SAOR方法的收敛性

1.引言 迭代求解线性方程组Ax=b的AOR方法已是众所周知.由AOR迭代很自然联想到构造对称AOR(SAOR)迭代,但目前讨论SAOR迭代的文章还不多见.中对系数矩阵为H阵的SAOR迭代,[6]中对系数矩阵为对称正定阵的SAOR迭代,均给出了收敛性定理.本文讨论系数矩阵为对角元素非零的相容次序阵时SAOR迭代的收敛性,得到了相应的收敛性定理,并给出了SAOR迭代矩阵谱半径表达式以及谱半径的一个上下界.     

AOR方法的收敛性

A.Hadjidimos在[1]中提出一个迭代求解线性方程组的 AOR方法(Accelerated Overre-laxation Method),并在方程组的系数矩阵为不可约弱对角优势、L-矩阵和相容有序矩阵的条件下,讨论了此方法的收敛性.在这篇文章里,我们将考虑系数矩阵是H-矩阵、正定矩阵以及L-矩阵的情况.所得结果表明,可以放宽在[1]的3,4两节中对参数所加的限制.     

具有完全并行度的SOR算法

1　引　　言Jacobi和 SOR迭代是求解线性方程组的两类基本的迭代方法 .并行计算机的出现使人们能立刻注意到它们在拥有并行处理性能上的显著差别 .Jacobi迭代因其各个分量的修正相互独立而具有十分明显的内在并行计算特性 .SOR则完全不同 ,其中诸分量的计算是逐个相关的 .由此而导致一般认为 SOR不适合并行处理 ,其内在并行性远不如 Jacobi迭代[1 ] [2 ] .由于 SOR多用于有限差分或有限元方法导致的大型稀疏方程组求解 ,因此 ,利用系数矩阵零元素或非零元素的特殊分布 ,采用红 -黑或多色排序成为实现 SOR并行处理的有效途径 .然而 ,…     

迭代求解线性方程组的FBAOR方法

本文在AOR方法的基础上构造出一种求解线性方程组的迭代格式,姑且称之为向前向后AOR方法(即FBAOR),它包含了熟知的Jacobi、Gauss—Seidel、SOR、SSOR、AOR以及SAOR等方法。本文主要讨论了当系数矩阵具有相容次序时FBAOR方法的若干性质,如收敛性、收敛速度等,证明了在一定条件下FBAOR方法能优于AOR方法。     

NEWTON—AOR方法的收敛性

本文将解线性方程组的AOR迭代法推广到解非线性方程组,构造和研究了Newton-AOR方法,建立了收敛性定理和比较定理,在一定条件下,从理论上证明了Newton—AOR方法比Newton—SOR方法收敛快,并给出了数值例子。文中所用有关概念和记号的意义见[1]。     

推广的迭代矩阵的收敛性

§1.引言 近若干年来,许多文献中讨论了线性代数方程组一些迭代格式的收敛性,亦即其系数矩阵A的各种分裂的收敛性和A为M阵或H阵的关系,例如Jacobi迭代、JOR迭代、SOR迭代、SSOR(对称SOR)迭代和AOR(快速SOR)迭代等等。在[4]中我们已将这样送代的迭代矩阵推广为 G_1=(D-RL)~(-1)[I-Ω)D (Ω-R)L ΩU], (1)这里D=diagA,L和U分别为-A的严格下三角矩阵和严格上三角矩阵,I为n阶单位阵,n为A的阶数,R和Ω为对角阵:     

循环矩阵和AOR方法的敛散区域

§1.引言解线性方程组通常采用迭代法 1978年,HadjidimosA.在[1]中提出了Accelerated Overrelaxation Method(简其中D、L、U分别为A的对角部分、严格下和严格上三角部分,则AOR方法的迭代矩阵为     

SOR和AOR迭代收敛的新判别准则

充分条件1和2,对于SOR迭代(0<ω≤1)和AOR迭代(0≤r≤1,0<ω≤1)也是适用的。那么,充分条件3对于SOR迭代和AOR迭代是否适用呢?迄今为止尚没有讨论过。这里我们给予肯定的回答。我们的结论基于如下两个引理。 引理1 如果A的主对角元全不为零,且满足条件(4),则detA≠0。 引理2 对线性方程组(1)的迭代法     

块AOR迭代法的收敛性

本文推广了解线性方程组的AOR迭代法,给出了块AOR迭代法(BAOR迭代法).文中引进了块M-矩阵,块H-矩阵,块严格对角优势矩阵,块Hermite正定矩阵,块相容次序矩阵和广义块相容次序矩阵等概念.在线性方程组的系数矩阵分别具有上述性质的假设下,讨论了BAOR迭代法的敛散性.     

