Numerical Analysis of the Progressive Iterative Approximation Method

Based on the expansion of the matrix power series of a class of special matrices,we deduce the equivalence of the progressive iterative approximation method and the algebraic interpolation method.Furthermore,due to the slow convergence rate of the PIA method caused by the ill-conditioned collocation matrix,we present a new method to speed up the convergence rate of this method,which uses the QR method to decompose collocation matrix,and then uses a transform matrix to optimize the spectral radius of the collocation matrix.Due to the uncertain computational efficiency caused by the different parameterization methods,we propose to choose the centripetal method to insure the numerical efficiency by optimizing the collocation matrix.Some numerical examples are given to show that the theoretical reasoning is correct and the methods in this paper are effective.