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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 Symmetric level-index arithmetic ： ウィキペディア英語版 Symmetric level-index arithmetic The level-index (LI) representation of numbers, and its algorithms for arithmetic operations, were introduced by Clenshaw and Olver in 1984. The symmetric form of the LI system and its arithmetic operations were presented by Clenshaw and Turner. Anuta, Lozier, Schabanel and Turner developed the algorithm for symmetric level-index (SLI) arithmetic, and a parallel implementation of it. There has been extensive work on developing the SLI arithmetic algorithms and extending them to complex and vector arithmetic operations. ==Definition== The idea of the level-index system is to represent a non-negative real number X as $X=e^\}\}$ where and the process of exponentiation is performed l times, with $l\backslash geq\; 0$. l and f are the level and index of X respectively. x = l + f is the LI image of X. For example, $X=1234567=e^$ so its LI image is $x=l+f=3+0.9711308=3.9711308\; \backslash ,$. The symmetric form is used to allow negative exponents, if the magnitude of X is less than 1. One takes sgn(log(X)) or sgn(|X|-|X|−1) and stores it (after substituting +1 for 0 for the reciprocal sign since for X=1=e0 the LI image is x=1.0 and uniquely defines X=1 and we can do away without a third state and use only 1 bit for the 2 states -1 and +1) as the reciprocal sign rX. Mathematically, this is equivalent to taking the reciprocal (multiplicative inverse) of a small magnitude number, and then finding the SLI image for the reciprocal. Using one bit for the reciprocal sign enables the representation of extremely small numbers. A sign bit may also be used to allow negative numbers. One takes sgn(X) and stores it (after substituting +1 for 0 for the sign since for X=0 the LI image is x=0.0 and uniquely defines X=0 and we can do away without a third state and use only 1 bit for the 2 states -1 and +1) as the sign sX. Mathematically, this is equivalent to taking the inverse (additive inverse) of a negative number, and then finding the SLI image for the inverse. Using one bit for the sign enables the representation of negative numbers. The mapping function is called the generalized logarithm function. It is defined as 1+ \psi (\ln X) & \mathrm \quad X \geq 1 \end \right. and it maps $[0,\backslash infty\; )$ onto itself monotonically and so it is invertible on this interval. The inverse, the generalized exponential function, is defined by e^ & \mathrm \quad x\geq 1 \end \right. The density of values X represented by x has no discontinuities as we go from level l to l+1 (a very desirable property) since: $\backslash frac|\_\; =\; \backslash frac|\_$. The generalized logarithm function is closely related to the iterated logarithm used in computer science analysis of algorithms. Formally, we can define the SLI representation for an arbitrary real X (not 0 or 1) as $X=s\_\backslash phi\; (x)^=\backslash text(X),\; r\_=\backslash text(|X|-|X|^),\; x=\backslash psi\; (\backslash max\; (|X|,|X|^))=\backslash psi\; (|X|^=+1,\; r\_=+1,\; x=0.0\; \backslash ,$, $s\_=+1,\; r\_=+1,\; x=1.0\; \backslash ,$. For example, $X=-\backslash dfrac=-e^$ and its SLI representation is $x=-\backslash phi\; (3.9711308)^\; \backslash ,$. 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「Symmetric level-index arithmetic」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.