二阶系统基于奇异值的保结构模型降阶方法（英文）

本文提出了一种二阶系统基于奇异值的保结构模型降阶方法.该方法通过脉冲响应的移位勒让德多项式展开系数直接计算得到二阶系统的可控和可观格拉姆矩阵的低秩因子.然后基于二阶系统的奇异值结构,构造相应的平衡系统,进而通过截断较小奇异值对应的状态得到保结构的降阶系统.最后,通过两个数值实验展示了所提方法的有效性.     

对角占优矩阵奇异-非奇异的充分必要判据

本文研究对角占优矩阵奇异-非奇异的充分必要条件.基于Taussky定理,本文得出,可约对角占优矩阵的奇异性由其独立Frobenius块的奇异性决定,从而将这一问题化为不可约对角占优矩阵的奇异-非奇异性问题;运用Taussky定理研究奇异不可约对角占优矩阵的相似性和酉相似性,获得这类矩阵元素辐角间的关系;并与Taussky定理给出的这类矩阵元素模之间的关系结合在一起,研究不可约对角占优矩阵奇异的充分必要条件;最后给出不可约对角占优矩阵奇异-非奇异性的判定方法.     

关于“一类非奇异H-矩阵判定的新条件”一文的注记

通过构造新的正对角因子元素，本文给出了几个判定非奇异H-矩阵新的充分条件，改进和推广了"一类非奇异H-矩阵判定的新条件"一文的主要结果，并用数值例子说明了文中结果判定范围的更加广泛性.     

线性奇异系统的解耦问题

本文讨论在状态反馈下的线性奇异系统的解耦问题，利用奇异系统的受限等价变换，给出了输出，输入维数相同的方线性奇异系统解耦问题有解的充分必要条件。     

奇摄动向量Robin问题的对角化方法

本文使用对角化的方法和技巧,把一个二阶的非线性系统变换成二个一阶的近似对角线的系统,获得奇异摄动类型的向量二阶非线性Robin问题解的存在性和渐近估计式.     

Broer-Kaup系统的达布变换及其孤子解

根据Broer-Kaup系统的Lax对, 借助Broer-Kaup系统的谱问题的规范变换, 一个包含多参数的达布变换被构造出. 以一个平凡解作为种子解, 利用达布变换, 可以求得Broer-Kaup系统的非平凡解的一般表达式. 并且讨论了N=1和N=2两种孤子解的情形. 这是一种与2X2谱问题有关的孤子碰撞图像的新类型.     

《数值计算与计算机应用》2021年第42卷第4期摘要

潘佳佳,李会元,二阶椭圆问题的弱迦辽金四边形谱元方法[J].数值计算与计算机应用,2021,42(4):303-322.摘要:本文对二阶椭圆方程特征值问题的弱伽辽金谱元方法开展相关数值研究.与弱有限元方法类似,弱伽辽金谱元方法的逼近函数空间包括各个单元上的独立内部分量、并辅以各单元边界分量作为单元与单元间的联系.本文聚焦任意凸四边形网格剖分下的弱伽辽金四边形谱元方法,弱逼近函数中的各内部分量与边界分量分别由参考正方形单元与参考单元边界上的正交多项式通过双线性变换来构造;而弱梯度逼近空间则由参考正方形上的正交多项式通过Piola变换构造.在此基础上,本文提出了二阶椭圆方程特征值问题的弱伽辽金四边形谱元方法逼近格式和实现算法,并通过对离散弱梯度核空间的系统研究。     

二阶半线性奇摄动边值问题的渐近解

研究不满足法向双曲条件的二阶半线性非自治奇摄动Dirichlet边值问题.首先,利用边界层函数法,构造了问题在两个区间端点的代数边界层,获得了形式渐近解;接着,利用上下解方法,证明了解的存在性、渐近解的一致有效性以及渐近解与精确解之间的误差估计.研究表明:通过对奇异摄动参数进行适当的尺度变换,一定条件下可处理任意退化的二阶半线性非自治奇摄动边值问题.最后,通过一个典型例子验证了理论结果的正确性.     

非奇异H-矩阵新的含参数细分迭代判别法

结合矩阵自身的元素,构造了含参数的迭代公式,进而细分了矩阵非对角占优行指标集.利用广义严格α-对角占优矩阵与非奇异H-矩阵的关系,给出了非奇异H-矩阵一组新的细分迭代判定准则,推广和改进了已有的结果,通过数值算例说明了结果的优越性.     

