解线性不适定方程隐式迭代法的r-步拟残差准则

本文给出了解不适定算子方程隐式迭代法后验选取步数的一类准则,称为r-步拟残差准则,证明了它们总导致最佳收敛阶.这类准则包含著名的Morozov残差准则和Gfrerer准则作为特例.     

解线性不适定方程隐式迭代法的,γ-步拟残差准则

本文给出了解不适定算子方程隐式迭代法后验选取步数的一类准则,称为,ｒ－步拟残差准则,证明了它们总导致最佳收敛阶,这类准则包含著名的 Ｍｏｒｏｚｏｖ残差准则和 Ｇｆｒｅｒｅｒ 准则作为特例．     

求解非线性不适定问题的隐式迭代法

将处理线性不适定算子方程的隐式迭代法推广到非线性不适定问题,证明了迭代解误差序列的单调性,并进一步利用迭代误差的单调性得出求解非线性不适定问题隐式迭代法对精确方程和扰动方程的收敛性.     

求解热传导反问题的一种正则化Newton型迭代法

讨论热传导方程求解系数的一个反问题.把问题归结为一个非线性不适定的算子方程后,考虑该方程的Newton型迭代方法.对线性化后的Newton方程用隐式迭代法求解,关键的一步是引入了一种新的更合理的确定(内)迭代步数的后验准则.对新方法及对照的Tikhonov方法和Bakushiskii方法进行了数值实验,结果显示了新方法具有明显的优越性.     

关于一个多项式重根迭代法的收敛性

讨论一个同时求解多项式重根的迭代法,给出其收敛性定理及其简洁证明,数值结果是满意的.     

求解时谐涡流模型鞍点问题的分块交替分裂隐式迭代算法的改进

分块交替分裂隐式迭代方法是求解具有鞍点结构的复线性代数方程组的一类高效迭代法.本文通过预处理技巧得到原方法的一种加速改进方法,称之为预处理分块交替分裂隐式迭代方法·理论分析给出了新方法的收敛性结果.对于一类时谐涡旋电流模型问题,我们给出了若干满足收敛条件的迭代格式.数值实验验证了新型算法是对原方法的有效改进.     

求解M-矩阵代数Riccati方程的两种不动点迭代法（英文）

M-矩阵代数Riccati方程由于广泛的应用,已成为近年来的热点问题之一,有关其理论和数值方法的研究层出不穷.本文研究M-矩阵代数Riccati方程的数值解法,给出求解其最小非负解的两种新的不动点迭代法.理论分析表明新的不动点迭代法相比现有的不动点迭代法收敛速度快,数值实验也验证了新方法的有效性.     

一种基于Chebyshev迭代解非线性方程组的方法

本文根据牛顿迭代和Chebyshev迭代法给出了一种新的迭代方法,该方法有较高的收敛阶,并在理论上给予了证明.最后给出了四个实例,将本文的实验结果与现有的几种方法的实验结果进行比较,表明我们的方法迭代次数少,有明显的优势.     

求解一类隐式互补问题的加速模系矩阵分裂迭代方法

本文提出求解一类隐式互补问题的加速模系矩阵分裂迭代法.通过将隐式互补问题重新表述为一个等价的不动点方程,建立一类新的基于模系的两步矩阵分裂方法,并在一定条件下证明了方法的收敛性.数值实验表明,该方法在迭代步数上优于传统的模系矩阵分裂迭代方法.     

关于Jain的隐式迭代法的注

本文利用嵌入法思想构造了一类求解非线性方程组的隐式迭代法，分析了方法的收敛阶，给出了具体的计算格式，最后的计算结果表明了方法的有效性．     

两步定常线性迭代法的收敛区域及最优参数选取

１．引言 给定一线性系统 Ａｘ＝ｂ,（１．１）其线性两步定常迭代方法可表示为 ｘｎ＋１＝ ｘｎ＋ αｒｎ＋ β（ｘｎ－ ｘｎ－１）,（１．２）其中 ｒｎ＝ｂ－Ａｘｎ（１．３）是剩余向量, ｘ０, ｘ１是任意的（ｃｆ．Ｙｏｕｎｇ［１,ｐ．４８７］）．本文我们将研究迭代式（１．２）的收敛条件及参数α,β如何选取问题．关于此问题已有一些结果,如［２－４］,本文将从方程根的角度讨论最一般的情况,即在复数域上来讨论此问题,同时作为其特例来讨论复 ＳＯＲ、 ＭＳＯＲ的收敛性． 下文中除了特别说明,Ａ是复矩阵,α,β是复…     

Application of orthogonal expansions for approximate integration of ordinary differential equations

An approximate method to solve the Cauchy problem for normal and canonical systems of second-order ordinary differential equations is proposed. The method is based on the representation of a solution and its derivative at each integration step in the form of partial sums of series in shifted Chebyshev polynomials of the first kind. A Markov quadrature formula is used to derive the equations for the approximate values of Chebyshev coefficients in the right-hand sides of systems. Some sufficient convergence conditions are obtained for the iterative method solving these equations. Several error estimates for the approximate Chebyshev coefficients and for the solution are given with respect to the integration step size.     

Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.     

Equivalent iterative methods for p-cyclic matrices

An overview is given of the simplifications which arise when p-cyclic systems are solved by iterative methods. Besides the classic iterative methods, we treat the Chebyshev semi-iterative method and the OR and MR variants of the class of Krylov subspace methods. Particular emphasis is given to equivalent iterations applied to the cyclically reduced system.     

Monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given. Numerical experiments are presented.     

The semi‐iterative method applied to the hyper‐power iteration

The semi‐iterative method (SIM) is applied to the hyper‐power (HP) iteration, and necessary and sufficient conditions are given for the convergence of the semi‐iterative–hyper‐power (SIM–HP) iteration. The root convergence rate is computed for both the HP and SIM–HP methods, and the quotient convergence rate is given for the HP iteration. Copyright © 2005 John Wiley & Sons, Ltd.     

解不适定算子方程的一个定常二步隐式迭代法

On the basis of an implicit iterative method for ill-posed operator equations,we introduce a relaxation factor and a weighted factor , and obtain a stationarytwo-step implicit iterative method. The range of the factors which guarantee theconvergence of iteration is explored.We also study the convergence properties and ratesfor both non-perturbed andperturbed equations.An implementable algorithm is presented by using Morozov discrepancy principle.The theoretical results show that the convergence rates of the new methods always lead to optimal convergentrates which are superior to those of the original one after choosing suitable relaxation and weightedfactors. Numerical examplesare also given, which coincide well with the theoretical results.     

线性方程组的异步松弛迭代法

本文考虑解线性方程组经典迭代法的异步形式，对系数矩阵为H矩阵，给出了异步迭代过程收敛性的充分条件，这不仅降低了文献［3］对系数矩阵的要求，而且收敛区域比文献［3］的大．     

非Hermitian正定线性方程组的外推的HSS迭代方法

为了高效地求解大型稀疏非Hermitian正定线性方程组，在白中治、Golub和Ng提出的Hermitian和反Hermitian分裂（HSS）迭代法的基础上，通过引入新的参数并结合迭代法的松弛技术，对HSS迭代方法进行加速，提出了一种新的外推的HSS迭代方法（EHSS），并研究了该方法的收敛性.数值例子表明：通过参数值的选择，新方法比HSS方法具有更快的收敛速度和更少的迭代次数，选择了合适的参数值后，可以提高HSS方法的收敛效率.