Abstract:
A preconditioner of the type for speeding up convergence of the successive overrelaxation (SOR) iterative method for solving linear system is proposed. Two forms of the preconditioned SOR iteration are obtained and implemented, under limited conditions imposed on the coefficient matrix of the original linear system. Convergence properties are analyzed and established in conformity with standard procedures. The rates of convergence of the preconditioned iterations are shown to supersede that of the SOR method. Numerical experiments confirmed the established theoretical results.

Abstract:
-(-)matrices appear in many areas of science and engineering, for example, in the solution of the linear complementarity problem (LCP) in optimization theory and in the solution of large systems for real-time changes of data in fluid analysis in car industry. Classical (stationary) iterative methods used for the solution of linear systems have been shown to convergence for this class of matrices. In this paper, we present some comparison theorems on the preconditioned AOR iterative method for solving the linear system. Comparison results show that the rate of convergence of the preconditioned iterative method is faster than the rate of convergence of the classical iterative method. Meanwhile, we apply the preconditioner to -matrices and obtain the convergence result. Numerical examples are given to illustrate our results.

Abstract:
Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is proved. And the convergence and comparison theorem for any preconditioner are indicated. This comparison theorem indicates the possibility of finding new preconditioner and splitting. The purpose of this paper is to show that the preconditioned iterative method yields a new splitting satisfying the regular or weak regular splitting. And new combination preconditioners are proposed. In order to denote the validity of the comparison theorem, some numerical examples are shown.

Abstract:
In this paper, the improved symmetric SOR (ISSOR) iterative method is introduced to solve augmented systems. Convergence properties of the proposed method are studied. Some numerical experiments of the ISSOR method are given to compare with that of the well-known SOR-like and MSSOR methods.

Abstract:
We present a preconditioned mixed-type splitting iterative method for solving the linear system , where is a Z-matrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give one numerical example to illustrate our results. 1. Introduction For solving linear system, where is an square matrix and and are -dimensional vectors, the basic iterative method is where and is nonsingular. Thus, (2) can be written as where and . Assuming that has unit diagonal entries, let , where is the identity matrix and and are strictly lower and strictly upper triangular parts of , respectively. Transform the original system (1) into the preconditioned form as follows: Then, we can define the basic iterative scheme as follows: where and is nonsingular. Thus, the equation above can also be written as where and . In paper [1], Cheng et al. presented the mixed-type splitting iterative method as follows: with the following iterative matrix: where is an auxiliary nonnegative diagonal matrix, is an auxiliary strictly lower triangular matrix, and . In this paper, we will establish the preconditioned mixed-type splitting iterative method with the preconditioners , , and for solving linear systems. And we obtain some comparison results which show that the rate of convergence of the preconditioned mixed-type splitting iterative method with is faster than that of the preconditioned mixed-type splitting iterative method with or . Finally, we give one numerical example to illustrate our results. 2. Preconditioned Mixed-Type Splitting Iterative Method For the linear system (1), we consider its preconditioned form as follows: with the preconditioner ; that is, We apply the mixed-type splitting iterative method to it and have the corresponding preconditioned mixed-type splitting iterative method as follows: that is, So, the iterative matrix is where , , and are the diagonal, strictly lower, and strictly upper triangular matrices obtained from , is an auxiliary nonnegative diagonal matrix, is an auxiliary strictly lower triangular matrix, and . If we choose , we have the following corresponding iterative matrix: And if we choose , we have the following corresponding iterative matrix: If we choose certain auxiliary matrices, we can get the classical iterative methods as follows.(1)The PSOR method is (2)The PAOR method is We need the following definitions and results. Definition 1 (see [2]). A matrix is a -matrix if , for all , such

Abstract:
Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but their convergence theory remains sparse and complex. We consider the simplest preconditioned eigensolver--the gradient iterative method with a fixed step size--for symmetric generalized eigenvalue problems, where we use the gradient of the Rayleigh quotient as an optimization direction. A sharp convergence rate bound for this method has been obtained in 2001--2003. It still remains the only known such bound for any of the methods in this class. While the bound is short and simple, its proof is not. We extend the bound to Hermitian matrices in the complex space and present a new self-contained and significantly shorter proof using novel geometric ideas.

Abstract:
In this paper, we present a preconditioned variant of the generalized successive overrelaxation (GSOR) iterative method for solving a broad class of complex symmetric linear systems. We study conditions under which the spectral radius of the iteration matrix of the preconditioned GSOR method is smaller than that of the GSOR method and determine the optimal values of iteration parameters. Numerical experiments are given to verify the validity of the presented theoretical results and the effectiveness of the preconditioned GSOR method.

Abstract:
The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results.

本文利用块预条件技术考虑了解线性方程组Ax=b的块预条件AOR迭代法。当方程组的系数矩阵A是H-矩阵时，得出了该方法的收敛性结果。
We consider block AOR preconditioned iterative method for solving the linear system Ax=b , using the preconditioning technology. When the coefficient matrix A is an H-matrix, the conver-gence results of the presented method are given.

Abstract:
Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using multilevel preconditioning for elliptic differential operators or graph Laplacian eigenvalue problems, exhibit almost optimal complexity in practice, i.e., their computational costs to calculate a fixed number of eigenvalues and eigenvectors grow linearly with the matrix problem size. Theoretical justification of their optimality requires convergence rate bounds that do not deteriorate with the increase of the problem size. Such bounds were pioneered by E. D'yakonov over three decades ago, but to date only a handful have been derived, mostly for symmetric eigenvalue problems. Just a few of known bounds are sharp. One of them is proved in [doi:10.1016/S0024-3795(01)00461-X] for the simplest preconditioned eigensolver with a fixed step size. The original proof has been greatly simplified and shortened in [doi:10.1137/080727567] by using a gradient flow integration approach. In the present work, we give an even more succinct proof, using novel ideas based on Karush-Kuhn-Tucker theory and nonlinear programming.