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In this study, we are concerned with the interpolation problem that interpolates not only a given set of points but also derivatives at endpoints by using Bézier curves. The progressive iterative approximation (PIA) is proposed to interpolate the given points and derivatives. To speed up the convergence rate of PIA, we exploit the preconditioned PIA (PPIA), in which the diagonally compensated reduction is used to construct the preconditioner. The convergence of PIA and PPIA is also analyzed in this study. The proposed PPIA is applied to approximate higher order or/and rational Bézier curves. Numerical examples are given to illustrate the effectiveness of the proposed methods.

PIA and PPIA for Interpolating Points and Derivatives at Endpoints by Bézier Curves