求解Hilbert矩阵线性方程组的SOR迭代预处理方法

选择了求解Hilbert矩阵线性方程组的三种数值解方法,提出了SOR迭代中的松弛因子的预处理方法,比较了高斯-赛德尔迭代和SOR迭代数值解的迭代收敛次数,并给出了SOR迭代收敛最快时的松弛因子取值.最后通过SOR迭代解分量及误差范围,说明了提出的SOR迭代预处理方法是有效的.     

一类特殊矩阵方程的并行预处理变形共轭梯度算法

研究了求解一类矩阵方程AXB=C,提出了一种并行预处理变形共轭梯度法．该方法给出一种迭代法的预处理模式．首先给出的预处理矩阵是严格对角占优矩阵,构造并行迭代求解预处理矩阵方程的迭代格式,进而使用变形共轭梯度法并行求解．通过数值试验,预处理变形共轭梯度法与直接使用变形共轭梯度法相比较,该算法不仅有效提高了收敛速度,而且具有很高的并行性．     

解线性互补问题的预处理加速模Gauss-Seidel迭代方法

本文提出了解线性互补问题的预处理加速模系Gauss-Seidel迭代方法,当线性互补问题的系统矩阵是M-矩阵时证明了方法的收敛性,并给出了该预处理方法关于原方法的一个比较定理.数值实验显示该预处理迭代方法明显加速了原方法的收敛.     

二维Helmholtz方程的联合紧致差分离散方程组的预处理方法

对于二维的Helmholtz方程,本文用联合紧致差分格式(CCD)离散,该差分格式具有六阶精度,三点差分和隐式的特点.本文基于CCD格式离散得到的线性系统和循环矩阵的快速傅里叶变换,提出了一种循环型预处理算子用于广义极小残量迭代算法(GMRES).给出了循环型预处理子的求解算法,证明了该预处理算子能使迭代算法具有较快的收敛速度.本文还与其他算法的预处理算子作比较,数值结果表明本文提出的循环型预处理算子具有更好的稳定性,并且对于较大的波数k,收敛速度也更快.     

非埃尔米特正定线性系统的m步预处理的斜埃尔米特和反埃尔米特分裂方法（英文）

进一步研究了非埃尔米特正定线性系统的斜埃尔米特和反埃尔米特迭代方法,并在预处理的斜埃尔米特和反埃尔米特迭代方法的基础上,引入了m步多项式预处理子,证明了预处理的斜埃尔米特和反埃尔米特迭代方法在一定条件下是收敛的,而且得到了预处理的斜埃尔米特和反埃尔米特迭代方法的收缩因子.通过数值例子说明,对于非埃尔米特正定线性系统m步的预处理有效地加速了Krylov子空间方法,例如GMRES.     

关于鞍点问题的预处理HSS-SOR交替分裂迭代方法

为了高效地求解大型稀疏鞍点问题,在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂(PHss)迭代法的基础上,通过结合SOR-like迭代格式对原有迭代算法进行加速,提出了一种预处理HSS-SOR交替分裂迭代方法,并研究了该算法的收敛性.数值例子表明:通过参数值的选择,新算法比SOR-like和PHSS算法都具有更快的收敛速度和更少的迭代次数,选择了合适的参数值后,可以提高算法的收敛效率.     

一种新型压缩预处理CGS算法

CGS算法是求解大型非对称线性方程组的常用算法，然而该算法无极小残差性质，因此它常因出现较大的中间剩余向量而出现典型的不规则收敛行为．本根据IRA方法提出了一种压缩预处理CGS方法，数值实验表明这种算法在一定程度上减小了迭代算法在收敛过程中的剩余问题，从而使得算法具有更好的稳定性，该法构造简单，减少了收敛次数，加快了收敛速度．     

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．     

求解时谐涡流模型鞍点问题的分块交替分裂隐式迭代算法的改进

分块交替分裂隐式迭代方法是求解具有鞍点结构的复线性代数方程组的一类高效迭代法.本文通过预处理技巧得到原方法的一种加速改进方法,称之为预处理分块交替分裂隐式迭代方法·理论分析给出了新方法的收敛性结果.对于一类时谐涡旋电流模型问题,我们给出了若干满足收敛条件的迭代格式.数值实验验证了新型算法是对原方法的有效改进.     

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.     

Preconditioned GAOR methods for solving weighted linear least squares problems

In this paper, we present the preconditioned generalized accelerated overrelaxation (GAOR) method for solving linear systems based on a class of weighted linear least square problems. Two kinds of preconditioning are proposed, and each one contains three preconditioners. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the convergence rate of the preconditioned GAOR methods is indeed better than the rate of the original method, whenever the original method is convergent. Finally, a numerical example is presented in order to confirm these theoretical results.     

New preconditioned AOR iterative method for Z-matrices

In this paper, we present a new preconditioned AOR-type iterative method for solving the linear system Ax=b, where A is a Z-matrix, and prove its convergence. Then we give some comparison theorems to show that the rate of convergence of the preconditioned AOR-type iterative method is faster than the rate of convergence of the AOR-type iterative method. Finally, we give two numerical examples to illustrate our results.     

Comparison results on preconditioned SOR-type iterative method for Z-matrices linear systems

In this paper, we present some comparison theorems on preconditioned iterative method for solving Z-matrices linear systems, Comparison results show that the rate of convergence of the Gauss–Seidel-type method is faster than the rate of convergence of the SOR-type iterative method.     

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.     

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

Based on the HSS (Hermitian and skew-Hermitian splitting) and preconditioned HSS methods, we will present a generalized preconditioned HSS method for the large sparse non-Hermitian positive definite linear system. Our method is essentially a two-parameter iteration which can extend the possibility to optimize the iterative process. The iterative sequence produced by our generalized preconditioned HSS method can be proven to be convergent to the unique solution of the linear system. An exact parameter region of convergence for the method is strictly proved. A minimum value for the upper bound of the iterative spectrum is derived, which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting. An efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method. The optimality and efficiency are effectively testified by some comparisons with numerical results.     

关于鞍点问题的广义预处理HSS-SOR交替分裂迭代方法

本文研究了鞍点问题的迭代法. 在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂（PHSS）迭代法的基础上,通过结合GSOR迭代格式,利用两个参数加速,提出了一种广义预处理HSS-SOR交替分裂迭代法,并研究了该方法的收敛性.数值结果表明本文所给方法是有效的.     

Sine transform based preconditioning techniques for space fractional diffusion equations

We study the preconditioned iterative methods for the linear systems arising from the numerical solution of the multi-dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed according to the symmetric and skew-symmetric splitting of the Toeplitz factor in the resulting coefficient matrix. Theoretical analyses show that the upper bound of relative residual norm of the GMRES method when applied to the preconditioned linear system is mesh-independent which implies the linear convergence. Numerical experiments are carried out to illustrate the correctness of the theoretical results and the effectiveness of the proposed preconditioning technique.