严格对角占优Z-矩阵的多级预条件AOR迭代法

为了加快线性方程组的迭代法求解速度,提出了一类新预条件子,分析了相应的预条件AOR迭代法的收敛性。给出了当系数矩阵为严格对角占优的Z-矩阵时,AOR和预条件AOR迭代法收敛速度的比较结论。同时也给出了多级预条件迭代法的相关比较结果,推广了现有的结论。数值算例验证了文中结果。     

稀疏近似逆并行预条件子

1．引言考虑求解线性方程组AX一b,X,bE＊”,山其中A二（a;小＿是大型稀疏非对称矩阵．通常使用迭代法求解式（1）,如GMRESBICGSTAB,CGSTFQMRCGSZ等Kryl0V子空间迭代法．直接使用迭代法的收敛速度有时特别慢,或根本不收敛,需使用预条件以加速迭代法的收敛速度．通常使用左或右预条件子M使式（1）变成易于求解的形式＊M9一6,X二M队或＊AX二＊6．由然后用迭代法求解式（2）,M的选择要使得AM（或M则近似等于单位矩阵．构造预条件子的方法有很多,如不完全分解方法、SSOR方法、多项式方法等,不完全分解方法和SSOR…     

求解鞍点问题的一般加速超松弛方法

针对大型稀疏鞍点问题给出了一种含有待定参数的新迭代解法,将其称之为一般加速松弛方法,简记为GAOR方法．当参数α=时,新迭代方法是变成由Golub等人给出的SOR-Like方法．该迭代法的构成是基于对系数矩阵进行的一种分裂．迭代法需要选择一个预处理矩阵和待定参数,通过适当选取预处理矩阵和待定参数,新迭代法是收敛的,并且以定理的形式给出了新迭代方法的迭代矩阵的特征值和参数之间的基本等式,从而也导出了迭代法收敛的充分和必要条件．理论结果表明新方法更具有广泛性,并且适当的选择参数可以使新方法较SOR-Like方法具有更快的收敛速度．在文中的最后给出了迭代法的数值试验结果．     

一种求解鞍点问题的广义预条件对称一反对称分裂迭代法

鞍点问题的来源和应用都很广泛，如计算流体力学，约束最优化，约束加权最小二乘问题等。寻求快速有效地求解这类问题的算法具有很重要的现实意义．在白中治，Golub和潘建瑜提出的预条件对称／反对称分裂迭代法（PHSS）的基础上，通过引入新的待定参数对原有迭代算法进行加速的思想，本文提出了一种解鞍点问题的具有两个待定参数的广义预条件对称／反对称分裂迭代法（GPHSS），并给出了该算法收敛性的条件．数值例子表明：通过最优参数值的选择，新算法比PHSS算法具有更快的收敛速度和更小的迭代次数，选择了最优参数值后，可以提高算法的收敛效率．     

步长加速法优化B样条参数的离散数据点拟合

采用迭代法拟合离散数据点时,数据点的参数化会同时影响逼近的效果和逼近的速度,为此,提出一种通过迭代调整优化控制顶点和数据点参数的方法,其收敛速度较快且拟合得到曲线更贴合控制点.首先,选取初始控制顶点,通过自适应的BFGS方法优化控制顶点得到拟合曲线;其次,保持控制顶点不变,利用步长加速法优化数据点对应的参数;最后,利用新参数值重新优化控制顶点并得到新的拟合曲线.数值实例表明,所提方法在迭代前期步骤中,收敛速度快于现有的基于控制顶点迭代法,且优化后的曲线更加逼近离散的数据点,拟合误差更小.     

加速松弛法最佳参数的确定

§1.引言 求线性方程组的数值解有直接法和迭代法。在迭代法中,超松弛迭代占有重要地位。文[1]把超松弛迭代推广到双参数的情况(称加速松弛法),对在什么条件下方法收敛的问题进行了讨论,并指出如何确定加速松弛法的最佳参数是有待今后解决的问题。本文确定了加速松弛法的最佳参数,使迭代矩阵的谱半径达到最小,并在各种情况下对加速松弛法与超松弛法的收敛速度进行了比较。     

H-矩阵方程组的预条件迭代法

针对系数矩阵A为H-矩阵的线性方程组Ax=b,引入了预条件矩阵I＋S_α~β,通过对系数矩阵施行初等行变换,提出了求解线性方程组Ax=b的一种新的预条件Gauss-Seidel方法.论文中首先证明了若A为H-矩阵,则（I＋S_α~β）A仍然是H-矩阵;其次,以定理的形式给出了新的预条件Gauss-Seidel方法收敛的充分条件,即给出了为保证新的预条件Gauss-Seidel方法收敛时参数所需满足的条件;然后从理论上证明了新的预条件Gauss-Seidel迭代方法较经典的Gauss-Seidel迭代方法收敛速度快,论文中提出的新的预条件Gauss-Seidel迭代方法推广了文[1-2]中提出的预条件方法;最后又通过数值算例说明了新的预条件Gauss-Seidel迭代方法的有效性.     

