Accurate and Fast Evaluation of Elementary Symmetric Functions

Abstract

This paper is concerned with the fast and accurate evaluation of elementary symmetric functions. We present a new compensated algorithm by applying error-free transformations to improve the accuracy of the so-called Summation Algorithm, which is used, by example, in the MATLAB's poly function. We derive a forward round off error bound and running error bound for our new algorithm. The round off error bound implies that the computed result is as accurate as if computed with twice the working precision and then rounded to the current working precision. The running error analysis provides a shaper bound along with the result, without increasing significantly the computational cost. Numerical experiments illustrate that our algorithm runs much faster than the algorithm using the classic double-double library while sharing similar error estimates. Such an algorithm can be widely applicable for example to compute characteristic polynomials from eigen values. It can also be used into the Rasch model in psychological measurement.