Semiconvergence of P-regular splittings for solving singular linear systems

We investigate necessary and sufficient conditions for semiconvergence of a splitting for solving singular linear systems, where the coefficient matrix A is a singular EP matrix. When A is a singular Hermitian matrix, necessary and sufficient conditions for semiconvergence of P-regular splittings are given, which generalize known results. As applications, the necessary and sufficient conditions for semiconvergence of block AOR and SSOR iterative methods are derived. A numerical example is given to illustrate the theoretical results. The work is supported by the National Natural Science Foundation of China under grant 10371056, the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China under grant 200720 and the Natural Science Foundation of Jiangsu Province of China under grant BK2006725.     

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。     

DCT- and DST-based splitting methods for Toeplitz systems

New splitting iterative methods for Toeplitz systems are proposed by means of recently developed matrix splittings based on discrete sine and cosine transforms due to Kailath and Olshevsky [Displacement structure approach to discrete-trigonometric transform-based preconditioners of G. Strang type and of T. Chan type, SIAM J. Matrix Anal. Appl. 26 (2005), pp. 706–734]. Theoretical analysis shows that new splitting iterative methods converge to the unique solution of a symmetric Toeplitz linear system. Moreover, an upper bound of the contraction factor of our new splitting iterations is derived. Numerical examples are reported to illustrate the effectiveness of new splitting iterative methods.     

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.     

An acceleration of preconditioned generalized accelerated over relaxation methods for systems of linear equations

In this paper, a new type of preconditioners are proposed to accelerate the preconditioned generalized accelerated over relaxation methods presented by Zhou et al. [Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math. 224 (2009), pp. 242–249] for the linear system of the generalized least-squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rates of the proposed methods are better than those of the original methods. Finally, numerical experiments are provided to confirm the results obtained in this paper.     

线性时不变系统PID–型迭代学习控制律的单调收敛形态

传统的迭代学习控制机理中,积分补偿是典型的策略之一,但其跟踪效用并不明确.本文针对连续线性时不变系统,对传统的PD–型迭代学习控制律嵌入积分补偿,利用分部积分法和推广的卷积Young不等式,在Lebesguep范数意义下,理论分析一阶和二阶PID–型迭代学习控制律的收敛性态.结果表明,当比例、积分和导数学习增益满足适当条件时,一阶PID–型迭代学习控制律是单调收敛的,二阶PID–型迭代学习控制律是双迭代单调收敛的.数值仿真验证了积分补偿可有效地提高系统的跟踪性能.     

Convergence properties concerning Lebesgue-p norm of iterative learning control for a class of fractional differential systems

This paper investigates the monotonic convergence and speed comparison of first- and second-order proportional-α-order-integral-derivative-type ( ${\mathit{PI}}^{\alpha }D-$ type) iterative learning control (ILC) schemes for a linear time-invariant (LTI) system, which is governed by the fractional differential equation with order $\alpha \in \left(1,2\right)$. By introducing the Lebesgue-p ( $L^P$) norm and utilizing the property of the Mittag-Leffler function and the boundedness feature of the fractional integration operator, the sufficient condition for the monotonic convergence of the first-order updating law is strictly analyzed. Therewith, the sufficient condition of the second-order learning law is established using the same means as the first one. The obtained results objectively reveal the impact of the inherent attributes of system dynamics and the constructive mode of the ILC rule on convergence. Based on the sufficient condition of first/second-order updating law, the convergence speed of first- and second-order schemes is determined quantitatively. The quantitative result demonstrates that the convergence speed of second-order law can be faster than the first one when the learning gains and weighting coefficients are properly selected. Finally, the effectiveness of the proposed methods is illustrated by the numerical simulations.     

Multipoint efficient iterative methods and the dynamics of Ostrowski's method

ABSTRACT

In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martínez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369–2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.     

不确定离散线性系统的鲁棒单调反馈–前馈迭代学习控制

针对一类不确定离散线性系统,提出一种沿迭代方向鲁棒单调收敛和沿时间方向有界输入有界输出(bouned-input bounded-output,BIBO)稳定的反馈–前馈迭代学习控制策略.首先,将不确定反馈–前馈迭代学习系统表示为不确定二维Roesser模型系统;然后,把二维系统沿迭代方向的鲁棒单调收敛问题转化成一维系统的H∞干扰抑制控制问题,并给出系统的稳定性证明和用线性矩阵不等式(linear matrix inequality,LMI)表示的沿迭代方向鲁棒单调收敛的充分条件,该LMI充分条件不仅可以用于确定反馈–前馈控制器的增益矩阵,而且还可以保证系统沿时间轴方向是BIBO稳定的;最后,仿真结果证明了该反馈–前馈迭代学习控制策略的有效性.     

Trigonometric transform splitting methods for real symmetric Toeplitz systems

In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix $A$. Theoretical analyses show that if the generating function $f$ of the $n\times n$ Toeplitz matrix $A$ is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large $n$. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved.Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix $A$ is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.     

Modified iterative methods for nonnegative matrices and M-matrices linear systems

The purpose of this paper is to present new preconditioning techniques for solving nonnegative matrices linear system and M-matrices linear system Ax = b based on the I + S(α) type preconditioning matrices provided by Hadjidimos et al. ^[1] and Evans et al. ^[2]. Convergence analysis of the proposed methods are given. Numerical results are presented, which show the improvements on the convergence rate of the Jacobi type and Gauss-Seidel type preconditioned iterative methods.     

