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

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.