不确定性问题中逻辑关系方程的置换矩阵解法

本文给出了在不确定性问题中逻辑关系方程有解 ,有唯一解的充分必要条件 ,并把求解逻辑关系方程的问题转化为求解一些系数矩阵是置换矩阵的逻辑方程组问题 ,从而给出一种求解逻辑关系方程的新算法     

主子阵约束下矩阵方程AX=B的对称最小二乘解

本文主要讨论主子阵约束下矩阵方程AX=B的对称最小二乘解．基于投影定理,巧妙的把最小二乘问题转化为等式问题求解,并利用奇异值分解的方法,给出了该对称最小二乘解的一般表达式．此外,文章还考虑了此对称最小二乘解集合对任一给定矩阵的最佳逼近问题,得到了最佳逼近解,并给出了相应的算法步骤和数值例子．     

利用并行方法解AX+XB=C型线性矩阵方程

提出了一种新的递推算法用于求解AX+XB=C型线性矩阵方程,这种算法可以用脉 动阵列结构并行实现,该算法和结构还可求解其它几种类似的线性矩阵方程,特殊情况下求解 方程的阵列结构可进一步简化.仿真结果表明,这种并行方法有较高的加速比及效率.     

矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2的迭代算法

矩阵方程组的求解在结构设计、参数识别、生物学、电学、分子光谱学、固体力学、自动控制理论、振动理论、有限元、线性最优控制等领域都有着重要应用。本文从解线性代数方程组的共轭梯度法中受到启示,不是采用传统的矩阵分解的方法,而是采用迭代算法给出了求矩阵方程组A1XB1=C1,A2XB2=C2的解、极小范数解及其最佳逼近解的方法。     

Excel的逆矩阵法

本刊98年第4期的“用Excel求解线性方程组”,利用高斯消元法和Excel的粘贴功能对方程组求解。这里介绍逆矩阵方法。 我们知道,所有线性方程组都可以表示为: AX=B或X=A~(-1)B 利用Excel提供的矩阵求逆函数MINVERSE,可以直接求出A~(-1),然后利用逆矩阵乘法函数MMULT,算出A~(-1)与B矩阵的乘积,即可得出方程组的解。假设有一方程组:     

梯度神经网络解线性矩阵方程之收敛性分析

为了求解线性矩阵方程问题,应用一种基于负梯度法的递归神经网络模型,并探讨了该递归神经网络实时求解线性矩阵方程的全局指数收敛问题.在讨论渐近收敛性基础上,进一步证明了该类神经网络在系数矩阵满足有解条件的情况下具有全局指数收敛性,在不能满足有解条件的情况下具有全局稳定性.计算机仿真结果证实了相关理论分析和该网络实时求解线性矩阵方程的有效性.     

几种特殊循环矩阵的PROCRUSTES问题

本文研究了在Hankel—循环矩阵和Hankel—反循环矩阵的约束下矩阵方程组AX=B,XC=D的最小二乘解问题.结合最优化理论和循环矩阵的性质,将其转化为简单的线性方程Qy=b的求解问题,得到了通解的表达式.进一步,证得系数矩阵Q是一个与所求矩阵X相关联的循环矩阵,从而找到了解唯一的充分必要条件并给出了解的表达式.此外,借助于矩阵的广义1—范数,给出了有唯一解的判定条件.最后,给出了具体的算法和算例.     

求多变量线性矩阵方程组自反解的迭代算法

利用矩阵分解的方法求多变量线性矩阵方程组的自反解是很困难的.本文建立了一种迭代方法来解决这个问题,利用此迭代方法可以判断多变量线性矩阵方程组的可解性,且当矩阵方程组相容时,可以在有限步迭代后得到其自反解.选取特殊的初始矩阵时,能够求得矩阵方程组的极小范数自反解.进一步,通过求新的线性矩阵方程组的极小范数自反解,能够求得给定矩阵的最佳逼近矩阵.数值算例表明,迭代算法是有效的.     

非对称强非线性振动特征分析

提出了构造一类非线性振子解析逼近周期解的的初值变换法.用Ritz-Galerkin法,将描述动力系统的二阶常微分方程,化为以振幅、角频率和偏心距为独立变量的不完备非线性代数方程组;关键是考虑初值变换,增加补充方程,构成了以角频率、振幅和偏心距为变量的完备非线性代数方程组.作为例子利用初值变换法求解了相对论修正轨道方程的六种分岔周期解.给出了非对称振动的幅频曲线和偏频(偏心距与角频率的关系)曲线.发现了固有角频率漂移现象.     

