AKNS方程的新双Wronski解

通过构造双Wronski行列式元素的矩阵方法, 导出二阶AKNS方程的孤子解、 有理解、Matveev解和Complexiton解, 其中后三类解是新解, 并借助约化 求得非线性Schrödinger方程的有理解.     

约束矩阵方程的解的表示及行列式公式

本文利用Ｂｏｔｔ－Ｄｕｆｆｉｎ逆导出约束矩阵方程的唯一解或通解的表示及其较为简洁的行列式公式，并由之给出了某些特殊矩阵方程与约束性方程组的解以及几个重要广义逆矩阵的表示及其较是洁的行列式公式，我们的结果改进与推广了文（２－１１，１４，１５）中的有关结果。     

矩阵方程AX＋XB=C的双对称解及其最佳逼近

提出一种求解线性矩阵方程AX＋XB=C双对称解的迭代法.该算法能够自动地判断解的情况,并在方程相容时得到方程的双对称解,在方程不相容时得到方程的最小二乘双对称解.对任意的初始矩阵,在没有舍入误差的情况下,经过有限步迭代得到问题的一个双对称解.若取特殊的初始矩阵,则可以得到问题的极小范数双对称解,从而巧妙地解决了对给定矩...     

非自治ABS方程的双线性化和Casorati行列式解

本文给出可积非自治Adler-Bobenko-Suris(ABS)链方程,并通过适当的变量变换将其转化为非自治离散双线性方程,从而得到Casorati行列式解.为完成解的验证,在附录中,本文给出一系列Casorati行列式平移公式.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

一类Riccati矩阵方程广义自反解的双迭代算法

本文研究了一类Riccati矩阵方程广义自反解的数值计算问题.利用牛顿算法将Riccati矩阵方程的广义自反解问题转化为线性矩阵方程的广义自反解或者广义自反最小二乘解问题,再利用修正共轭梯度法计算后一问题,获得了求Riccati矩阵方程的广义自反解的双迭代算法.拓宽了求解非线性矩阵方程的迭代算法.数值算例表明双迭代算法是有效的.     

双矩阵变量Riccati矩阵方程对称解的迭代算法

研究一类双矩阵变量Riccati矩阵方程(R-ME)对称解的数值计算问题.运用牛顿算法求R-ME的对称解时,会导出求双矩阵变量线性矩阵方程的对称解或者对称最小二乘解的问题,采用修正共轭梯度法解决导出的线性矩阵方程约束解问题,可建立求R-ME的对称解的迭代算法.数值算例表明,迭代算法是有效的.     

正弦-Gordon方程n孤子解的一致性

正弦-Gordon方程是一种重要的非线性波动方程,其n孤子解具有Hirota表示与Wronski行列式表示形式,利用行列式的性质说明正弦-Gordon方程的这两种n孤子解的表示是一致的.     

含高次逆幂的矩阵方程对称解的双迭代算法

本文研究了在控制理论和随机滤波等领域中遇到的一类含高次逆幂的矩阵方程的等价矩阵方程对称解的数值计算问题.采用牛顿算法求等价矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立了求这类矩阵方程对称解的双迭代算法,数值算例验证了双迭代算法是有效的.     

矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解及其最佳逼近

基于共轭梯度法的思想,通过特殊的变形,建立了一类求矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解的迭代算法．对任意的初始双对称矩阵．在没有舍人误差的情况下,经过有限步迭代得到它的双对称最小二乘解;在选取特殊的初始双对称矩阵时,能得到它的的极小范数双对称最小二乘解．另外,给定任意矩阵,利用此方法可得到它的最佳逼近双对称解,数值例子表明,这种方法是有效的．     

Generalized double Casoratian solutions to the four-potential isospectral Ablowitz–Ladik equation

Generalized solutions in double Casoratian form of the four-potential isospectral Ablowitz–Ladik equation possessing bilinear form are derived through a matrix method for constructing double Casoratian entries. A novel class of explicit solutions, such as soliton, rational-like, Matveev, Complexiton and interaction solutions, are obtained by letting the general matrix be some special cases. Interestingly, a periodic solution is deduced from the Complexiton solution.     

New double Wronskian solutions of the AKNS equation

Soliton solutions, rational solutions, Matveev solutions, complexitons and interaction solutions of the AKNS equation are derived through a matrix method for constructing double Wronskian entries. The latter three solutions are novel. Moreover, rational solutions of the nonlinear Schrodinger equation are obtained by reduction.     

A New Algorithm for Constrained Matrix Least Squares Approximations

This paper considers the problem of approximating a given symmetric matrix by a symmetric matrix with a prescribed spectrum so that the Frobenius norm of the matrix difference is minimized. By the introduction of a variable search direction, a new convergent algorithm for solving the problem is derived, which is guaranteed to be convergent and is capable of achieving a fast rate of convergence. It is shown that the set of fixed points of the proposed algorithm coincides with the set of equilibrium points of the original double bracket equation. A numerical example is presented to demonstrate superior performance of the proposed algorithm over a standard double bracket algorithm.     

模糊线性系统的扰动分析

使用谱范数分析了模糊线性系统在三种情形下的扰动: (1)右端模糊向量有扰动, 系数矩阵不变; (2)系数矩阵有扰动,右端模糊向量不变; (3)系数矩阵和右端模糊向量都有扰动,并通过数值实例验证给出的扰动界的估计.     

Double Series Identity and Some Laurent Type Hypergeometric Generating Relations

The main aim of this article is to obtain certain Laurent type hypergeometric generating relations. Using a general double series identity, Laurent type generating functions(in terms of Kampéde Fériet double hypergeometric function) are derived. Some known results obtained by the method of Lie groups and Lie algebras, are also modified here as special cases.     

Some results on functions of matrix argument

A number of integrals of scalar variables are extended to the case of matrix variables. Functions where the argument is a positive definite symmetric matrix are considered in this article. With the help of a generalized matrix transform a number of results are derived which are extensions of the corresponding results in the univariate cases.     

On a partially debonded rigid line inclusion penetrating a circular inhomogeneity

We derive closed-form solutions to the mixed boundary value problem of a partially debonded rigid line inclusion penetrating a circular elastic inhomogeneity under antiplane shear deformation. The two tips of the rigid line inclusion are just mutual mirror images with respect to the inhomogeneity/matrix interface, and the upper part of the rigid line inclusion is debonded from the surrounding materials. By using conformal mapping and the method of image, closed-form solutions are derived for three loading cases: (i) the matrix is subjected to remote uniform stresses; (ii) the matrix is subjected to a line force and a screw dislocation; and (iii) the inhomogeneity is subjected to a line force and a screw dislocation. In the mapped ξ-plane, the solutions for all the three loading cases are interpreted in terms of image singularities. For the remote loading case, explicit full-field expressions of all the field variables such as displacement, stress function and stresses are obtained. Also derived is the near tip asymptotic elastic field governed by two generalized stress intensity factors. The generalized stress intensity factors for all the three loading cases are derived.