Study Of The Semi-Iterative Method For A Class Of Nonsymmetric Matrix

How to solve the large linear equations is the core of the large-scale science and engineering project computation. With the development of the practice, direct method has been substituted by iterative method and iterative method has become one kind of the important methods for solving large linear equations. Semiiteration is one of the iterative methods. Compared with the usual methods, semiiterative method not only can improve the convergence rate of the linear equations, but also converge for the original equations which may be divergence. From 1957, the term "semi-iterative methods" was first used by Varga, many authors have studied it(see[1]-[16]). In this paper, we disscuss mainly with the convergence of the semi-iterative method for a class of nonsymmetric matrix.In [17], the convergence of the semi-iterative method for the linear equations Ax=b was analyzed under the assumptions that the iterative matrix is symmetric matrix(its eigenvalues are reals). In chapter 2, following the similar method and using Chebyshev polynomial and its properties, we study the convergence of the semiiterative method for the linear equations when the iterative matrix is antisymmetric matrix(its eigenvalues are imaginaty or zero). Thus, we extend the usual convergence condition to a new situation. In§2. 3, examples are given for illustrate the effect and practicability of our new results.In [18], the application of optimal semi-iteration to the standard successive overrelaxation (SOR) iterative method, with any real relaxation factorω, was analyzed under the assumptions that the associated Jacobi matrix B is consistently ordered and weakly cyclic of index 2 and that the spectrum,σ(B~2),of B~2 satisfies In chapter 3, we extend the results of [18], using analogous assumptions and techniques, starting the functional relationship to analyze the application of SSOR iterative method under the assimptions that the associated Jacobi matrix B is consistently ordered and weakly cyclic in index 2 and that the spectrum,σ(B~2), of B~2 satifies(This is the so-called "nonnegative case"). SSOR-SI method is studied. It is shown in Theorem 3.3.1 that there is semi-iterative method applied to the SSOR method with optimal relaxation parameterω=ω_b. Moreover, we obtain an interested result [see Theorem 3.4.1]: a smaller asymptotic convergence factor can be achieved by using a semi-iterative method together with a suboptimal relaxation factor (ω_s＜ω_b) when more profound information onσ(B~2), of the form is at hand.Under the assumptions that the associated Jacobi matrix B is consistently ordered and weakly cyclic in index 2 and that the spectrum,σ(B~2), of B~2 satifies (This is the so-called "nonpositive case"). In chapter 4, the semi-iterative SSOR method is studied and the result is the same as theorem 3.3.1.