分布时滞系统的反馈镇定

求解特征矩阵是镇定时滞系统的关键问题,本文给出了系统的特征根的代数重复度与几 何重复度均为一般值情况下特征矩阵的求法,即把它归结为求解一组线性代数方程的问题,并 得到了该方程组有解及对应于同一特征值的解向量组线性独立的充分条件.此外,还提出了 一种算法,用以处理系统对应于不同特征值的左特征向量线性相关情况下系统的镇定问题.     

一种基于逻辑关系方程解的属性约简算法

粗糙集理论中所有的概念与运算都是通过代数学的等价关系和集合运算来定义的.在这种定义下,粗糙集理论的很多概念与运算的直观性较差.从逻辑代数的角度出发,建立了属性集与布尔矩阵以及逻辑关系方程之间的关系,给出了逻辑关系方程有解、有惟一解、有多个解的充分必要条件,在逻辑关系方程解的基础上给出了一种新的高效的属性约简算法.     

On positive definite solutions to the algebraic Riccati equation

Complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic Riccati equation are given. It is also shown that when a positive definite solution exists, it is either unique, or else there are uncountably many such solutions.     

真度方程组及其应用

基于计量逻辑学和真度方程的思想,提出了真度方程组的概念。给出了真度方程组的同型解,探讨了真度方程组解集合的相容性。将这一理论成功地运用于多重广义MP问题的研究,求出了多重广义MP问题的三I真度解与α-三I真度解,为进一步探讨模糊推理的逻辑基础提供了一个可行的途径。     

一般阻尼动力系统非奇异摄动法

借助矩阵摄动理论,将模态叠加法运用于一般阻尼矩阵的动力学方程求解结构的动响应是一种较为理想的方法.但当系统的外荷载激振频率接近于系统的固有频率时,直接将阻尼矩阵作为摄动矩阵,会使解产生奇异,并导致求解失败或误差过大,这是因为模态坐标下的动力学方程是无阻尼方程.为了解决这一问题,本文考虑在模态坐标的动力学方程中保留一定的阻尼.即将阻尼做分解,代入振动方程,得到不同阶次摄动方程,再将摄动方程变换到模态坐标,即采用非奇异摄动方法.最后通过数值算例,得到一阶、二阶摄动,将其与精确解进行比较.精度明显得到改善,基本趋于精确解.从而验证了本方法的精确性和有效性.     

Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive-definite solutions of the matrix equation X^s+A*X^?tA=Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q=I is discussed. The error estimations are found.     

Analytical forms for most likely matrices derived from incomplete information

Consider a rectangular matrix describing some type of communication or transportation between a set of origins and a set of destinations, or a classification of objects by two attributes. The problem is to infer the entries of the matrix from limited information in the form of constraints, generally the sums of the elements over various subsets of the matrix, such as rows, columns, etc., or from bounds on these sums, down to individual elements. Such problems are routinely addressed by applying the maximum entropy method to compute the matrix numerically, but in this article we derive analytical, closed-form solutions. For the most complicated cases we consider the solution depends on the root of a non-linear equation, for which we provide an analytical approximation in the form of a power series. Some of our solutions extend to 3-dimensional matrices. Besides being valid for matrices of arbitrary size, the analytical solutions exhibit many of the appealing properties of maximum entropy, such as precise use of the available data, intuitive behaviour with respect to changes in the constraints, and logical consistency.     

采用线性不定方程组的Hamiltonian存在判定条件

本文基于{0，1}线性不定方程组和顶边关联矩阵．提出了一个基于无向Hamiltonian图的充要判定定理。并证明了满足该定理的不定方程组解向量对应给定圄的Hamiltonian回路中边的集合，本文还推导出两个可以基于矩阵秩的Hamiltonian回路存在的必要判据。     

Field analysis of bridge grids with non-warping torsion

Exact closed form solutions are found for rationally based mathematical models of rectangular grids with simple end supports and flexible side supports. Micro discrete field mechanics techniques are employed to obtain the formulas for non-prismatic membered, regular grids. A deformation or stiffness approach is used to formulate the mathematical model in the form of a partial difference equation with the nodal or joint displacements being the unknowns. The solutions are valid for general loadings and are in the form of double finite trigonometric series. A general solution for a doubly symmetric, regular, bridge grid is obtained by superposition of solutions for: (1) simple side supports case; (2) imposed boundary deflections case; (3) imposed boundary rotations case; and (4) imposed boundary moments case. Definitions of the imposed boundary function coefficients are obtained by satisfying equilibrium of the side boundary stringers. Axial effects and warping torsion effects are not accounted for. Computer programs written in Fortran and tested by the first author are available upon request by any interested person.     

Gradient-based maximal convergence rate iterative method for solving linear matrix equations

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.