Convergence Proof of GS-PIA Algorithm

The GS-PIA algorithm for non-uniform cubic B-spline curve interpolation has the advantages of simplicity,stability,fast convergence and so on.In this paper,we elaborate the detailed geometric meaning of the GS-PIA algorithm and prove the convergence of the algorithm.We first define comparison matrix of configuration matrix of the algorithm.And the convergence of the iterative matrix corresponding to the comparison matrix is proved by the regular splittings of the matrices.Using the similarity of the matrices,we prove that the GS-PIA algorithm for non-uniform cubic B-spline curve interpolation is convergent.It lays out a theoretical foundation for further research of GS-PIA algorithm and applications in computer graphics and related fields.