On The Study Of Iterative Methods For Two Classes Of Positive Definite Linear Systems

The efficient schemes for solving large sparse linear systems of equations lie at the heart of massive-scale scientific and engineering computing.This thesis focuses on how to efficiently solve large sparse positive definite linear systems of equations.First of all,inspired by the idea of GHSS method[9],we present a kind of positive definite and positive definite splitting(PPS)iteration method for solving large sparse positive definite linear systems of equations,and prove that the PPS method converges to the unique solution of the linear system of equations unconditionally.A numerical test demonstrates that the PPS method is more effective in solving the two-dimensional convection-diffusion problems than the celebrated HSS method[6].Next,we study the solution of the positive definite CUPL-Toeplitz linear systems of equations derived from the Markov chain.Recently,Fu et al.[26]proposed two fast algorithms by employing the inverse formula of a Toeplitz matrix and the ShermanMorrison-Woodbury formula.However,two Toeplitz linear systems of equations need to be solved in each algorithm.In order to improve the computing efficiency,we develop two algorithms whose computational complexities are less than those of Fu’s algorithms.Numerical experiments are provided to illustrate the effectiveness.