AOR迭代SOR迭代和JOR迭代的收敛性

1 引言和术语 已知n阶线性方程组 A_x=b, (1.1)其中A是n×n复或实矩阵,b是n维复或实向量。 本文研究当A是对角优势矩阵时,求解(1.1)的AOR迭代、SOR迭代和JOR迭     

Convergence analysis of the parallel multisplitting TOR method

In this paper, we propose the parallel multisplitting TOR method, for solving a large nonsingular systems of linear equations Ax = b. These new methods are a generalization and an improvement of the relaxed parallel multisplitting method (Formmer and Mager, 1989) and parallel multisplitting AOR Algorithm (Wang Deren, 1991). The convergence theorem of this new algorithm is established under the condition that the coefficient matrix A of linear systems is an H-matrix. Some results also yield new convergence theorem for TOR method.     

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.     

Extended convergence regions for the AOR method

In this paper, we give sufficient conditions for the convergence of the (AOR) method, when the matrix A for Ax = b is a strictly diagonally dominant matrix. These results improve the conclusions obtained in the Theorem 4 [10].With the notion of generalized diagonal dominant matrix, we enlarge the convergence regions given in Theorem 9 [10], when A is a nonsingular H-matrix.In the last section we generalize theorem 6 of Robert [11] and we present some results which extend the convergence regions for the (AOR) method.     

一类矩阵的AOR迭代收敛性分析及其与SOR迭代的比较

1 引言 许多实际问题最后常归结为解一个或一些矩阵的线性代数方程组Ax=b （1.1）这里讨论A为（1,1）相容次序矩阵的情形。     

Convergence of two-stage multisplitting method using AOR or SSOR multisplittings

In this paper, we study the convergence of two-stage multisplitting method using AOR or SSOR multisplittings as inner splittings and an outer splitting for solving a linear system whose coefficient matrix is an H-matrix. We also introduce an application of the two-stage multisplitting method.     

A new optimized iterative method for solving M-matrix linear systems

In this paper, we present a new iterative method for solving a linear system, whose coefficient matrix is an M-matrix. This method includes four parameters that are obtained by the accelerated overrelaxation (AOR) splitting and using the Taylor approximation. First, under some standard assumptions, we establish the convergence properties of the new method. Then, by minimizing the Frobenius norm of the iteration matrix, we find the optimal parameters. Meanwhile, numerical results on test examples show the efficiency of the new proposed method in contrast with the Hermitian and skew-Hermitian splitting (HSS), AOR methods and a modified version of the AOR (QAOR) iteration.

     

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.     

Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices

In this paper, some improvements on Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133] are presented for bounds of the spectral radius of _lω,r$l_{\omega ,r}$, which is the iterative matrix of the generalized AOR (GAOR) method. Subsequently, some new sufficient conditions for convergence of GAOR method will be given, which improve some results of Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133].     

A class of two‐stage iterative methods for systems of weakly nonlinear equations

The discretizations of many differential equations by the finite difference or the finite element methods can often result in a class of system of weakly nonlinear equations. In this paper, by applying the two-tage iteration technique and in accordance with the special properties of this weakly nonlinear system, we first propose a general two-tage iterative method through the two-tage splitting of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a two-tage AOR method, which particularly uses the AOR iteration as the inner iteration and is substantially a relaxed variant of the afore-presented method. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only B-differentiable. When the system matrix is either a monotone matrix or an H-matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new two-tage iteration methods, and investigate the influence of the matrix splittings as well as the relaxation parameters on the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for solving the system of weakly nonlinear equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

并行区间矩阵多分裂AOR算法及其收敛性

本文提出了一类求解大型区间线性方程组的并行区间矩阵多分裂松弛算法,并在系数矩阵是区间H-矩阵的条件下,建立了这类算法的收敛理论。