A New Class AOR Preconditioner for $L$-Matrices

Hadjidimos(1978) proposed a classical accelerated overrelaxation(AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant, L-matrices, and consistently orders matrices. Several preconditioned AOR methods have been proposed to solve system of linear equations Ax = b, where A ∈ R~(n×n) is an L-matrix. In this work, we introduce a new class preconditioners for solving linear systems and give a comparison result and some convergence result for this class of preconditioners. Numerical results for corresponding preconditioned GMRES methods are given to illustrate the theoretical results.     

Generalized consistent orderings and the Accelerated Overrelaxation method

In this paper, the behavior of the block Accelerated Overrelaxation (AOR) iterative method, when applied to linear systems with a generalized consistently ordered coefficient matrix, is investigated. An equation, relating the eigenvalues of the block Jacobi iteration matrix to the eigenvalues of its associated block AOR iteration matrix, as well as sufficient conditions for the convergence of the block AOR method, are obtained.     

广义块Toeplitz特征值问题的基于sine变换的预处理子

在求块Toeplitz矩阵束（Amn，Bmn）特征值的Lanczos过程中，通过对移位块Toepltz矩阵Amn-ρBmn进行基于sine变换的块预处理，从而改进了位移块Toeplitz矩阵的谱分布，加速了Lanczos过程的收敛速度．该块预处理方法能通过快速算法有效快速执行．本文证明了预处理后Lanczos过程收敛迅速，并通过实验证明该算法求解大规模矩阵问题尤其有效．     

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

We present the convergence analysis of the inexact infeasible path-following (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of the interior-point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis. Communicated by Y. Zhang.     

The AOR iterative method for new preconditioned linear systems

In this paper we apply the AOR method to preconditioned linear systems different from those considered in Evans and Martins (Internat. J. Comput. Math. 5 (1995) 69–76), Gunawardena et al. (Linear Algebra Appl. 154–156 (1991) 123–143) and Li and Evans (Technical Report No. 901, Department of Computer Studies, University of Loughborough, 1994). Our results show that some improvements in the convergence rate of this iterative method can be obtained.     

On HSS-based constraint preconditioners for generalized saddle-point problems

For the generalized saddle-point problems with non-Hermitian (1,1) blocks, we present an HSS-based constraint preconditioner, in which the (1,1) block of the preconditioner is constructed by the HSS method for solving the non-Hermitian positive definite linear systems. We analyze the invertibility of the HSS-based constraint preconditioner and prove the convergence of the preconditioned iteration method. Numerical experiments are used to demonstrate the efficiency of the preconditioner as well as the corresponding preconditioned iteration method, especially when the (1,1) block of the saddle-point matrix is essentially non-Hermitian.     

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

     

Semiconvergence of extrapolated iterative methods for singular linear systems

In this paper, we discuss convergence of the extrapolated iterative methods for solving singular linear systems. A general principle of extrapolation is presented. The semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coefficient matrix A is a singular M-matrix with ‘property c’ and an irreducible singular M-matrix, respectively. Since the (generalized, block) JOR and AOR methods are respectively the extrapolated methods of the (generalized, block) Jacobi and SOR methods, so the semiconvergence of the (generalized, block) JOR and AOR methods for solving general singular systems are proved. Furthermore, the semiconvergence of the extrapolated power method, the (block) JOR, AOR and SOR methods for solving Markov chains are discussed.     

求解加权线性最小二乘问题的一类预处理GAOR方法

为了快速求解一类来自加权线性最小二乘问题的2×2块线性系统,本文提出一类新的预处理子用以加速GAOR方法,也就是新的预处理GAOR方法.得到了一些比较结果,这些结果表明当GAOR方法收敛时,新方法比原GAOR方法和之前的一些预处理GAOR方法有更好的收敛性.而且,数值算例也验证了新预处理子的有效性．     

PRECONDITIONING BLOCK LANCZOS ALGORITHM FOR SOLVING SYMMETRIC EIGENVALUE PROBLEMS

1. Introduction;The Lanczos process is an effective method [1, 2, 14, 21] for computing a feweigenValues and corresponding eigenvectors of a large sparse symmetric matrix A ERnxn. If it is practical to factor the matrix A -- PI for one or more values of p near thedesired eigenvalues, the Lanczos method can be used with the inverted operator andconvergence will be very rapid[5,10,22]. In practical applications, however, the matrixA is usually large and sparse, so factoring A is either impos…     

广义鞍点问题基于PSS的约束预条件子

对于(1,1)块为非Hermitian阵的广义鞍点问题,本文给出了一种基于正定和反对称分裂(Positive definite andskew-Hermitian splitting, PSS)的约束预条件子.该预条件子的(1,1)块由求解非Hermitian正定线性方程组时的PSS迭代法所构造得到.文中分析了PSS约束预条件子的一些性质并证明了预处理迭代法的收敛性.最后用数值算例验证了该预条件子的有效性.     

The effect of block red-black ordering on blockILU preconditioner for sparse matrices

It is well known that the ordering of the unknowns can have a significant effect on the convergence of a preconditioned iterative method and on its implementation on a parallel computer. To do so, we introduce a block red-black coloring to increase the degree of parallelism in the application of the blockILU preconditioner for solving sparse matrices, arising from convection-diffusion equations discretized using the finite difference scheme (five-point operator). We study the preconditioned PGMRES iterative method for solving these linear systems.     

An Automatic Procedure for Updating the Block Size in the Block Conjugate Gradient Method for Solving Linear Systems

The paper considers the problem of constructing an efficient automatic procedure for reducing the block size in the block conjugate gradient method insuring that the resulting rate of convergence is comparable with that of the block conjugate gradient method with constant block size. The numerical results provided show that, independently of the type of distribution of the smallest eigenvalues of the preconditioned matrix, the procedure suggested always leads to a decrease of the arithmetic costs with respect to those of the block method with constant block size. Bibliography: 8 titles.     

Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis

A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method. Received March 20, 1997 / Revised version received January 14, 1998     

求解2×2块对称不定线性方程组迭代法的收敛性分析(英文)

本文研究求解系数矩阵为2×2块对称不定矩阵时的线性方程组,提出了一种新的分裂迭代法,并通过研究迭代矩阵的谱半径,详细讨论了新方法的收敛性.最后,我们也讨论了预条件矩阵特征根的几条性质.     

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.     

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for ordinary differential equations require the solution of non‐symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation

Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD-id using a restricted formulation of the PSD-id. More importantly, we extend the new convergence analysis of the PSD-id to a practically preferred block version of the PSD-id, or BPSD-id, and show the cluster robustness of the BPSD-id. Numerical examples are provided to validate the theoretical estimates.     

Band Toeplitz preconditioners for block Toeplitz systems

We consider the solutions of block Toeplitz systems with Toeplitz blocks by the preconditioned conjugate gradient (PCG) method. Here the block Toeplitz matrices are generated by nonnegative functions f(x,y). We use band Toeplitz matrices as preconditioners. The generating functions g(x,y) of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing (f − g)/f∞. We prove that the condition number of the preconditioned system is O(1). An a priori bound on the number of iterations for convergence is obtained.