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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 FEE method ： ウィキペディア英語版 FEE method In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba〔E.A. Karatsuba, Fast evaluations of transcendental functions. Probl. Peredachi Informat., Vol. 27, N 4, (1991)〕〔D.W. Lozier and F.W.J. Olver, Numerical Evaluation of Special Functions. Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics, W. Gautschi, eds., Proc. Sympos. Applied Mathematics, AMS, Vol.48 (1994).〕 and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel $E$ -functions, and in particular, $e^x.$ A class of functions, which are 'similar to the exponential function' was given the name 'E-functions' by Siegel.〔C.L. Siegel, ''Transcendental numbers''. Princeton University Press, Princeton (1949).〕 Among these functions are such special functions as the hypergeometric function, cylinder, spherical functions and so on. Using the FEE, it is possible to prove the following theorem Theorem: Let $y=f(x)$ be an elementary Transcendental function, that is the exponential function, or a trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the inverses. Then : $$ s_f(n) = O(M(n)\log^2n). \, Here $s\_f(n)$ is the complexity of computation (bit) of the function $f(x)$ with accuracy up to $n$ digits, $M(n)$ is the complexity of multiplication of two $n$-digit integers. The algorithms based on the method FEE include the algorithms for fast calculation of any elementary Transcendental function for any value of the argument, the classical constants e, $\backslash pi,$ the Euler constant $\backslash gamma,$ the Catalan and the Apéry constants,〔Karatsuba E. A., Fast evaluation of $\backslash zeta(3)$, Probl. Peredachi Informat., Vol. 29, N 1 (1993)〕 such higher transcendental functions as the Euler gamma function and its derivatives, the hypergeometric,〔Ekatharine A. Karatsuba, Fast evaluation of hypergeometric function by FEE. Computational Methods and Function Theory (CMFT'97), N. Papamichael, St. Ruscheweyh and E.B. Saff, eds., World Sc.Pub. (1999)〕 spherical, cylinder (including the Bessel)〔Catherine A. Karatsuba, Fast evaluation of Bessel functions. Integral Transforms and Special Functions, Vol. 1, N4 (1993)〕 functions and some other functions for algebraic values of the argument and parameters, the Riemann zeta function for integer values of the argument〔E. A. Karatsuba, Fast Evaluation of Riemann zeta-function $\backslash zeta(s)$ for integer values of argument $s$. Probl. Peredachi Informat., Vol. 31, N 4 (1995).〕〔J.M. Borwein, D.M. Bradley and R.E. Crandall, Computational strategies for the Riemann zeta function. J. of Comput. Appl. Math., Vol. 121 ,N 1-2 (2000).〕 and the Hurwitz zeta function for integer argument and algebraic values of the parameter,〔E.A. Karatsuba, Fast evaluation of Hurwitz zeta function and Dirichlet $L$-series, Problem. Peredachi Informat., Vol. 34, N 4, pp. 342–353, (1998).〕 and also such special integrals as the integral of probability, the Fresnel integrals, the integral exponential function, the trigonometric integrals, and some other integrals〔E.A. Karatsuba, Fast computation of some special integrals of mathematical physics. Scientific Computing, Validated Numerics, Interval Methods, W.Kramer, J.W.von Gudenberg, eds.(2001).〕 for algebraic values of the argument with the complexity bound which is close to the optimal one, namely :$$ s_(n)= O(M(n)\log^2 n). \, At present, only the FEE makes it possible to calculate fast the values of the functions from the class of higher transcendental functions,〔E. Bach, The complexity of number-theoretic constants. Info. Proc. Letters, N 62 (1997).〕 certain special integrals of mathematical physics and such classical constants as Euler's, Catalan's〔E.A. Karatsuba, Fast computation of $\zeta(3)$ and of some special integrals ,using the polylogarithms, the Ramanujan formula and its generalization. J. of Numerical Mathematics BIT, Vol. 41, N 4 (2001).〕 and Apéry's constants. An additional advantage of the method FEE is the possibility of parallelizing the algorithms based on the FEE. == FEE-computation of classical constants == For fast evaluation of the constant $\backslash pi,$ one can use the Euler formula $\backslash frac\; =\; \backslash arctan\; \backslash frac12\; +\; \backslash arctan\; \backslash frac13,$ and apply the FEE to sum the Taylor series for : $$ \arctan \frac12 = \frac - \frac+ \cdots + \frac} + R_1 , : $$ \arctan \frac13 = \frac - \frac+ \cdots + \frac} + R_2 , with the remainder terms $R\_1,$ $R\_2,$ which satisfy the bounds : $|R\_1|\; \backslash leq\; \backslash frac45\backslash frac\backslash frac\backslash frac\backslash frac.$ To calculate $\backslash pi$ by the FEE it is possible to use also other approximations〔D.H. Bailey, P.B. Borwein and S. Plouffe, On the rapid computation of various polylogarithmic constants. Math. Comp.,Vol. 66 (1997).〕 In all cases the complexity is : $s\_\backslash pi\; =\; O(M(n)\backslash log^2\; n).\; \backslash ,$ To compute the Euler constant gamma with accuracy up to $n$ digits, it is necessary to sum by the FEE two series. Namely, for : $m=6n,\; \backslash quad\; k\; =\; n,\; \backslash quad\; k\; \backslash geq\; 1,\; \backslash ,$ : $$ \gamma = - \log n \sum_^ \frac + \sum_^ \frac + O(2^) . The complexity is : $s\_\backslash gamma\; =\; O(M(n)\backslash log^2\; n).\; \backslash ,$ To evaluate fast the constant $\backslash gamma$ it is possible to apply the FEE to other approximations.〔R.P. Brent and E.M. McMillan, Some new algorithms for high-precision computation of Euler's constant. Math. Comp., Vol.34 (1980).〕 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「FEE 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.