means of fast Fourier transform-related algorithms for the DCT.〔W. Morven Gentleman, "Implementing Clenshaw-Curtis quadrature I: Methodology and experience," ''Communications of the ACM'' 15(5), p. 337-342 (1972).〕〔Jörg Waldvogel, "(Fast construction of the Fejér and Clenshaw-Curtis quadrature rules )," ''BIT Numerical Mathematics'' 46 (1), p. 195-202 (2006).〕 ==General method== A simple way of understanding the algorithm is to realize that Clenshaw–Curtis quadrature (proposed by those authors in 1960)〔C. W. Clenshaw and A. R. Curtis "(A method for numerical integration on an automatic computer ) ''Numerische Mathematik'' 2, 197 (1960).〕 amounts to integrating via a change of variable ''x'' = cos(θ). The algorithm is normally expressed for integration of a function ''f''(''x'') over the interval () (any other interval can be obtained by appropriate rescaling). For this integral, we can write: :$\backslash int\_^1\; f(x)\backslash ,\; dx\; =\; \backslash int\_0^\backslash pi\; f(\backslash cos\; \backslash theta)\; \backslash sin(\backslash theta)\backslash ,\; d\backslash theta\; .$ That is, we have transformed the problem from integrating $f(x)$ to one of integrating $f(\backslash cos\; \backslash theta)\; \backslash sin\; \backslash theta$. This can be performed if we know the cosine series for $f(\backslash cos\; \backslash theta)$: :$f(\backslash cos\; \backslash theta)\; =\; \backslash frac\; +\; \backslash sum\_^\backslash infty\; a\_k\; \backslash cos\; (k\backslash theta)$ in which case the integral becomes: :$\backslash int\_0^\backslash pi\; f(\backslash cos\; \backslash theta)\; \backslash sin(\backslash theta)\backslash ,\; d\backslash theta\; =\; a\_0\; +\; \backslash sum\_^\backslash infty\; \backslash frac\; .$ Of course, in order to calculate the cosine series coefficients :$a\_k\; =\; \backslash frac\; \backslash int\_0^\backslash pi\; f(\backslash cos\; \backslash theta)\; \backslash cos(k\; \backslash theta)\backslash ,\; d\backslash theta$ one must again perform a numeric integration, so at first this may not seem to have simplified the problem. Unlike computation of arbitrary integrals, however, Fourier-series integrations for periodic functions (like $f(\backslash cos\backslash theta)$, by construction), up to the Nyquist frequency $k=N$, are accurately computed by the $N+1$ equally spaced and equally weighted points $\backslash theta\_n\; =\; n\; \backslash pi\; /\; N$ for $n\; =\; 0,\backslash ldots,N$ (except the endpoints are weighted by 1/2, to avoid double-counting, equivalent to the trapezoidal rule or the Euler–Maclaurin formula).〔J. P. Boyd, ''Chebychev and Fourier Spectral Methods'', 2nd ed. (Dover, New York, 2001).〕〔See, for example, S. G. Johnson, "(Notes on the convergence of trapezoidal-rule quadrature )," online MIT course notes (2008).〕 That is, we approximate the cosine-series integral by the type-I discrete cosine transform (DCT): :$a\_k\; \backslash approx\; \backslash frac\; \backslash left$ for $k\; =\; 0,\backslash ldots,N$ and then use the formula above for the integral in terms of these $a\_k$. Because only $a\_$ is needed, the formula simplifies further into a type-I DCT of order ''N''/2, assuming ''N'' is an even number: :$a\_\; \backslash approx\; \backslash frac\; \backslash left)\; \backslash right\backslash \}\; \backslash cos\backslash left(\backslash frac\backslash right)\; \backslash right]$ From this formula, it is clear that the Clenshaw–Curtis quadrature rule is symmetric, in that it weights ''f''(''x'') and ''f''(−''x'') equally. Because of aliasing, one only computes the coefficients $a\_$ up to ''k''=''N''/2, since discrete sampling of the function makes the frequency of 2''k'' indistinguishable from that of ''N''–2''k''. Equivalently, the $a\_$ are the amplitudes of the unique bandlimited trigonometric interpolation polynomial passing through the ''N''+1 points where ''f''(cos ''θ'') is evaluated, and we approximate the integral by the integral of this interpolation polynomial. There is some subtlety in how one treats the $a\_$ coefficient in the integral, however—to avoid double-counting with its alias it is included with weight 1/2 in the final approximate integral (as can also be seen by examining the interpolating polynomial): :$\backslash int\_0^\backslash pi\; f(\backslash cos\; \backslash theta)\; \backslash sin(\backslash theta)\backslash ,\; d\backslash theta\; \backslash approx\; a\_0\; +\; \backslash sum\_^\; \backslash frac\; +\; \backslash frac.$ 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「Clenshaw–Curtis 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.