A class of iterative methods for computing polar decomposition

In this article, there is offered a parametric class of iterative methods for computing the polar decomposition of a matrix. Each iteration of this class needs only one scalar-by-matrix and three matrix-by-matrix multiplications. It is no use computing inversion, so no numerical problems can be created because of ill-conditioning. Some available methods can be included in this class by choosing a suitable value for the parameter. There are obtained conditions under which this class is always quadratically convergent. The numerical comparison performed among six quadratically convergent methods for computing polar decomposition, and a special method of this class, chosen based on a specific value for the parameter, shows that the number of iterations of the special method is considerably near that of a cubically convergent Halley's method. Ten n×n matrices with n=5, 10, 20, 50, 100 were chosen to make this comparison.     

Scheduling parallel iterative methods on multiprocessor systems

The paper describes the implementation of the Successive Overrelaxation (SOR) method on an asynchronous multiprocessor computer for solving large, linear systems. The parallel algorithm is derived by dividing the serial SOR method into noninterfering tasks which are then combined with an optimal schedule of a feasible number of processors. The important features of the algorithm are: (i) achieves a speedup S_p O(N/3) and an efficiency E_p 2/3 using P = [N/2] processors, where N is the number of the equations, (ii) contains a high level of inherent parallelism, whereas on the other hand, the convergence theory of the parallel SOR method is the same as its sequential counterpart and (iii) may be modified to use block methods in order to minimise the overhead due to communication and synchronisation of the processors.     

Convergence of relaxation methods for two-point boundary-value linear problems

The solution of linear differential problems, with explicit two-point boundary conditions, can sometimes be obtained by a relaxation method of computation. This paper shows that the convergence of iterations is linked to the spectral radius value of an integral operator. The equivalence between eigenvalues research and critical lengths calculation of a differential system is demonstrated. In this context, we present a case of optimal control law calculation of a linear system. The efficiency of this preliminary convergence calculation is illustrated by a numerical example.     

线性广义系统P型迭代学习控制离散频域收敛性

针对一类线性广义系统,研究其P型迭代学习控制在离散频域中的收敛性态。在离散频域中,对广义系统进行奇异值分解后,利用傅里叶级数系数的性质和离散的Parseval能量等式,推演了一阶P型迭代学习控制律跟踪误差的离散能量频谱的递归关系和特性,获得了学习控制律收敛的充分条件;讨论了二阶P型迭代学习控制律的收敛条件。仿真实验验证了理论的正确性和学习律的有效性。     

分数阶迭代学习控制的收敛性分析

本文将传统的迭代学习控制时域和频域分析方法扩展到一类针对分数阶非线性系统的分数阶迭代学习控制时域分析方法.提出了一类新的分数阶迭代学习控制框架并简化了收敛条件,且证明了常增益情况下两类分数阶迭代学习控制收敛条件的等价性问题.该讨论进一步引出了如下两个结果:分数阶不确定系统的分数阶自适应迭代学习控制的可学习区域以及理想带阻型分数阶迭代学习控制的框架.上述结果均得到了仿真验证.     

Parallelizable restarted iterative methods for nonsymmetric linear systems. part I: Theory

Large sparse nonsymmetric problems of the form A u = b are frequently solved using restarted conjugate gradient-type algorithms such as the popular GCR and GMRES algorithms. In this study we define a new class of algorithms which generate the same iterates as the standard GMRES algorithm but require as little as half of the computational expense. This performance improvement is obtained by using short economical three-term recurrences to replace the long recurrence used by GMRES. The new algorithms are shown to have good numerical properties in typical cases, and the new algorithms may be easily modified to be as numerically safe as standard GMRES. Numerical experiments with these algorithms are given in Part II, in which we demonstrate the improved performance of the new schemes on different computer architectures.     

Singular H∞ control in nonlinear systems: positive semidefinite storage functions and separation principle

A framework is developed for the general nonlinear H_∞ output feedback control problem, in which two major restrictions are relaxed, i.e., the non-singular penalty in H_∞ cost and the positive definite solution of Hamilton–Jacobi inequality at present state space nonlinear H_∞ control literatures. As illustrated in an example, positive semidefinite solution simplifies the structure of the H_∞ controller. Based on this framework, some sufficient conditions are derived. While specialized to linear systems, the controller reduces to the so-called central controller. © 1997 by John Wiley & Sons, Ltd.     

Constructing third-order derivative-free iterative methods

In this work, we develop nine derivative-free families of iterative methods from the three well-known classical methods: Chebyshev, Halley and Euler iterative methods. Methods of the developed families consist of two steps and they are totally free of derivatives. Convergence analysis shows that the methods of these families are cubically convergent, which is also verified through the computational work. Apart from being totally free of derivatives, numerical comparison demonstrates that the developed methods perform better than the three classical methods.     

Convergence of normalized iterative identification of Hammerstein systems

An iterative identification algorithm of Hammerstein systems needs a proper initial condition to guarantee its convergence. In this paper, we propose a new algorithm by fixing the norm of the parameter estimates. The normalized algorithm ensures the convergence property under arbitrary nonzero initial conditions. The proofs of the property also give a geometrical explanation on why the normalization guarantees the convergence. An additional contribution is that the static function in the Hammerstein system is extended to square-integrable functions.