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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 Risch algorithm ： ウィキペディア英語版 Risch algorithm The Risch algorithm, named after Robert Henry Risch, is an algorithm for indefinite integration (i.e., finding antiderivatives in calculus). The algorithm is used in some symbolic computation (computer algebra) programs. The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding ''whether'' a function has an elementary function as an indefinite integral; and also, if it does, determining it. The Risch algorithm is summarized in ''Algorithms for Computer Algebra'' by Keith O. Geddes, Stephen R. Czapor and George Labahn. The Risch–Norman algorithm (after A. C. Norman), a faster but less powerful technique, was developed in 1976. ==Description== The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations (+ − × ÷). Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions. The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s. Liouville formulated the problem that is solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution ''g'' to the equation ''g''′ = ''f'' then there exist constants α''i'' and functions ''ui'' and ''v'' in the field generated by ''f'' such that the solution is of the form :$g\; =\; v\; +\; \backslash sum\_\; \backslash alpha\_i\; \backslash ,\backslash ln\; (u\_i)$ Risch developed a method that allows one to consider only a finite set of functions of Liouville's form. The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function ''f'' e''g'', where ''f'' and ''g'' are differentiable functions, we have : $\backslash left(f\; \backslash cdot\; e^g\backslash right)\text{'}\; =\; \backslash left(f^\backslash prime\; +\; f\backslash cdot\; g^\backslash prime\backslash right)\; \backslash cdot\; e^g,\; \backslash ,$ so if e''g'' were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as : $\backslash left(f\; \backslash cdot(\backslash ln\; g)^n\backslash right)\text{'}\; =\; f^\backslash prime\; (\backslash ln\; )^n\; +\; n\; f\; \backslash frac\; (\backslash ln)^$ then if (ln ''g'')''n'' were in the result of an integration, then only a few powers of the logarithm should be expected. 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「Risch algorithm」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.