Research On PIA Algorithm For Curve Interpolation

The progressive iterative approximation is a classical iterative method for interpolating a given set of data points.It has its clear geometric meaning,stable convergence and simple iterative format.Because of these characteristics,the PIA has intrigued researchers for decades.Based on the Jacobi-PIA algorithm and the GS-PIA algorithm for non-uniform cubic B-spline curves interpolation,in this thesis,we propose the weighted Jacobi-PIA(WJacobi-PIA)algorithm and the weighted GS-PIA(WGS-PIA)algorithm for cubic B-Spline curve interpolation under totally positive bases.First,we prove that for any totally positive basis,there exists a weight parameterβ such that the WJacobi-PIA algorithm always converges,and then derive the optimal weight ω,so that the spectral radius of the iterative matrix is minimized.The numerical experiments show that the WJacobi-PIA algorithm outperforms the Jacobi-PIA algorithm and PIA algorithm for cubic B-spline curve-interpolation,and the WJacobiPIA algorithm is superior to the PIA algorithm for Bernstein curve-interpolation.Secondly,for cubic B-Spline basis,we prove that there is always an interval(0,ζ),the WGS-PIA algorithm converges if ω belas in this interval.At the same time,we derive an upper bound of the iteration contraction factor less than 1 and the optimal weight to minimize the upper bound.In addition,we prove that the convergence rate of the GS-PIA algorithm is faster than Jacobi-PIA algorithm for cubic B-Spline curve interpolation,and the numerical experiments are shown that for a given termination tolerance,the number of iteration steps and the CPU time required by the WGSPIA algorithm are less than those required by GS-PIA algorithm,and at the same iteration step,the interpolation error of WGS-PIA algorithm is less than that of GSPIA algorithm.