On The Study Of Preconditioned Krylov Subspace Iterative Methods And Their Application

The efficient method for solving large sparse linear equations is a very important subject in computational mathematics,which is widely used in the field of scientific and engineering computation.This paper mainly studies how to efficiently solve large sparse positive definite linear equations.Firstly,based on the original alternating LU algorithm,ALU iterative method is proposed and its convergence is analyzed.Secondly,the ALU preconditioner is designed,the spectral property of the preconditioned matrix and the convergence of preconditioned Krylov subspace iteration method are given.Numerical experiments demonstrate that the ALU preconditioner greatly accelerates the convergence of Krylov iteratve method and the effect is better than the symmetric Gauss-Seidel preconditioner similar to the ALU preconditioner.Finally,the ALU preconditioned Krylov subspace iterative method is applied to bicubic uniform B-spline surface fitting.The experimental results show that the ALU preconditioned Krylov subspace iterative method is superior to the conmonly methods for surface fitting,that is,the progressive iterative approximation method and the weighted progressive iterative approximation method.This thesis consists of four chapters:In Chapter 1,we briefly introduce the research background and status of large sparse positive definite linear equations as well as innovations of this paper.In Chapter 2,we review some definitions and lemmas used in this thesis.In Chapter 3,we deeply study a new ALU preconditioning technique,which can improve the computational efficiency of Krylov subspace iterative method for solving large sparse positive definite linear equations.In Chapter 4,we apply Krylov subspace iterative method preconditioned by ALU preconditioner to bicubic uniform B-spline surface fitting.