On the stability of transformations between power and Bernstein polynomial forms

Abstract

We show that in the ||{radical dot}||1 and ||{radical dot}||∞ norms, the condition number of the matrix T that maps the coefficients of a polynomial P(t) of degree n in the Bernstein basis on t ∈[0,1] to those in the power basis about t = 0, or vice-versa, assumes the simple form κ(T) = (n+1)(nν)2ν where ν denotes the largest integer not exceeding 2(n + 1)/3 (for n ≥ 3, the expression κ(T){reversed tilde equals}3n+1 n + 1 2 π gives an excellent approximation). The growth of κ(T) with n suggests a potentially severe loss of accuracy during the basis conversion of high-degree curves and surfaces, a practice that should be avoided whenever possible. © 1991.