On Preconditioners For Block Tridiagonal Symmetric Indefinite Linear Systems

In practical application, people often face the problems of solving symmetric in-definite linear systems. In general, these problems are symmetric tridiagonal indefiniteand with large sparse structure. The purpose of this paper lies in giving proper pre-conditioners of block tridiagonal symmetric indefinite linear systems, improving thecalculation of the condition number, so that the convergence rate of iterative method isgreatly improved.It is unlikely to solve the large definite system with the direct solution because of itshigh storage requirements. It is solved by iterative method usually. Iterative method canmake full use of sparsity of the matrix and saves a lot of storage space. In recent years,the iterative method of tridiagonal linear system has a new development, especially forthe reference of preconditioners. It greatly improves the convergence speed of iterativemethod to fulfill the computing needs. Perturbation analysis illustrates the distancebetween exact solution of new problem and origin problem, which plays a importantrole in study of numerical calculation.This paper illustrates a kind of preconditioning methods of block tridiagonal sym-metric indefinite linear systems. The importance of matrix decomposition in precon-ditioners’ selection is paid attention to under the condition of the strong understandingof preconditioning methods of saddle point problem. A kind of matrix decompositionmethod of saddle point problem is generalized to block tridiagonal symmetric indefinitelinear systems. Firstly, it makes a research on generalized Cholesky decomposition ofblock tridiagonal symmetric indefinite linear systems in this paper as well as precon-ditioners are designed in this way. The new preconditioners make a smaller conditionnumber of linear equations, which is proved by Wely theorem. Secondly, based on thematrix perturbation theory, it makes a research on perturbation analysis of generalized Cholesky decomposition and gets a few normwise perturbation bounds and component-wise perturbation bounds of these matrix decompositions. At the end of article, there isa numerical example which, through the iteration of conjugate gradient method, illus-trates different choice of preconditioners directly affect the final number and the speedof the iterations. The numerical example proves the effectiveness of the given numericalmethod.