A Class of Filtering Superoptimal Preconditioners for Highly Ill-Conditioned Linear Systems

Popular preconditioners for conjugate gradient methods often reveal poor regularization properties that make them useless for very ill-conditioned linear systems arising in inverse problems. Recent results have awakened the interest towards the Tyrtyshnikov superoptimal preconditioners since it has been demonstrated that they exhibit good filtering capabilities. Here, in order to improve the regularizing behaviour, we generalize the definition of superoptimal preconditioner. Later on, by means of this more general definition, we develop a particular family of preconditioners for Toeplitz highly ill-conditioned linear systems.     

病态代数系统求解的精细迭代方法

提出了病态代数系统求解的精细迭代方法．首先利用一个小参数对病态矩阵加以改良,将原病态系统的求解问题转化为该改良系统的求解问题．然后利用精细积分法给出了改良矩阵求逆的高精度方法．该方法具有高精度、高效率的优点,且对改良参数的适应性较好,具有良好的应用前景．理论和数值分析证明了该方法的有效性．     

用遗传算法求解病态线性方程组

众所周知 ,病态方程组的条件数较大 ,当输入数据有微小扰动或计算过程中的舍入误差都可能引起输出数据的很大扰动 ,使得解严重失真 ,因此求解此类方程组是相当困难的 .本文尝试使用遗传算法来求解病态线性方程组 ,得到了较好的结果 ,并与传统的求解方法作了简单的比较     

灰色系统模型病态矩阵的双正则化算法

灰色系统模型矩阵会存在病态问题.为消除其病态性,基于病态矩阵的双正则化方法,建立了正则化灰色系统模型中灰参数求解的表达式,给出了其导出方式;提出了正则参数α的选择原则.从而避免了灰参数求解过程中矩阵的病态问题.数值试验分析说明,灰色系统模型的双正则化算法是正确和适用的.     

求解病态线性方程组的收缩Lanczos方法

Lanczos方法是求解大型线性方程组的常用方法.遗憾的是,在Lanczos过程中通常会发生算法中断或数值不稳定的情况.将给出求解大型对称线性方程组的收缩Lanczos方法,即DLanczos方法.新算法将采用增广子空间技术,在Lanczos过程中向Krylov子空间加入少量绝对值较小的特征值所对应的特征向量进行收缩.数值实验表明,新算法比Lanczos方法收敛速度更快,并且适合求解病态对称线性方程组.     

求解病态线性方程组的模拟退火算法

病态方程组的条件数较大,当输入数据有微小扰动或计算过程中的舍入误差都可能引起输出数据的很大扰动,使得解严重失真,因此求解此类方程组是相当困难的.本文尝试使用模拟退火算法来求解病态线性方程组,得到了较好的结果,并与传统的求解方法作了简单的比较.     

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.     

Dynamical Systems Gradient Method for Solving Ill-Conditioned Linear Algebraic Systems

A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An a priori and a posteriori stopping rules are justified. An algorithm for computing the solution using a spectral decomposition of the left-hand side matrix is proposed. Numerical results show that when a spectral decomposition of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization.     

病态线性方程组新的Jacobi迭代解法

给出了解病态线性方程组的一种新的Jacobi迭代算法,并证明了算法的收敛性;通过具体算例说明了算法的实用性和有效性.     

动力系统方法解决无界线性算子方程

针对反问题中出现的第一类算子方程Au=f,其中A是实Hilbert空间H上的一个无界线性算子利用动力系统方法和正则化方法,求解上述问题的正则化问题的解:u'(t)=-A~*(Au(t)-f)利用线性算子半群理论可以得到上述正则化问题的解的半群表示,并证明了当t→∞时,所得的正则化解收敛于原问题的解.     

Uniform Stability for Time-Varying Infinite-Dimensional Discrete Linear Systems

Stability for time-varying discrete linear systems in a Banachspace is investigated. On the one hand is established a fairlycomplete collection of necessary and sufficient conditions foruniform asymptotic equistability for input-free systems. Thisincludes uniform and strong power equistability, and uniformand strong l_p-equistability, among other technical conditionswhich also play an essential role in stability theory. On theother hand, it is shown that uniform asymptotic equistabilityfor input-free systems is equivalent to each of the followingconcepts of uniform stability for forced systems: l_p-input l_p-state,e_o-input e_o-state, bounded-input bounded-state, l_p-input bounded-state(with p>1), e_o-input bounded-state, and convergent-inputbounded-state; these are also equivalent to their nonuniformcounterparts. For time-varying convergent systems, the aboveis also equivalent to convergent-input convergent-state stability.The proofs presented here are all ‘lementary’ inthe sense that they are based essentially only on the Banach–Steinhaustheorem.     

线性广义系统时滞相关渐近稳定的一个新判据

研究线性广义时滞系统时滞相关的渐近稳定性问题.利用Lyapunov-Krasovskii泛函理论和自由权矩阵方法,得到严格线性矩阵不等式(LMI)形式的时滞相关渐近稳定新判据,并没有采用模型变换和界定交叉项方法.最后通过数值算例验证新判据的有效性.     

Approximate Inverse Preconditioning for Shifted Linear Systems

In this paper we consider the problem of preconditioning symmetric positive definite matrices of the form A =A+I where >0. We discuss how to cheaply modify an existing sparse approximate inverse preconditioner for A in order to obtain a preconditioner for A . Numerical experiments illustrating the performance of the proposed approaches are presented.     

A Mathod to Solve Systems of Fuzzy Linear Equations

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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.     

求多变量线性系统的逆向系统的一种新方法

对多变量线性系统,本文给出了求其逆向系统的一种新方法,这种方法将逆向系统计算中高阶矩阵的求逆转化为通过初等变换求低阶矩阵的规范型,比以往的方法更加简单有效且易于编程计算。本文结合系统的可观测空间与不可观测空间的情况,给出了一种特定的等价变换,得到了比通常更低阶的逆向系统。     

A Survey of Reachability and Controllability for Positive Linear Systems

This paper is a survey of reachability and controllability results for discrete-time positive linear systems. It presents a variety of criteria in both algebraic and digraph forms for recognising these fundamental system properties with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control, manpower planning, scheduling and other areas of operations research) but also in studying properties of reachable sets, in feedback control problems, and others. The paper highlights the intrinsic combinatorial structure of reachable/controllable positive linear systems and reveals the monomial components of such systems. The system matrix decomposition into monomial components is demonstrated by solving some illustrative examples.     

奇异离散线性系统的开关切换控制

In this paper, we study the stability of discrete linear singular systems by switching controller. Using some recent results on multiple-Lyapunov function technique, we obtain two sufficient conditions of linear singular systems.     

带有未知输入的降阶观测器的设计

本文讨论了带未知输入的线性时不变系统的状态重构问题 .在一种特定的坐标变换下 ,给定了该系统存在降阶观测器的充分必要条件 .与已有结果相比 ,该条件比相应的文献所给出的条件弱 ,并且本文的条件是系统本身的 ,因而也更合理     

On Linear Systems of Curves on Rational Scrolls

In this paper we prove a conjecture on the dimension of linear systems, with base points of multiplicity 2 and 3, on an Hirzebruck surface.