关于线性系统解的对称性算法的强稳定性的探索

关于线性方程解的算法方面,本文在对称非奇异矩阵类和对称正定矩阵类上给出了强稳定性算法.     

高校应用数学学报第16卷(2001年)B辑(英文版)第1期目次和提要

三幂等符号模式矩阵的结构李炯生　高玉斌 (中国科学技术大学数学系 )元素为 + ,- ,0的矩阵称为符号模式矩阵 .设 A为 n阶符号模式方阵 ,如果 A3 =A,则称 A为三幂等符号模式矩阵 .该文对 n阶 ( n≥ 2 )三幂等符号模式矩阵的结构进行了刻划 ,同时也给出了一个符号模式矩阵是三幂等的但不是幂等的充分必要条件 .关于二维光滑线性方程无解的注记边保军　李俊杰 (浙江大学数学系 )给出了一个无解的含两个自变量的复线性偏微分方程 .在此基础上 ,得到了一个实线性方程无解的例子 .零点有次线性项的椭圆问题的变号解吴绍平　孙义静 (浙江大学数…     

线性齐次常微分方程(组)的λ-矩阵求解法

本文在文 [2 ]的基础上 ,应用λ-矩阵及微分算子性质给出了一种变系数齐次常微分线性方程 (组 )的λ-矩阵求解法 ,对文 [2 ]作出了更一般的推广 .     

非线性分析中的全量刚度矩阵与增量刚度矩阵

本文详细导出了非线性分析中的全量(割线)刚度矩阵和增量(切线)刚度矩阵的一般表达式,并由此进一步讨论了它们二者之间的数学关系.本文的结果对于非线性方程的求解,及非线性、线性稳定性分析都具有重要帮助作用.     

非线性方程分枝解理论在线性闭环系统极点摄动量估计中的应用

本文利用非线性方程小分枝解的理论,研究多变量定常线性控制系统的状态矩阵与控制矩阵受到摄动时,闭环系统的极点所受到的摄动量的估计问题.     

求解非线性方程七阶收敛的牛顿迭代修正格式（英文）

本文研究了非线性方程求解的问题.利用泰勒公式和耦合方法,获得了一种求解非线性方程的加速收敛的七阶迭代改进格式,该格式不需要计算高阶导数,且具有更大的收敛半径,大大提高了计算效率.     

求解非线性方程七阶收敛的牛顿迭代修正格式

本文研究了非线性方程求解的问题.利用泰勒公式和耦合方法,获得了一种求解非线性方程的加速收敛的七阶迭代改进格式,该格式不需要计算高阶导数,且具有更大的收敛半径,大大提高了计算效率.     

Fuzzy双线性方程最大解的一种简便算法

引进Fuzzy双线性方程A·X=B·X中间矩阵的概念, 并利用此概念得出了这类方程最大解的一种快速算法.最后通过实例说明了该算法的优越性.     

具有终端不等式约束的线性二次控制问题

本文研究具有终端不等式约束的线性二次控制问题,得到了最优控制的反馈形式.所得到的反馈形式与相应的无约束问题的Riccati微分方程和一个函数矩阵的线性方程的解有关.     

区间动力系统Robust稳定性进展

本义综述了区间动力系统robust稳定性进展,内容涉及区间多项式,矩阵的Hurwitz稳定性、robust度、Schur稳定性、robust度,不确定时滞系统及控制系统的robust稳定性。1 引官 近二十年来,先后从计算数学,数理统计,控制理论等不同分支提出了区间分析和所谓稳健性(robustness)问题。例如:在计算数学中,解非线性方程:     

The alternative Legendre Tau method for solving nonlinear multi-order fractional differential equations

In this paper, the alternative Legendre polynomials (ALPs) are used to approximate the solution of a class of nonlinear multi-order fractional differential equations (FDEs). First, the operational matrix of fractional integration of an arbitrary order and the product operational matrix are derived for ALPs. These matrices together with the spectral Tau method are then utilized to reduce the solution of the mentioned equations into the one of solving a system of nonlinear algebraic equations with unknown ALP coefficients of the exact solution. The fractional derivatives are considered in the Caputo sense and the fractional integration is described in the Riemann-Liouville sense. Numerical examples illustrate that the present method is very effective for linear and nonlinear multi-order FDEs and high accuracy solutions can be obtained only using a small number of ALPs.     

Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations

The aim of this article is to present an analytical approximation solution for linear and nonlinear multi-order fractional differential equations (FDEs) by extending the application of the shifted Chebyshev operational matrix. For this purpose, we convert FDE into a counterpart system and then using proposed method to solve the resultant system. Our results in solving four different linear and nonlinear FDE, confirm the accuracy of proposed method.     

