ASYNCHRONOUS RELAXED ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS

In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.

The AOR Iterative Method for Preconditioned Linear Systems

The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterative method does not converge. The results in corresponding references are improved and perfected.

A New Iterative Solution Method for Solving Multiple Linear Systems

In this paper, a new iterative solution method is proposed for solving multiple linear systems A(i)x(i)=b(i), for 1≤ i ≤ s, where the coefficient matrices A(i) and the right-hand sides b(i) are arbitrary in general. The proposed method is based on the global least squares (GL-LSQR) method. A linear operator is defined to connect all the linear systems together. To approximate all numerical solutions of the multiple linear systems simultaneously, the GL-LSQR method is applied for the operator and the approximate solutions are obtained recursively. The presented method is compared with the well-known LSQR method. Finally, numerical experiments on test matrices are presented to show the efficiency of the new method.

A BLOCK VARIANT OF THE GMRES METHOD FOR UNSYMMETRIC LINEAR SYSTEMS

Iterative methods that take advantage of efficient block operations and block communications are popular research topics in parallel computation. These methods are especially important on Massively Parallel Processors (MPP). This paper presents a block variant of the GMRES method for solving general unsymmetric linear systems. It is shown that the new algorithm with block size s, denoted by BVGMRES(s,m), is theoretically equivalent to the GMRES(s. m) method. The numerical results show that this algorithm can be more efficient than the standard GMRES method on a cache based single CPU computer with optimized BLAS kernels. Furthermore, the gain in efficiency is more significant on MPPs due to both efficient block operations and efficient block data communications. Our numerical results also show that in comparison to the standard GMRES method, the more PEs that are used on an MPP, the more efficient the BVGMRES(s,m) algorithm is.

BLOCK BIDIAGONALIZATION METHODS FOR MULTIPLE NONSYMMETRIC LINEAR SYSTEMS

The symmetric linear system gives us many simplifications and a possibility to adapt the computations to the computer at hand in order to achieve better performance. The aim of this paper is to consider the block bidiagonalization methods derived from a symmetric augmented multiple linear systems and make a comparison with the block GMRES and block biconjugate gradient methods.

A Class of Generalized Approximate Inverse Solvers for Unsymmetric Linear Systems of Irregular Structure Based on Adaptive Algorithmic Modelling for Solving Complex Computational Problems in Three Space Dimensions

A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex computational problems in three space dimensions. The proposed class of approximate inverse is chosen as the basis to yield systems on which classic and preconditioned iterative methods are explicitly applied. Optimized versions of the proposed approximate inverse are presented using special storage (k-sweep) techniques leading to economical forms of the approximate inverses. Application of the adaptive algorithmic methodologies on a characteristic nonlinear boundary value problem is discussed and numerical results are given.

IMinpert:An Incomplete Minimum Perturbation Algorithm for Large Unsymmetric Linear Systems

This paper gives the truncated version of the Minpert method:the incomplete minimum perturbation algorithm(IMinpert).It is based on an incomplete orthogonal- ization of the Krylov vectors in question,and gives a quasi-minimum backward error solution over the Krylov subspace.In order to make the practical implementation of IMinpert easy and convenient,we give another approximate version of the IMinpert method:A-IMinpert.Theoretical properties of the latter algorithm are discussed.Nu- merical experiments are reported to show the proposed method is effective in practice and is competitive with the Minpert algorithm.

USING THE SKEW-SYMMETRIC ITERATIVE METHODS FOR SOLUTION OF AN INDEFINITE NONSYMMETRIC LINEAR SYSTEMS

The concept of the field of value to localize the spectrum of the iteration matrices of the skew-symmetric iterative methods is further exploited. Obtained formulas are derived to relate the fields of values of the original matrix and the iteration matrix. This allows us to determine theoretically that indefinite nonsymmetric linear systems can be solved by this class of iterative methods.

BLOCK ITERATIVE METHODS FOR LINEAR ALGEBRAIC EQUATION AND DOMAIN DECOMPOSITION METHOD FOR INCOMPRESSIBLE VISCOUS FLOW

It was proved numerically that the Domain Decomposition Method CDDM) with one layer overlapping grids is identical to the block iterative method of linear algebra equations. The results obtained by using DDM could be in reasonable agreement with the results of full-domain simulation. With the three dimensional solver developed by the authors, the flow field in a pipe was simulated by using the full-domain DDM with one layer overlapping grids and with patched grids respectively. Both of the two cases led to the convergent solution. Further research shows the superiority of the DDM with one layer overlapping grids to the DDM with patched grids. A comparison between the numerical results obtained by the authors and the experimental results given by Enayet[3] shows that the numerical results are reasonable.

Study on the Application of Iterative Learning Control to Terminal Control of Linear Time-varying Systems

An iterative learning control algorithm based on shifted Legendre orthogonal polynomials is proposed to address the terminal control problem of linear time-varying systems. First, the method parameterizes a linear time-varying system by using shifted Legendre polynomials approximation. Then, an approximated model for the linear time-varying system is deduced by employing the orthogonality relations and boundary values of shifted Legendre polynomials. Based on the model, the shifted Legendre polynomials coefficients of control function are iteratively adjusted by an optimal iterative learning law derived. The algorithm presented can avoid solving the state transfer matrix of linear time-varying systems. Simulation results illustrate the effectiveness of the proposed method.

