On The Semiconvergence Of Multisplitting Iterative Methods For Solving Singular Linear Systems

The parallel multisplitting iterative method for solving large linear system of equations was introduced by O'Leary and White [21] and has been further studied by many authors. In the past few years, many researchers have developed many multisplitting iterations for solving the large sparse linear systems. But the attention was mainly paid to monotone matrix (in particular M-matrix) and H-matrix (see [7, 10, 11, 16, 18, 20, 23, 32]). Only a few attention was paid to symmetric positive matrix (cf. [19, 21, 25, 32]), especially singular symmetric positive semidefinite matrix. In fact, many practical problems, such as the computation of steady state solutions for Markov processes and the discrete solution to elliptic partial differential equations subject to the Neumann boundary value condition (see [6]), have been involved in the singular case. Recently, some authors have studyied the parallel multisplitting methods for solving singular linear system(see [7, 16, 18, 26]).In this paper, we will mainly investigate the semiconvergence of multisplitting iterations for systems of singular linear equations in which the efficient matrix is Hermitian (symmetric) positive semidefinite. The arrangement of this paper is as follows.The development of the multisplitting iterative methods for the solution oflinear system in the past few years is simply introduced in the Chapter 1.In Chapter 2, a sort of parallel mutisplitting methods for solving consistent singular symmetric positive semidefinite linear system is presented using analogously diagonally compensated reduction. And we investigate the semiconvergence of the presented parallel multispliting methods not only in any positive semidefinite coefficient matrix A but also in constructed coefficient matrix A. Some of these new results generate results in [9]. Furthermore, in the end of this chapter we give a simple numerical experiment.In Chapter 3, we mainly discuss semiconvergence conditions for additive and multiplicative splitting iteration methods, i.e., two generalizations of the additive and the multiplicative Schwarz iterations, for singular Hermitian semidefinite systems of linear equations. We present new theories for singular linear equations which generate the results of [4] quite different from the existing theorems.