The Study Of Iterative Methods For Saddle Point Problems

The numerical solution of large sparse linear equations has been a classical research topic in the field of science and engineering computing.As a class of special linear equations,the study of numerical solution algorithm for saddle point problem has extremely important theoretical and practical significance.Therefore,how to solve the saddle point problem quickly and effectively has become an important research object of many scholars.In practical problems.the coefficient matrix of many saddle point problem is a large sparse matrix with special properties.The solution to such problems is usually an iterative method,which is also the focus of this paper.This paper proposes a generalized positive-definite and skew-Hermitian splitting method algorithm to solve the large sparse saddle-point problems based on the positive-definite splitting.The method first uses the positive definite splitting of matrix to construct two splitting forms of the saddle-point matrix.Then,the two kinds of splitting formats are used to construct the Generalized Positive-definite and Skew-Hermitian Splitting(GPSS)iteration.Next hen the convergence of the algorithm is analyzed,and the necessary and sufficient conditions for the convergence of the iterative are given.Finally,some numerical comparison experiments are carried out and it shows that the GPSS is more effective than "Positive-definite and skew-Hermitian Splitting"(PSS)and "Hermitian and skew-Hermitian Splitting",(HSS)methods.In order to further improve the algorithm solution rate,this paper further proposes asymmetric generalized positive definite and Hermitian splitting(AGPSS)algorithm and lopsided generalized positive definite and Hermitian splitting(LGPSS)algorithm for solving the saddle point problem,and compares the algorithm with the classical solution algorithm through numerical experiments to prove the effectiveness of the algorithm.Finally,the paper summarizes the full text.and further analyzes the research direction.