The Analysis On Convergence Of The SAOR For One Type Matrix And Semiconvergence Of The Second Quasi-nonnegative Splitting

The solutions of many problems in mathematics, physics, mechanics, engineering and so on are sumed up to the solutions of one or some large sparse linear systems which usually are solved by iterative method. The critical matter of the study of iterative methods is the convergence and the convergence rate of the iterative methods. Of course, the divergent methods will not be adopted. If the method has a low rate of convergence, the time of the human and machines will be wasted and the answer are not surely attainable. So, we must look for the methods with the high rate of convergence or try to settle some parameters of the iteration methods (for instance the overrelaxation parameter of SOR iterative method). In general, the convergence of the iterative method is closely related to the property of the coefficient matrix of linear systems, for instance, nonnegative matrices, cyclic matrices, M matrices, H matrices and so on. If the matrices are different, then the research methods of iterative methods will also be different. Therefore, it's often to study the convergence for the given matrices, of the iterative method. One type matrix in the title this paper just is (1, 1) consistently ordered matrix.This paper contains three chapters. The main results of chapters as following:Chapter 1 Preliminaries. This part mainly makes preparations for Chapter 2 and Chapter 3. Chapter 1 mainly introduce some elementary concepts about matrix: consistently ordered matrix, matrix norm, nonnegative matrix and famous Perron-Frobenius theorem. They are important theoretical foundation of the study about nonnegative matrices' spectral radius. Meanwhile, basic knownledge about Drazin inverse is simplely presented.Chapter 2 The convergence analysis on the SAOR interative method for consistently ordered matrices. The chapter is the main results part of this paper. The convergence of the SAOR interative method is discussed according to the formula about the relation between eigenvalues of the SAOR interative matrix and eigenvalues of its Jacobi matrix. When the coefficient matrix of a linear system is (1, 1) consistently ordered and all eigenvalues of its Jacobi matrix are real numbers, we compute the range of real parametersγandω, that is convergence area of the SAOR iterative method. At last, in the case thatγis 2 andωis complx, when all eigenvalues of its Jacobi matrix are real numbers or all pure imaginaries, we discuss the convergence and the optimum parameters of its SAOR method.Chapter 3 The semiconvergence analysis on the second quasi-nonnegative splitting. Firstly, we introduce some background knowledge about singular linear systems, give the definition and equivalence conditions of the semiconvergence. Based on this, we introduce a new concept of the second quasi-nonnegative splitting that is from quasi-nonnegative splitting and the second weak splitting. Finally, we study the equivalence theorem and the comparison theorem of semiconvergence for the second quasi-nonnegative splitting.