Krylov子空间方法在一类PDE上的数值实验与分析

１．引言本文重点进行迭代算法的数值比较,利用数值实验来分析求解非对称线性系统的Ｋｒｙｌｏｖ子空间方法（如：ＧＭＲＥＳ,Ｏｒｔｈｏｍｉｎ,ＱＭＲ,ＣＧＳ,ＢＩＣＧＳＴＡＢ等）及其预条件算法（ＩＬＵ（一１）,ＩＬＵ（０）;ＩＬＵ（１）;ＩＬＵ（２）,左预条件,右预条件）的迭代求解效果（收敛速度）;迭代收敛行为的比较（剩余向量ＬＺ模的下降速度及下降曲线的光滑性）,迭代参数的选取（正交向量的个数的选取及对算法的影响）;迭代收敛速度受问题规模的影响等等．目的是对各种预条件算法的优缺点进行数值分析和评价,为…     

三维对流扩散方程的稀疏存储及预条件迭代

基于四阶紧致格式对三维对流扩散方程进行离散,并给出所得到的离散线性方程组的块三角稀疏矩阵形式。以带双阈值的不完全因子化LU分解[(ILUT(τ,s))]作为预条件子,分别用FGMRES、BICGSTAB和TFQMR作为迭代加速器,对离散线性方程组进行求解验证了格式精度并比较了不同迭代法的CPU时间和迭代步。此外,通过比较传统迭代法和预条件迭代法的计算效率,表明预条件迭代法不仅能够保证格式的四阶精度,还能极大地提高收敛效率。     

用割线法迭代求解对称三对角矩阵特征值问题

为对称三对角矩阵特征值问题，提出一种新的分而治之的算法。新算法以二分法，割线法迭代为基础，不同于Ｃｕｐｐｅｎ的方法和Ｌａｎｇｕｅｒｒｅ迭代法。理论分析和数据实验的结果表明：新算法的收敛速度明显比文［１］中的Ｌａｇｕｅｒｒｅ迭代法快。     

On Accelerated Iterative Methods for the Solution of Systems of Linear Equations

In literature, it has been reported that the convergence of some preconditioned stationary iterative methods using certain type upper triangular matrices as preconditioners are faster than the basic iterative methods. In this paper, a new preconditioned iterative method for the numerical solution of linear systems has been introduced, and the convergence analysis of the proposed method and an existing one have been done. Some numerical examples have also been given, which show the effectiveness of both of the methods.     

Comparison results on the preconditioned GAOR method for generalized least squares problems

Recently, Zhou et al. [Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math. 224 (2009), pp. 242–249] have proposed the preconditioned generalized accelerated over relaxation (GAOR) methods for solving generalized least squares problems and studied their convergence rates. In this paper, we propose a new type of preconditioners and study the convergence rates of the new preconditioned GAOR methods for solving generalized least squares problems. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Zhou et al. whenever these methods are convergent. Lastly, numerical experiments are provided in order to confirm the theoretical results studied in this paper.     

New preconditioners based on symmetric-triangular decomposition for saddle point problems

In this paper, we construct some new triangular preconditioners for saddle point problems based on the symmetric and triangular (ST) decomposition. Furthermore, we obtain some estimations on the condition number for the preconditioned systems and give the quasi-optimal parameters. Numerical experiments on the Stokes problems are given to illustrate fast convergence of the associated conjugate gradient method.     

求解大型非对称线性系统的一种新的预处理方法

针对大型稀疏非对称正定线性方程组,本文提出了新的预处理GMRES方法,并分析了谱半径和最优参数α的选取.最后通过数值例子比较GMRES方法,HSS预处理和新的预处理GMRES方法,发现新的预处理方法具有更好的收敛率.     

复奇异鞍点问题预条件修正AHSS法的半收敛性

提出了求解一类复奇异鞍点问题的预条件修正AHSS法。研究了所提出的新方法的半收敛性。对任意的正迭代参数,得到了所提出的新方法的半收敛定理。数值实验说明,新方法比HSS法求解鞍点问题时更有效。     

An efficient two-step iterative method for solving a class of complex symmetric linear systems

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.     

Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems

In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method.     

大型复线性方程组预处理双共轭梯度法

当复线性方程组的规模较大或系数矩阵的条件数很大时,系数矩阵易呈现病态特性,双共轭梯度法存在不收敛和收敛速度慢的潜在问题,采用适当的预处理技术,可以改善矩阵病态特性,加快收敛速度。从实型不完全Cholesky分解预处理方法出发,构造了一种针对复线性方程组的预处理方法,结合双共轭梯度法,给出了一种预处理双共轭梯度法。数值算例表明该算法求解速度快,可靠高效,能够应用于大型复线性方程组的求解。     

On the restrictively preconditioned conjugate gradient method for solving saddle point problems

Bai et al. [2003, IMA J Numer. Anal. 23, 561–580] proposed the restrictively preconditioned conjugate gradient (RPCG) method. In this paper, based on the special structure of saddle point systems, we consider the RPCG method and propose a new format. This new format can be obtained by applying the classical PCG method to a simpler system instead of the original format, which greatly reduces computational cost. The new format of the RPCG method can often attain almost the same convergence rate as the original one. In particular, for some practical problems, the former converges faster than the latter. Numerical experiments show the efficiency of the proposed format.     

Algebraic multilevel iterative preconditioning methods for h-matrices

A purely algebraic method is presented to construct preconditioned for symmetric positive definite H-matrices. The main technique is H-compatible splitting and diagonal compensation reduction. Associated with some special matrix polynomials, under certain condition, this method is optimal with respect to the rate of convergence and computational complexity. Numerical results that illustrate these properties are provided.