无穷迭代函数系统的遍历定理

度量空间的压缩映射的一个集合称为一个迭代函数系统．凝聚迭代函数系统可以被看成无穷迭代函数系统．研究了紧度量空间上的无穷迭代函数系统．利用Banach极限的特性和均匀压缩性,证明了紧度量空间上无穷迭代函数系统的随机迭代算法满足遍历性．于是,凝聚迭代函数系统的随机迭代算法也满足遍历性．     

关于非线性方程的牛顿迭代格式初始值选取的注记

基于构造非线性方程的牛顿迭代格式简便和牛顿迭代格式具有收敛快的特点,在解决实际问题时,牛顿迭代格式显得尤为重要,但是,牛顿迭代格式的初始值选取具有很大的局限性.利用泰勒级数展开,对牛顿迭代格式的收敛性进行分析,从而提出改进牛顿迭代格式的初始值选取方案,并利用不同的数值算例验证牛顿迭代格式收敛区域的改进方案的可行性,同时数值算例表明该方法具有操作简单的特点.     

一种新的Bregman迭代方法在信号恢复中的应用

本文结合残量Bregman迭代方法以及不动点迭代方法提出一种新迭代方法,将其应用于信号恢复问题.数值试验表明,新方法避免了Bregman迭代方法产生的停滞现象且较线性Bregman迭代方法更稳定、快速、有效.     

一种基于牛顿迭代改进的新的三阶预估校正格式

基于对牛顿迭代公式的改进及预估校正迭代的思想,提出了一种求解非线性方程的新的三阶预估-校正迭代格式.迭代公式无须计算函数的导数值,且理论上证明了它至少是三阶收敛的.数值实验验证了该迭代公式的有效性.     

求解一类分块二阶线性方程组的QHSS迭代方法

本文将QHSS迭代方法运用于求解一类分块二阶线性方程组. 通过适当地放宽QHSS迭代方法的收敛性条件,我们给出了用QHSS迭代方法求解一类分块二阶线性方程组的具体迭代格式,并证明了当系数矩阵中的(1,1)块对称半正定时该QHSS迭代方法的收敛性.我们还用数值实验验证了QHSS迭代方法的可行性和有效性.     

关于广义Φ-压缩映射几种迭代序列收敛的等价性

在广义Φ-压缩映射条件下,分别得到了Picard迭代序列与Krasnoselskii迭代序列以及Mann迭代序列与Ishikawa迭代序列收敛的等价性.     

外推的MHSS迭代法求解一类大型稀疏的复对称线性系统

本文对改良的Hermitian和反Hermitian分裂迭代方法 (MHSS)使用了外推技术,构造了外推的MHSS(EMHSS)迭代法.从理论上给出了EMHSS迭代方法的迭代矩阵与MHSS迭代方法的迭代矩阵之间的关系,并讨论了EMHSS迭代方法的收敛条件.最后用数值实验验证了所提方法的有效性.     

一类复线性系统的局部HSS迭代方法

本文提出求解一类复线性系统的局部HSS (LHSS)迭代方法.讨论迭代方法的收敛性,分析了最优迭代参数的选取.结合最优控制问题验证LHSS迭代方法的理论结果,并从迭代次数和计算时间上证明新方法的可行性和有效性.     

含参无导数有记忆迭代方法与其动力系统的构造

借鉴含导数两步迭代格式转化成不含导数两步迭代格式的思想,提出了一种更通用的两步无导数迭代格式,通过权值保证了两步无导迭代格式达到最优阶;利用自加速参数和Newton(牛顿)插值多项式得到了两参和三参有记忆迭代格式,并与已有的两参和三参有记忆迭代格式进行比较;给出了几个格式的吸引域,比较了几个迭代格式的性能.     

一致凸Banach空间非扩张映像具误差的Ishikawa迭代

研究一致凸 Banach空间中非扩张映像迭代序列的收敛问题 ,使用了基于 Ishikawa迭代的一种具误差的 Ishikawa迭代 ,证明了非扩张映像的具误差的 Ishikawa迭代收敛定理 .     

