Research and Advances
Systems and Networking

Minimax logarithmic error

Posted

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.

View this article in the ACM Digital Library.

Join the Discussion (0)

Become a Member or Sign In to Post a Comment

The Latest from CACM

Shape the Future of Computing

ACM encourages its members to take a direct hand in shaping the future of the association. There are more ways than ever to get involved.

Get Involved

Communications of the ACM (CACM) is now a fully Open Access publication.

By opening CACM to the world, we hope to increase engagement among the broader computer science community and encourage non-members to discover the rich resources ACM has to offer.

Learn More