解线性方程组的松弛方法及其应用

线性方程组在科学和工程领域中有着重要的应用,松弛方法是求解线性方程组的有效算法之一.本文在著名的Gauss-Seidel迭代法的基础上,研究了一种有效的松弛方法.理论分析表明,该方法能收敛到线性方程组的唯一解.此外,我们还将该方法应用在鞍点问题和PageRank问题的求解上,并得出了相应的数值结果.结果表明该方法比现有的松弛方法更有效.     

Skew-Hermitian Triangular Splitting Iteration Methods for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts

By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

Convergence Domains of AOR Type Iterative Matrices for Solving Non-Hermitian Linear Systems

We discuss AOR type iterative methods for solving non-Hermitian linear systems based on Hermitian splitting and skew-Hermitian splitting. Convergence domains of iterative matrices are given and optimal parameters are investigated for skew-Hermitian splitting. Numerical examples are presented to compare the effectiveness of the iterative methods in different points in the domain. In addition, a model problem of three-dimensional convection-diffusion equation is used to illustrated the application of our results.     

Block Projection Methods for Linear Systems

The aim of this paper is to provide a theory of block projection methods for the solution of a system of linear equations with multiple right-hand sides. Our approach allows to obtain recursive algorithms for the implementation of these methods.     

Relaxation Methods for Strictly Convex Regularizations of Piecewise Linear Programs

We give an algorithm for minimizing the sum of a strictly convex function and a convex piecewise linear function. It extends several dual coordinate ascent methods for large-scale linearly constrained problems that occur in entropy maximization, quadratic programming, and network flows. In particular, it may solve exact penalty versions of such (possibly inconsistent) problems, and subproblems of bundle methods for nondifferentiable optimization. It is simple, can exploit sparsity, and in certain cases is highly parallelizable. Its global convergence is established in the recent framework of B -functions (generalized Bregman functions). Accepted 29 October 1996     

Symplectic Multistep Methods for Linear Hamiltonian Systems

1.IntroductionProhaorFengKangadvancedtheprincipleforconstructiollofsymplecticalgrvrithm8forHarniloniansystemsI11andpointedout.thatsymplecticalgoritlunscanre-ffedmainhauresofHtalltonianSystems,thereforetheyaremoreavailable.Plentyoft~talandrnunericalresultshaveprovedthesepoints.PrO~FengKangalsodiscussedtheaPproalmationproblemsbyalgebraicfull-tfonS.Theconclusionsaxestatedasfolfows[2l:1.wenoteop(f)=p(()/a(f).AmulistepmethodM(p,a)issymplecticforlinearIhailonhaSy8tems(wecallitlinearsymPlectic…     

非Hermite线性方程组的若干预处理迭代算法

非Hermite线性方程组在科学和工程计算中有着重要的理论研究意义和使用价值,因此如何高效求解该类线性方程组,一直是研究者所探索的方向.通过提出一种预处理方法,对非Hermite线性方程组和具有多个右端项的复线性方程组求解的若干迭代算法进行预处理,旨在提高原算法的收敛速度.最后通过数值试验表明,所提出的若干预处理迭代算法与原算法相比较,预处理算法迭代次数大大降低,且收敛速度明显优于原算法.除此之外,广义共轭A-正交残量平方法(GCORS2)的预处理算法与其他算法相比,具有良好的收敛性行为和较好的稳定性.     

奇异线性方程组的并行多分裂迭代法

改进了奇异M-矩阵的线性方程组的并行多分裂法的一些最近结果,给出了并行多分裂迭代方法的一些收敛性的理论结果。     

Linear Systems Associated with Numerical Methods for Constrained Optimization

Linear systems associated with numerical methods for constrained optimization are discussed in thia paper ,It is shown that the corresponding subproblems arise in most well-known methods,no matter line search methods or trust region methods for constrained optimization can be expressed as similar systems of linear equations.All these linear systems can be viewed as some kinds of approximation to the linear system derived by the Lagrange-Newton method .Some properties of these linear systems are analyzed.     

