非Hermitian正定线性方程组的外推的广义HSS方法

探讨了如何高效求解非Hermitian正定线性方程组,提出了一种外推的广义Hermitian和反Hermitian (EGHSS) 迭代方法。首先,根据矩阵的广义Hermitian和反Hermitian分裂,构造出了一种新的非对称的二步迭代格式。接着,理论分析了新方法的收敛性,并给出了新方法收敛的充要条件。数值实验结果表明,在处理某些问题时,EGHSS迭代方法比GHSS迭代方法和EHSS迭代方法更有效。     

基于Newton迭代法的最小二乘渐进迭代逼近

最小二乘渐进迭代逼近(LSPIA)是一种有效的大规模数据拟合方法.针对LSPIA的加速问题,基于Newton迭代法,本文提出曲线曲面的两类最小二乘渐进迭代逼近格式.首先构造一个以控制顶点为变量的多元函数,其Hessian矩阵为正定矩阵,多元函数存在极小值,且其极小值所对应的控制顶点与LSPIA的收敛结果一致.对多元函数...     

求解带状线性方程组的一种并行算法

提出了一种在MIMD分布式存储环境下求解带状线性方程组的交替方向迭代并行算法。利用系数矩阵的结构特点分裂矩阵,使整个计算过程只在相邻处理机间通信两次。给出了系数矩阵分别为Hermite正定矩阵和M-矩阵时算法收敛的充分条件。最后,在HP rx2600集群系统上进行的数值计算表明,该算法与多分裂方法相比具有较高的加速比和并行效率。     

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．     

神经网络中处理鞍点的LMBP改进算法

针对目前神经网络中的Levenberg-Marquardt反向传播(LMBP)算法在训练过程中有可能迭代到鞍点的问题,提出一种能有效克服鞍点的LMBP改进算法。计算鞍点处雅克比矩阵的正特征值对应的特征向量并将其作为新的搜索方向。通过实例对比传统LMBP算法与改进LMBP算法的效果,证明改进的算法能有效地脱离鞍点并进一步收敛到极小点处。     

两正定矩阵联合对角化盲分离算法

针对具有时间结构的盲分离问题,提出了一种基于两正定矩阵精确联合对角化的盲分离算法。利用多个不同时延统计量构造了两个正定矩阵,以提取出数据的时间结构;再利用所提算法联合对角化构造的两个正定矩阵,得到分离矩阵,进而估计出源信号。所提算法克服了已有算法因采用多个矩阵联合对角化导致的计算量大和采用单个矩阵导致的分离精度低的缺点。计算机仿真结果表明了在有或无噪声情况下,所提算法性能均优于其他对比算法。     

求解对偶Riccati方程的矩阵符号函数法

本文介绍一种求解对偶代数Riccati方程正定(负定)稳定(反稳定)解的方法——矩阵符号函数法,给出这些解的唯一存在的充分必要条件和算法实现。     

解不定信赖域子问题的Heun三阶算法

针对信赖域子问题,当Hessian矩阵不正定时,利用Bunch-Parlett法对矩阵进行修正,构造了对称正定的矩阵,将不定子问题转化为正定子问题,用新的折线来逼近最优解曲线,给出了求解的Heun三阶算法。通过对Heun三阶折线路径性质的分析,理论上证明了算法的适定性。利用两个测试函数进行了数值实验,结果表明该算法有效。     

带状线性方程组的并行交替方向算法

提出了分布式存储环境下求解带状线性方程组的并行交替方向迭代算法。充分利用系数矩阵的结构特点,给出了在系数矩阵分别为Hermite正定矩阵和M-矩阵时算法的充分条件,并针对采用的分裂方式,讨论了参数的收敛范围,最后在HPrx2600集群系统上进行了数值计算,结果表明实算与理论相一致,算法简便可行且具有良好的并行性。     

黎曼核局部线性编码

最近的研究表明:在许多计算机视觉任务中,将对称正定矩阵表示为黎曼流形上的点能够获得更好的识别性能.然而,已有大多数算法仅由切空间局部逼近黎曼流形,不能有效地刻画样本分布.受核方法的启发,提出了一种新的黎曼核局部线性编码方法,并成功地应用于视觉分类问题.首先,借助于最近所提出的黎曼核,把对称正定矩阵映射到再生核希尔伯特空间中,通过局部线性编码理论建立稀疏编码和黎曼字典学习数学模型;其次,结合凸优化方法,给出了黎曼核局部线性编码的字典学习算法;最后,构造一个迭代更新算法优化目标函数,并且利用最近邻分类器完成测试样本的鉴别.在3个视觉分类数据集上的实验结果表明,该算法在分类精度上获得了相当大的提升.     

On convergence of splitting iteration methods for non-Hermitian positive-definite linear systems

A new splitting iteration method is presented for the system of linear equations when the coefficient matrix is a non-Hermitian positive-definite matrix. The spectral radius, the optimal parameter, and some norm properties of the iteration matrix for the new method are discussed in detail. Based on these results, the new method is convergent under reasonable conditions for any non-Hermitian positive-definite linear system. Finally, the numerical examples show that the new method is more effective than the Hermitian and skew-Hermitian splitting iterative (or positive-definite and skew-Hermitian splitting iterative) method in central processing unit time.     

Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method

It is known that the Hermitian and skew-Hermitian splitting (HSS) iteration method is an efficient solver for non-Hermitian positive-definite linear system of equations. Benzi [A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 360–374] proposed a generalized HSS (GHSS) iteration method. In this paper, we present a two-parameter version of the GHSS (TGHSS) method and investigate its convergence properties. To show the effectiveness of the proposed method the TGHSS iteration method is applied to image restoration and convection–diffusion problems and the results are compared with those of the HSS and GHSS methods.     

A variant of the HSS preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems, we establish a new variant of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is not only a better approximation to the block two-by-two real coefficient matrix than the well-known HSS preconditioner, but also resulting in an unconditional convergent fixed-point iteration. The quasi-optimal parameter, which minimizes an upper bound of the spectral radius of the iteration matrix, is analyzed. Eigen-properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the efficiency of the new preconditioner.     

Modified HSS iteration methods for a class of complex symmetric linear systems

In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method.     

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.     

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.     

On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems

For the singular, non-Hermitian, and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the semi-convergence of the Hermitian and skew-Hermitian splitting (HSS) iteration methods. We then investigate the semi-convergence factor and estimate its upper bound for the HSS iteration method. If the semi-convergence condition is satisfied, it is shown that the semi-convergence rate is the same as that of the HSS iteration method applied to a linear system with the coefficient matrix equal to the compression of the original matrix on the range space of its Hermitian part, that is, the matrix obtained from the original matrix by restricting the domain and projecting the range space to the range space of the Hermitian part. In particular, an upper bound is obtained in terms of the largest and the smallest nonzero eigenvalues of the Hermitian part of the coefficient matrix. In addition, applications of the HSS iteration method as a preconditioner for Krylov subspace methods such as GMRES are investigated in detail, and several examples are used to illustrate the theoretical results and examine the numerical effectiveness of the HSS iteration method served either as a preconditioner for GMRES or as a solver.     

Variants of the deteriorated PSS preconditioner for saddle point problems

Two new preconditioners, which can be viewed as variants of the deteriorated positive definite and skew-Hermitian splitting preconditioner, are proposed for solving saddle point problems. The corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks. The choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated Cayley transform, which result in faster convergence rate and more clustered spectrum. Compared with some preconditioners of similar structures, the new preconditioners have better convergence properties and spectrum distributions. In addition, more practical preconditioning variants of the new preconditioners are considered. Numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate GMRES.