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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 Tanh-sinh quadrature ： ウィキペディア英語版 Tanh-sinh quadrature Tanh-sinh quadrature is a method for numerical integration introduced by Hidetosi Takahasi and Masatake Mori in 1974. It uses hyperbolic functions in the change of variables :$x\; =\; \backslash tanh(\backslash tfrac12\; \backslash pi\; \backslash sinh\; t)\backslash ,$ to transform an integral on the interval ''x'' ∈ (−1, +1) to an integral on the entire real line ''t'' ∈ (−∞,+∞). After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula. For a given step size ''h'', the integral is approximated by the sum :$\backslash int\_^1\; f(x)\; \backslash ,\; dx\; \backslash approx\; \backslash sum\_^\backslash infty\; w\_k\; f(x\_k),$ with the abscissas :$x\_k\; =\; \backslash tanh(\backslash tfrac12\; \backslash pi\; \backslash sinh\; kh)$ and the weights :$w\_k\; =\; \backslash frac.$ Like Gaussian quadrature, tanh-sinh quadrature is well suited for arbitrary-precision integration, where an accuracy of hundreds or even thousands of digits is desired. The convergence is exponential (in the discretization sense) for sufficiently well-behaved integrands: doubling the number of evaluation points roughly doubles the number of correct digits. Tanh-sinh quadrature is less efficient than Gaussian quadrature for smooth integrands, but unlike Gaussian quadrature tends to work equally well with integrands having singularities or infinite derivatives at one or both endpoints of the integration interval. A further advantage is that the abscissas and weights are relatively easy to compute. The cost of calculating abscissa–weight pairs for ''n''-digit accuracy is roughly ''n''2 log2 ''n'' compared to ''n''3 log ''n'' for Gaussian quadrature. Upon comparing the scheme to Gaussian quadrature and error function quadrature, Bailey et al. (2005) found that the tanh-sinh scheme "appears to be the best for integrands of the type most often encountered in experimental math research". Bailey and others have done extensive research on tanh-sinh quadrature, Gaussian quadrature and error function quadrature, as well as several of the classical quadrature methods, and found that the classical methods are not competitive with the first three methods, particularly when high-precision results are required. In a conference paper (July 2004) comparing tanh-sinh quadrature with Gaussian quadrature and error function quadrature, Bailey and Li found: "Overall, the tanh-sinh scheme appears to be the best. It combines uniformly excellent accuracy with fast run times. ''It is the nearest we have to a truly all-purpose quadrature scheme at the present time.''" Bailey (2006) found that: "The tanh-sinh quadrature scheme ''is the fastest known high-precision quadrature scheme'', especially when the time for computing abscissas and weights is considered. It has been successfully employed for quadrature calculations of up to 20,000-digit precision. It works well for functions with blow-up singularities or infinite derivatives at endpoints." ==Notes== 〔 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「Tanh-sinh 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.