The numerical condition of univariate and bivariate degree elevated Bernstein polynomials

Abstract

The numerical condition of the degree elevation operation on Bernstein polynomials is considered and it is shown that it does not change the condition of the polynomial. In particular, several condition numbers for univariate and bivariate Bernstein polynomials, and their degree elevated forms, are developed and it is shown that the condition numbers of the degree elevated polynomials are identically equal to their forms prior to degree elevation. Computational experiments that verify this theoretical result are presented. The results in this paper differ from those in [Comput. Aided Geom. Design 4 (1987) 191–216] and [Comput. Aided Geom. Design 5 (1988) 215–252], where it is claimed that degree elevation causes a reduction in the numerical condition of a Bernstein polynomial. It is shown, however, that there is an error in the derivation of this result.