Progressive iteration approximation and the geometric algorithm

Abstract

Two recent tools for constructing interpolating curves and surfaces are the progressive iteration approximation (PIA) and the geometric algorithm. This communication has two main goals. On the one hand, we show that the proof of Theorem 1 of [5] on the convergence of the geometric algorithm is not correct. On the other hand, we clarify the difference between several variants of the PIA iterations by introducing the idea of preconditioned PIA iterations.