Adaptive data distribution for concurrent continuation

Summary Continuation methods compute paths of solutions of nonlinear equations that depend on a parameter. This paper examines some aspects of the multicomputer implementation of such methods. The computations are done on a mesh connected multicomputer with 64 nodes.One of the main issues in the development of concurrent programs is load balancing, achieved here by using appropriate data distributions. In the continuation process, many linear systems have to be solved. For nearby points along the solution path, the corresponding system matrices are closely related to each other. Therefore, pivots which are good for theLU-decomposition of one matrix are likely to be acceptable for a whole segment of the solution path. This suggests to choose certain data distributions that achieve good load balancing. In addition, if these distributions are used, the resulting code is easily vectorized.To test this technique, the invariant manifold of a system of two identical nonlinear oscillators is computed as a function of the coupling between them. This invariant manifold is determined by the solution of a system of nonlinear partial differential equations that depends on the coupling parameter. A symmetry in the problem reduces this system to one single equation, which is discretized by finite differences. The solution of the discrete nonlinear system is followed as the coupling parameter is changed.This material is based upon work supported by the NSF under Cooperative Agreement No. CCR-8809615. The government has certain rights in this material.     

求解带Toeplitz矩阵的线性互补问题的一类预处理模系矩阵分裂迭代法

针对系数矩阵为对称正定Toeplitz矩阵的线性互补问题,本文提出了一类预处理模系矩阵分裂迭代方法.先通过变量替换将线性互补问题转化为一类非线性方程组,然后选取Strang或T.Chan循环矩阵作为预优矩阵,利用共轭梯度法进行求解.我们分析了该方法的收敛性.数值实验表明,该方法是高效可行的.     

The spectral method for solving systems of Volterra integral equations

This paper presents a high accurate and stable Legendre-collocation method for solving systems of Volterra integral equations (SVIEs) of the second kind. The method transforms the linear SVIEs into the associated matrix equation. In the nonlinear case, after applying our method we solve a system of nonlinear algebraic equations. Also, sufficient conditions for the existence and uniqueness of the Linear SVIEs, in which the coefficient of the main term is a singular (or nonsingular) matrix, have been formulated. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.     

Riccati equations and Lie series

Lie series and a special matrix notation for first-order differential operators are used to show that the Lie group properties of matrix Riccati equations arise in a natural way. The Lie series notation makes it evident that the solutions of a matrix Riccati equation are curves in a group of nonlinear transformations that is a generalization of the linear fractional transformations familiar from the classical complex analysis. It is easy to obtain a linear representation of the Lie algebra of the nonlinear group of transformations and then this linearization leads directly to the standard linearization of the matrix Riccati equations. We note that the matrix Riccati equations considered here are of the general rectangular type.     

A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients

This article develops an efficient solver based on collocation points for solving numerically a system of linear Volterra integral equations (VIEs) with variable coefficients. By using the Euler polynomials and the collocation points, this method transforms the system of linear VIEs into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Euler coefficients. A small number of Euler polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving VIEs with variable coefficients.     

A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation

A Taylor matrix method is proposed for the numerical solution of the two-space-dimensional linear hyperbolic equation. This method transforms the equation into a matrix equation and the unknown of this equation is a Taylor coefficients matrix. Solutions are easily acquired by using this matrix equation, which corresponds to a system of linear algebraic equations. As a result, the finite Taylor series approach with three variables is obtained. The accuracy of the proposed method is demonstrated with one example.     

An inexact non-interior continuation method for semidefinite programming: convergence analysis and numerical results

In this paper, an inexact non-interior continuation method is proposed for semidefinite Programs. By a matrix mapping, the primal-dual optimal condition can be inverted into a smoothed nonlinear system of equations. A linear system of equations with residual vector is eventually driven by solving the smoothed nonlinear system of equations and finally solved by the conjugate residual method. The global and locally superlinear convergence are verified. Numerical results and comparisons indicate that the proposed methods are very promising and comparable to several interior-point and other exact non-interior continuation methods.     

非线性系统动力分析的模态综合技术

各种模态综合方法已广泛应用于线性结构的动力分析,但是,一般都不适用于非线性系统. 本文基于[20][21]提出的方法,将一种模态综合技术推广到非线性系统的动力分析.该法应用于具有连接件耦合的复杂结构系统,以往把连接件简化为线性弹簧和阻尼器.事实上,这些连接件通常具有非线性弹性和非线性阻尼特性.例如,分段线性弹簧、软特性或硬特性弹簧、库伦阻尼、弹塑性滞后阻尼等.但就各部件而言,仍属线性系统.可以通过计算或试验或兼由两者得到一组各部件的独立的自由界面主模态信息,且只保留低阶主模态.通过连接件的非线性耦合力,集合各部件运动方程而建立成总体的非线性振动方程.这样问题就成为缩减了自由度的非线性求解方程,可以达到节省计算机的存贮和运行时间的目的.对于阶次很高的非线性系统,若能缩减足够的自由度,那么问题就可在普通的计算机上得以解决. 由于一般多自由度非线性振动系统的复杂性,一般而言,这种非线性方程很难找到精确解.因此,对于任意激励下系统的瞬态响应,可以采用数值计算方法求解缩减的非线性方程.