一种对称非奇异核边界元法

本文依据泛函分析的基本理论,提出利用完备直交的本征函数系构造对称非奇异的基本解,给出了又一种非奇异边界元法.     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

On the quadratic two-parameter eigenvalue problem and its linearization

We introduce the quadratic two-parameter eigenvalue problem and linearize it as a singular two-parameter eigenvalue problem. This, together with an example from model updating, shows the need for numerical methods for singular two-parameter eigenvalue problems and for a better understanding of such problems.There are various numerical methods for two-parameter eigenvalue problems, but only few for nonsingular ones. We present a method that can be applied to singular two-parameter eigenvalue problems including the linearization of the quadratic two-parameter eigenvalue problem. It is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils.     

Instability Degree and Singular Subspaces of Integral Isotropic Cones of Linear Systems of Differential Equations

The instability degree of linear systems of differential equations is estimated in terms of the dimensions of completely singular subspaces of integral cones of these systems. Special attention is given to the case where the linear system under study has first integrals of the type of nonsingular quadratic forms. General results are applied to a well-known problem concerning the gyroscopic stabilization of unstable equilibria of a mechanical system.     

Singular quadratic variational problems

The purpose of this paper is to show that the general theory of quadratic forms developed earlier by the author is applicable to singular variational problems as well as to nonsingular ones. In particular, this general theory is applicable to the singular variational problems associated with Legendre polynomials, associated Legendre polynomials, Jacobi polynomials, and Tchebysheff polynomials.     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.     

A Trust Region Method for Optimization Problem with Singular Solutions

In this paper, we propose a trust region method for minimizing a function whose Hessian matrix at the solutions may be singular. The global convergence of the method is obtained under mild conditions. Moreover, we show that if the objective function is LC ² function, the method possesses local superlinear convergence under the local error bound condition without the requirement of isolated nonsingular solution. This is the first regularized Newton method with trust region technique which possesses local superlinear (quadratic) convergence without the assumption that the Hessian of the objective function at the solution is nonsingular. Preliminary numerical experiments show the efficiency of the method. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 70302003, 10571106, 60503004, 70671100) and Science Foundation of Beijing Jiaotong University (2007RC014).     

Domain sensitivity for singular elliptic problems: Harmonic transformation approach

A method for computing the sensitivities of functionals depending on the solutions of elliptic equations defined over variable domains is presented. It is based on the material derivative approach and allows the uniform treatment of both singular and nonsingular cases. The novelty consists in defining the vector field connected with the domain transformation as the solution of an auxiliary elliptic equation. Such a choice does not restrict the range of admissible goal functionals and has many advantages from the numerical point of view. It allows one also to consider singular domain variations.     

Regularized Newton Methods for Convex Minimization Problems with Singular Solutions

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC², then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually. Limited numerical experiments are presented, which show the practical advantage of the method.     

The distribution of hermitian indefinite quadratic forms

Hermitian quadratic forms in complex Gaussian random variables is considered in this article. Various representations of the exact densities for the central, non central, singular and nonsingular cases are given. Particular cases are shown to agree with the results obtained by others in connection with various applications in communication theory, multichannel antennas, vehicular transportations and other related problems     

Nonsingular subsampling for regression S estimators with categorical predictors

Simple random subsampling is an integral part of S estimation algorithms for linear regression. Subsamples are required to be nonsingular. Usually, discarding a singular subsample and drawing a new one leads to a sufficient number of nonsingular subsamples with a reasonable computational effort. However, this procedure can require so many subsamples that it becomes infeasible, especially if levels of categorical variables have low frequency. A subsampling algorithm called nonsingular subsampling is presented, which generates only nonsingular subsamples. When no singular subsamples occur, nonsingular subsampling is as fast as the simple algorithm, and if singular subsamples do occur, it maintains the same computational order. The algorithm works consistently, unless the full design matrix is singular. The method is based on a modified LU decomposition algorithm that combines sample generation with solving the least squares problem. The algorithm may also be useful for ordinary bootstrapping. Since the method allows for S estimation in designs with factors and interactions between factors and continuous regressors, we study properties of the resulting estimators, both in the sense of their dependence on the randomness of the sampling and of their statistical performance.