ON STRUCTURED VARIANTS OF MODIFIED HSS ITERATION METHODS FOR COMPLEX TOEPLITZ LINEAR SYSTEMS

The Modified Hermitian and skew-Hermitian splitting （MHSS） iteration method was presented and studied by Bai, Benzi and Chen （Computing, 87（2010）, 93-111） for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical ex- periments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

TOTAL GENERALIZED MINIMUM BACKWARD ERROR ALGORITHM FOR SOLVING NONSYMMETRIC LINEAR SYSTEMS

This paper extendes the results by E.M. Kasenally([7]) on a Generalized Minimum Backward Error Algorithm for nonsymmetric linear systems Ax = b to the problem in which pertubations are simultaneously permitted on A and b. The approach adopted by Kasenally has been to view the approximate solution as the exact solution to a perturbed linear system in which changes are permitted to the matrix A only. The new method introduced in this paper is a Krylov subspace iterative method which minimizes the norm of the perturbations to both the observation vector b and the data matrix A and has better performance than the Kasenally's method and the restarted GMRES method([12]). The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoratical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given.

ANALYSIS OF A MECHANICAL SOLVER FOR LINEAR SYSTEMS OF EQUATIONS

In this contribution we analyse some fundamental features of an iterative method to solve systems of linear equations, following the approve introduced in a previous work[l]. Such questions range from optimal parameters and initial conditions to comparison with other methods. An interesting result is that a priori we can give an estimation of the number of iterations to get a given accuracy.

Stochastic Gradient Descent for Linear Systems with Missing Data

Traditional methods for solving linear systems have quickly become imprac-tical due to an increase in the size of available data.Utilizing massive amounts of data is further complicated when the data is incomplete or has missing entries.In this work,we address the obstacles presented when working with large data and incom-plete data simultaneously.In particular,we propose to adapt the Stochastic Gradient Descent method to address missing data in linear systems.Our proposed algorithm,the Stochastic Gradient Descent for Missing Data method(mSGD),is introduced and theoretical convergence guarantees are provided.In addition,we include numerical experiments on simulated and real world data that demonstrate the usefulness of our method.

A SIGNIFICANT IMPROVEMENT ON NEWTON’S ITERATIVE METHOD

For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .

The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations

We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.

计及动态频率响应约束的高比例风电电力系统日前-日内联合调度策略

高比例风电电力系统具有转动惯量低、调频能力弱等特征,容易引发频率安全问题。现有计及频率安全约束的调度策略研究中,调频手段主要为惯性响应与一次调频,鲜见在单机故障等大功率扰动下考虑系统旋转备用与紧急功率支撑的动态响应作用。对此,在日前-日内联合调度模型中引入预想小功率扰动及大功率扰动下的系统动态频率响应约束。首先,分别构建两种预想功率扰动下的系统动态频率响应模型,采用阶梯备用等方法进行模型降阶,并进一步推导两种模型下的频率指标解析式。其次,提出计及动态频率响应约束的日前-日内联合调度架构,并分别构建日前及日内调度模型。进而,针对调度模型中最大频差约束导致的非线性问题,采用两阶段迭代方法求解。最后,开展算例分析,结果表明,所提联合调度策略能够兼顾系统运行经济性要求与频率安全性要求。

Distributed Model Predictive Control with Actuator Saturation for Markovian Jump Linear System

This paper is concerned with the distributed model predictive control (MPC) problem for a class of discrete-time Markovian jump linear systems (MJLSs) subject to actuator saturation and polytopic uncertainty in system matrices. The global system is decomposed into several subsystems which coordinate with each other. A set of distributed controllers is designed by solving a min-max optimization problem in terms of the solutions of linear matrix inequalities (LMIs). An iterative algorithm is developed to achieve the online computation. Finally, a simulation example is employed to show the effectiveness of the proposed algorithm. © 2014 Chinese Association of Automation.

分段线性系统经典数学模型的修正与动力学分析

由于间隙的存在,很多机械系统可以简化为分段线性模型,而简化后的模型中副簧系统通常包含阻尼.在大多数文献建立的数学模型中,主、副系统的接触点与分离点固定在间隙处.文章研究发现,由于副簧系统中弹簧与阻尼的力学特性不同,主、副系统的接触点与分离点位置实际上是随着系统参数和运动状态而变化的.若忽视这一点,后续的包括分岔和混沌在内的动力学分析就会出现误差甚至错误.首先基于经典的数学模型,用数值方法揭示了在简谐激励下主系统未返回到间隙处就与副簧系统提前分离,证明了经典数学模型的不当之处.进一步研究发现主系统不仅会出现提前分离,还会出现接触滞后的现象.因此对系统的接触与分离条件提出了修正,得到了更合理的数学模型.研究发现修正后的接触点、分离点位置与修正前相差较大,修正后的幅频响应曲线与修正前存在一定差别;在复杂运动中,修正前后的运动性质也可能发生改变,证明了修正后的模型更加合理,更能反映工程实际.然后,采用平均法并对平均法的积分区间进行推广,求得了修正模型的幅频响应的解析解,并通过龙格库塔法验证了解析解的正确性.对解析解进行稳定性分析,得到了解析解的稳定性判别式.最后,探究了修正模型的副簧系统参数对主系统幅频响应的影响.