Minimax logarithmic error

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.

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