牛顿迭代法优于预测式迭代法

从两个方面说明牛顿迭代法优于预测式迭代法：１牛顿迭代法的收敛阶数高于预测式迭代法的收敛阶数．２从算法复杂性出发,采用 Ｏｓｔｒｏｗ ｓｋｉ给出的“迭代过程有效性指标的概念,得到牛顿迭代法的有效性指标是 ２１３ ,预测式迭代法的有效性指标是 ３１５ ．     

Frozen Landweber Iteration for Nonlinear Ill-Posed Problems

In this paper we propose a modification of the Landweber iteration termed frozen Landweberiteration for nonlinear ill-posed problems.A convergence analysis for this iteration is presented.The numericalperformance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared withthat of the Landweber iteration.We obtain a shorter running time of the frozen Landweber iteration based onthe same convergence accuracy.     

The use of approximate factorization in stiff ODE solvers

We consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration.     

一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.     

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.     

关于线性互补问题的模系矩阵分裂迭代方法

模系矩阵分裂迭代方法是求解大型稀疏线性互补问题的有效方法之一.本文的目标是归纳总结模系矩阵分裂迭代方法的最新发展和已有成果,主要内容包括相应的多分裂迭代方法, 二级多分裂迭代方法和两步多分裂迭代方法, 以及这些方法的收敛理论.     

A MODIFIED GAUSS-SEIDEL ITERATION FOR LINEAR INTERVAL EQUATIONS

In order to solve linear interval equations Ax=b. the interval Gauss-Seidel iteration is applied to a modified equations A'x = b', where A' is obtained by eliminating all elements in the first upper codiagonal part of A. It is shown that this modified Gauss-Seidel iteration converges when the usual Gauss-Seidel iteration converges, and the modified Gauss-Seidel iteration always has a smaller convergence factor. The converse is not true. It is also pointed out that the modified Gauss-Seidel iteration can obtain a belter result comparing with the usual Gauss-Seidel iteration in some cases. Several examples confirmed these conclusions.     

Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems

The matrix multisplitting iteration method is an effective tool for solving large sparse linear complementarity problems. However, at each iteration step we have to solve a sequence of linear complementarity sub-problems exactly. In this paper, we present a two-stage multisplitting iteration method, in which the modulus-based matrix splitting iteration and its relaxed variants are employed as inner iterations to solve the linear complementarity sub-problems approximately. The convergence theorems of these two-stage multisplitting iteration methods are established. Numerical experiments show that the two-stage multisplitting relaxation methods are superior to the matrix multisplitting iteration methods in computing time, and can achieve a satisfactory parallel efficiency.     

On inexact Newton methods based on doubling iteration scheme for non‐symmetric algebraic Riccati equations

Newton iteration method can be used to find the minimal non‐negative solution of a certain class of non‐symmetric algebraic Riccati equations. However, a serious bottleneck exists in efficiency and storage for the implementation of the Newton iteration method, which comes from the use of some direct methods in exactly solving the involved Sylvester equations. In this paper, instead of direct methods, we apply a fast doubling iteration scheme to inexactly solve the Sylvester equations. Hence, a class of inexact Newton iteration methods that uses the Newton iteration method as the outer iteration and the doubling iteration scheme as the inner iteration is obtained. The corresponding procedure is precisely described and two practical methods of monotone convergence are algorithmically presented. In addition, the convergence property of these new methods is studied and numerical results are given to show their feasibility and effectiveness for solving the non‐symmetric algebraic Riccati equations. Copyright © 2010 John Wiley & Sons, Ltd.     

The Quasi-Laguerre iteration

The quasi-Laguerre iteration has been successfully established, by the same authors, in the spirit of Laguerre's iteration for solving the eigenvalues of symmetric tridiagonal matrices. The improvement in efficiency over Laguerre's iteration is drastic. This paper supplements the theoretical background of this new iteration, including the proofs of the convergence properties.