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Author Dunham, Charles B.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Language English
Subject Keyword Square root ♦ Logarithmic error ♦ Transformed rational approximation
Abstract In this note we point out how rational approximations which are best with respect to maximum logarithmic error can be computed by existing algorithms. Let y be a quantity that we wish to approximate and y be an approximation, then the logarithmic error is defined to be log (y/y). In a recent paper [3] it is shown that minimax logarithmic approximations are optimal for square root calculations, making the minimax logarithmic problem of practical interest. Suppose we wish to approximate a positive continuous function ƒ by a positive rational function R, then the logarithmic error at a point x is log (ƒ(x)) - log (R(x)). Our approximation problem is thus equivalent to ordinary approximation of a continuous function g = log (ƒ) by log (R). This is contained in the more general theory of approximation by ϕ(R), ϕ monotonic which appears in [1]. Computational procedures (based on the Remez algorithm) for the general problem are given in [2, 5]. These are easily adapted to the special case of logarithmic approximation and can readily be coded by modification of a standard rational Remez algorithm.
Description Affiliation: Univ. of Western Ontario, London, Canada (Dunham, Charles B.)
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2005-08-01
Publisher Place New York
Journal Communications of the ACM (CACM)
Volume Number 12
Issue Number 10
Page Count 2
Starting Page 581
Ending Page 582


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Source: ACM Digital Library