Accurate Evaluation of Polynomials

TL;DR

This paper introduces a new polynomial evaluation method that significantly improves accuracy over Horner's method, with minimal additional computational cost, especially beneficial for repeated evaluations at nearby points.

Contribution

The paper presents a novel polynomial evaluation technique that is more accurate than Horner's method, with comparable computational effort, particularly for multiple evaluations.

Findings

100 to 1000 times more accurate than Horner's method

Requires twice the floating point operations for a single evaluation

Maintains similar efficiency for repeated nearby evaluations

Abstract

For a large class of polynomials, the standard method of polynomial evaluation, Horner's method, can be very inaccurate. The alternative method given here is on average 100 to 1000 times more accurate than Horner's Method. The number of floating point operations is twice that of Horner's method for a single evaluation. For repeated evaluations at nearby points, the number of floating point operations is only doubled for the first evaluation, and is the same as Horner's Method for all following evaluations. This new method is tested with random polynomials.