用特征值刻画分式线性迭代数列的渐近性态

利用分式线性迭代数列系数矩阵的特征值,刻画这类数列的敛散性及收敛速度.     

递推数列xn＋1=f（xn）的单调性与收敛性讨论

利用函数单调递增对递推数列xn＋1=f（xn）单调性进行讨论,在对递推数列收敛性作分析的基础上,得到使得递推数列收敛的初始迭代值的区域,讨论的方法可以用于类似问题的研究．     

递推数列x_(n+1)=f(x_n)的单调性与收敛性讨论

利用函数单调递增对递推数列xn+1=f(xn)单调性进行讨论,在对递推数列收敛性作分析的基础上,得到使得递推数列收敛的初始迭代值的区域,讨论的方法可以用于类似问题的研究.     

分式线性迭代数列的敛散性及收敛速度

若a_i,b_i0(i=1,2),|a_1 a_2b_1 b_2|≠0,则数列x_10,x_(n+1)=a_1x_n+a_2/b_1x_n+b_2收敛.若迭代过程中,xn(n=1,2,…)全不是φ(x)=a1x+a2/b1x+b2的不动点,则迭代数列{xn}线性收敛.     

几类迭代数列的收敛性

从一个简单的线性迭代数列出发,引出倒数迭代数列和分式迭代数列的极限结论.这三类迭代数列是常见的,得到的结论有较强的实用性.     

一个动力系统的经典模型的全局收敛性

研究了由函数f(x)=cosx迭代所得到的一个动力系统的经典模型,讨论了其全局收敛性.首先,证明了对于任意的正整数n,函数cos~nx都存在唯一的不动点;其次,证明了对任意初值x_0∈R,皮卡迭代数列{cos~nx_0}都收敛到同一个常数,此常数正好为函数f(x)=cosx的不动点,从而证明了由函数f迭代生成的离散动力系统{f~0,f~1,f~2,…}是全局收敛的.     

一种生成迭代数列的新方法

围绕两个典型迭代数列的构造问题,以问题为驱动,提出一种生成迭代数列的新方法,并通过数值实验或理论证明验证迭代数列的收敛性.     

二阶有理迭代数列

对一类二阶有理迭代数列,通过将其化为一阶有理迭代数列并利用一阶倒数迭代数列的结论,给出了其通项公式与收敛性.     

数列的子列及其分类

本文对数列的子数列进行分类,给出了余子列等概念,讨论数列收敛与其子列收敛的关系．     

Lipschitz Φ-半压缩映象的不动点迭代逼近

设X为一致光滑的Banach空间,K为X的非空凸子集,T：K→KLipschitzφ半压缩映象,设和为[0,1]中的实数列且满足一定条件,则Ishikawa迭代序列强收敛于T的唯一不动点。     

集值非扩张映象列迭代过程的收敛性及不动点

本文讨论了集值非扩张映象列的Ishikawa迭代过程的收敛性及确保迭代过程收敛到公共不动点的条件．所得结果是单值非扩张映射情形的推广和发展．     

同时具有点和区间半径序列收敛性的免导迭代法的进一步讨论

1 引 言 传统的求零点的迭代法只讨论迭代序列{xn}的收敛阶,近年来,G.Alefeld和F.A.Po-tra研究了含零点的区间半径序列的收敛性[2][3],而我们提出了同时具有点和区间半径序列均平方收敛的免导迭代法[1],即当n充分大时,序列{xn}和含零点区间的半径序列{(bn-an)}都是平方收敛的.通过进一步的分析,我们发现,文[1]中的结果仍可改进,并且,不需     

Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations

For the non‐symmetric algebraic Riccati equations, we establish a class of alternately linearized implicit (ALI) iteration methods for computing its minimal non‐negative solutions by technical combination of alternate splitting and successive approximating of the algebraic Riccati operators. These methods include one iteration parameter, and suitable choices of this parameter may result in fast convergent iteration methods. Under suitable conditions, we prove the monotone convergence and estimate the asymptotic convergence factor of the ALI iteration matrix sequences. Numerical experiments show that the ALI iteration methods are feasible and effective, and can outperform the Newton iteration method and the fixed‐point iteration methods. Besides, we further generalize the known fixed‐point iterations, obtaining an extensive class of relaxed splitting iteration methods for solving the non‐symmetric algebraic Riccati equations. Copyright © 2006 John Wiley & Sons, Ltd.     

The Kačanov method for some nonlinear problems

The Kačanov method is an iteration method for solving some nonlinear partial differential equation problems. In each iteration, a linear problem is solved. In this paper, we discuss the use of the Kačanov method in the context of two model problems. We show the convergence of the Kačanov iteration sequences, and derive a posteriori error estimates for the Kačanov iterates. Numerical examples are given showing the convergence of the method and the effectiveness of the a posteriori error estimates.     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Monotone enclosing of solutions of the Steffensen‐type

In this paper we consider two iterative methods of the Steffensen‐type. The iteration sequences which approximate the solution of f(x) = 0 from opposite sides are generated by Steffensen's method and a secant method, respectively. We show two enclosing theorems and establish orders of convergence. The iteration sequences converge with order two and three, respectively. Numerical examples complete the paper.     

关于Ishikawa迭代程序稳定性的注释

在实一致光滑Banach空间中,研究了一类具有值域有界、连续强伪压缩算子和连续强增生算子的Ishikawa迭代程序的稳定性;给出了迭代程序中参数所满足的条件,并证明了迭代过程的收敛性。所得结果改进和扩展近期相关结果,为进一步讨论带误差迭代程序的收敛性提供理论依据。     

An extragradient algorithm for solving bilevel pseudomonotone variational inequalities

We present an extragradient-type algorithm for solving bilevel pseudomonone variational inequalities. The proposed algorithm uses simple projection sequences. Under mild conditions, the convergence of the iteration sequences generated by the algorithm is obtained.     

Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities

We introduce a new iteration method and prove strong convergence theorems for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of monotone and Lipschitz-type continuous Ky Fan inequality. Under certain conditions on parameters, we show that the iteration sequences generated by this method converge strongly to the common element in a real Hilbert space. Some preliminary computational experiences are reported.     

A Mixed Mann Iteration Process for Fixed Points of Mixed Increasing Operators in Ordered Banach Spaces

In this paper, we develop a new mixed Mann iteration process to approximate the fixed points of mixed increasing operators in ordered Banach spaces and discuss the convergence analysis of the iterative sequences generated by the iteration process. As an application, we develop a new iterative method for solving a class of nonlinear integral equations.