Compensated de Casteljau algorithm in $K$ times the working precision

TL;DR

This paper introduces a family of compensated de Casteljau algorithms that enhance polynomial evaluation accuracy in Bernstein form by simulating higher precision through error correction techniques, validated by analysis and experiments.

Contribution

It proposes a novel family of compensated algorithms that improve polynomial evaluation accuracy without requiring actual higher precision arithmetic.

Findings

Algorithms achieve accuracy comparable to higher precision computations.

Error analysis confirms the effectiveness of compensation.

Numerical experiments demonstrate improved evaluation accuracy.

Abstract

In computer aided geometric design a polynomial is usually represented in Bernstein form. This paper presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. At each stage of computation, round-off error is passed on to first order errors, then to second order errors, and so on. After the computation has been "filtered" $(K - 1)$ times via this process, the resulting output is as accurate as the de Casteljau algorithm performed in $K$ times the working precision. Forward error analysis and numerical experiments illustrate the accuracy of this family of algorithms.