Monotone Convergence of Iterative Methods for Singular Linear Systems

In this paper, monotonicity of iterative methods for solving general solvable singularly systems is discussed. The monotonicity results given by Berman, Plemmons, and Semal are generalized to singular systems. It is shown that for an iterative method introduced by a nonnegative splitting of the coefficient matrix there exist some initial guesses such that the iterative sequence converges towards a solution of the system from below or from above. The monotonicity of the block Gauss-Seidel method for solving a p-cyclic system and Markov chain is considered.     

Methods for Studying Stability of Systems with Linear Delay

Siberian Mathematical Journal -     

Block Descent Methods and Hybrid Procedures for Linear Systems

In this paper, we define and study several types of block descent methods for the simultaneous solution of a system of linear equations with several right hand sides. Then, improved block EN methods will be proposed. Finally, block hybrid and minimal residual smoothing procedures will be considered.     

解线性方程组的预条件Gauss-Seidel型迭代法

给出了解线性方程组的预条件Gauss-Seidel型方法,提出了选取合适的预条件因子．并讨论了对Z-矩阵应用这种方法的收敛性,给出了收敛最快时的系数取值．最后给出数值例子,说明选取合适的预条件因子应用Gauss-Seidel方法求解线性方程组是有效的．     

MATLAB中大型线性方程组的非定常迭代法

科学研究和大型工程设计中很多问题以非线性数学模型来描述,而这些数学模型求解常常归结为各种大型线性方程组的求解,因而能否有效地求解大型线性方程组,特别是病态的方程组,是非常关键的.本文介绍了MATLAB中求解大型线性方程组常用的非定常迭代法,并以GMRES算法为例介绍了算法的数学描述.     

Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems

1.IlltroductiollInmanyareasofphysics,mechanics,etc.,HamiltoniansystemsofODEsplayaveryimportantrole.Suchsystemshavethefollowinggeneralform:where,bydenotingwithOfandimthenullmatrixandtheidentitymatrixofordermarespectively,SimplepropertiesofthematrixJZmarethefollowingones:Inequation(1)AH(~,t)isthegradientofascalarfunctionH(y,t),usuallycalledHamiltonian.InthecasewhereH(y,t)=H(y),thenthevalueofthisfunctionremainsconstantalongt.hesollltion7/(t),t,hatis'*ReceivedFebruaryI3,1995.l)Worksupporte…     

Initial-Boundary Value Problem for a Class of Linear Relaxation Systems in Arbitrary Space Dimensions

In this paper, we consider the characteristic initial-boundary value problem (IBVP) for the multi-dimensional Jin-Xin relaxation model in a half-space with arbitrary space dimension n?2. As in the one-dimensional case (n=1, see (J. Differential Equations, 167 (2000), 388-437), our main interest is on the precise structural stability conditions on the relaxation system, particularly the formulation of boundary conditions, such that the relaxation IBVP is stiffly well posed, that is, uniformly well posed independent of the relaxation parameter ε>0, and the solution of the relaxation IBVP converges, as ε→0, to that of the corresponding limiting equilibrium system, except for a sharp transition layer near the boundary. Our main result can be roughly stated as Stiff Kreiss Condition=Uniform Kreiss Condition for the relaxation IBVP we consider in this paper, which is in sharp contrast to the one-dimensional case (Z. Xin and W.-Q. Xu, J. Differential Equations, 167 (2000), 388-437). More precisely, we show that the Uniform Kreiss Condition (which is necessary and sufficient for the well posedness of the relaxation IBVP for each fixed ε), together with the subcharacteristic condition (which is necessary and sufficient for the stiff well posedness of the corresponding Cauchy problem), also guarantees the stiff well posedness of our relaxation IBVP and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate. Optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified.     

线性方程组的异步松弛迭代法*

本文考虑解线性方程组经典迭代法的异步形式,对系数矩阵为H矩阵,给出了异步迭代过程收敛性的充分条件,这不仅降低了文献[3]对系数矩阵的要求,而且收敛区域比文献[3]的大.     

Polynomial Acceleration Methods for Solving Singular Systems of Linear Equations

In this paper we study the polynomial acceleration methods for solving singular linear systems. We establish iterative schemes, show their convergence and find iteration error bounds.