广义严格对角占优矩阵的判定条件

广义严格对角占优矩阵是一类应用广泛的特殊矩阵,它在计算数学、数学物理、控制论等众多领域中都有着重要的作用。本文利用α-链对角占优矩阵的性质,结合不等式的放缩技巧,给出了广义严格对角占优矩阵新的判定条件,扩大了矩阵的判别范围,推广了一些已有的结论。     

广义双对角占优矩阵的Schur余

我们知道对角占优矩阵的Schur余是对角占优矩阵,对于双对角占优矩阵也有这样的性质,这种性质也可以推广到严格广义双对角占优矩阵的情况。本文研究了非严格广义双对角占优矩阵的Schur余,给出了广义双对角占优矩阵的Schur余仍可保持对角优势的特性。     

几类广义正定矩阵之间的关系及有关性质

本文得到几类广义正定矩阵之间的关系,以及这几类矩阵各自的和,平方、主子阵、逆矩阵,Schur补、行列式不等式等一些结果,并提出几个有待解决的问题。     

非奇H-矩阵的细分迭代判别准则?

非奇H-矩阵在矩阵理论、经济数学、数学物理和动力系统理论等方面有着重要的应用,因此很有必要研究其判定问题。本文根据广义严格α-对角占优矩阵的性质以及广义严格α-对角占优矩阵与非奇H-矩阵的关系,通过构造递进系数和细分区间的方法,给出了非奇H-矩阵的细分迭代判别准则,推广和改进了相关已有结果。数值算例说明了所得判别准则的有效性。     

非奇异H-矩阵的新判据

非奇异H-矩阵在矩阵分析和数值代数的研究中具有重要作用.本文利用广义α-对角占优矩阵、不可约α-对角占优矩阵和具非零元素链α-对角占优矩阵的概念和性质,通过对矩阵行标作划分的方法,首先给出了非奇异H-矩阵的两个新的判定条件.然后进一步将所得结果应用于比较矩阵和转置比较矩阵的和,得到了另一个更为实用的判据.最后,用数值例...     

Bézier曲线降阶的矩阵方法

本文根据升阶的逆过程并结合矩阵代数知识,给出了Bézier曲线降阶的矩阵向量表达式,并且通过分析得到降阶矩阵实际上是升阶矩阵的广义逆矩阵,同时给出了降阶曲线的误差分析.在某些情形下,我们得到降阶曲线更好的误差估计.     

广义严格对角占优矩阵与非奇异M-矩阵的判定

利用α-链对角占优矩阵的性质,结合不等式的放缩等技巧,给出了判定广义严格对角占优矩阵与非奇异M-矩阵的几个充分条件,改进了近期的一些结果,并用相应的数值实例说明了这些结果的有效性。     

M-矩阵的广义Perron余

本文主要考虑当n×n矩阵K为M-矩阵时它的广义Perron余的一些性质。首先证明不可约K矩阵的广义Perron余也是M-矩阵,接着介绍此型矩阵最小特征值的一些性质。     

非奇异H-矩阵的一组充分条件

非奇异H-矩阵在数值线性代数的理论与应用中起着重要作用,因此研究非奇异H-矩阵的判定条件有着非常重要的理论价值.本文根据广义严格α-链对角占优矩阵和广义严格α-对角占优矩阵的性质,通过引入迭代因子,给出了一组非奇异H-矩阵新的迭代判定条件.该判定条件推广和改进了相关已有结果,丰富和完善了非奇异H-矩阵的理论,最后用数值算例说明了其有效性.     

准正交矩阵与准对称矩阵

利用次转置阵给出了准正交阵与(反)准对称阵的概念;研究了它们的基本性质及其之间的联系;取得了许多新的结果,推广丫正交阵、对称阵与反对称阵间的相应结果,特别将正交阵的广义Cayley分解推广到了准正交阵上;将各类正交阵、对称阵及广义逆矩阵统一了起来。     

Properties of constitutive matrices for electrostatic and magnetostatic problems

Constitutive matrices allow one to write the constitutive relations, in an algebraic form, in some finite-volume-based numerical methods, such as the cell method and the finite integration technique. This paper shows some properties that link a general class of constitutive matrices to the finite-element stiffness matrices. One of these properties can be exploited to provide an alternative sufficient condition for the stability of a generalized finite-difference time-domain algorithm set up by these numerical methods.     

Transient responses of dynamical systems with random uncertainties

A new approach is presented for modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The probability model is deduced from the use of the entropy optimization principle whose available information involves the algebraic properties related to the generalized mass, damping and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. Finally, a simple example is presented.     

Multilevel dynamic substructures

The dynamic substructure method is extended to multilevel (recursive) substructures. The obvious distinction of the two approaches is that the stiffness and mass matrices before condensation are no longer frequency independent. The dynamic stiffness matrix at any substructure level is proved to be a function of the vibrating frequency in terms of some constant matrices which are derivable from the dynamic stiffness matrix at one lower substructure level. The method can accurately predict more modes than the number of degrees of freedom retained. The computational procedure, the generalized inverse iteration, the stationary principle of the system natural frequency and the generalized Rayleigh's quotient are derived for the frequency dependent matrices. Numerical examples are given to illustrate some engineering applications. A transcendental dynamic stiffness matrix can be transformed to a more convenient algebraic form by the present method.     

A practical rule for the derivation of transition finite elements

The derivation of shape functions of two‐dimensional regular and transition finite element using Pascal triangle concept is presented. A practical rule for the trial function selection from Pascal triangle is developed. The derived elements are tested using some case studies. The results are validated by ANSYS. The developed rule is also generalized for the three‐dimensional finite elements. Copyright © 2000 John Wiley & Sons, Ltd.     

非奇异H矩阵的一类新递进判别法

非奇异H矩阵在数学物理、控制论、电力系统理论等领域具有重要的实际应用价值.本文通过递进选取两个正对角矩阵因子的元素,利用不等式的放缩技巧,给出了非奇异H矩阵一类新的判定方法,并将其推广到不可约情形和非零元素链情形.通过比较分析,此判据放宽了对矩阵各行元素的条件限制,并举例说明了此方法的优越性.     

Computational structural mechanics and optimal control—the simulation of substructural chain theory to linear quadratic optimal control problems

The theory of optimal control and the theory of a substructural chain in static structural analysis are mutually simulated issues. From the minimum potential energy variational principle of the substructural chain, the generalized variational principle with two kinds of variables is derived first. By comparing that generalized variational principle with the variational principle in LQ control theory, the simulation relation is established. Based on that relation, the potential energy and mixed energy formulation of the algebraic Riccati equations are derived, then iterative algorithms are proposed which give the upper and lower bounds to the solution matrix. By using the solutions of the positive and negative co-ordinate algebraic Riccati equations, the canonical transformation matrices for the eigenproblems of the substructural chain and LQ control are constructed respectively, which reduce the eigenproblem to half-size. The properties of the solutions are analysed, which establishes the basis for expansion solutions.     

Beam element matrices derived from Vlasov's theory of open thin-walled elastic beams

A uniform beam element of open thin-walled cross-section is studied under stationary harmonic end excitation. An exact dynamic (transcendentally frequency-dependent) 14 × 14 element stiffness matrix is derived from Vlasov's coupled differential equations. Special attention is paid to the computational problems arising when coefficients vanish in these equations because of symmetric cross-section, zero warping stiffness, etc. The dynamic element stiffness matrix is established via a generalized linear eigenvalue problem and a system of linear algebraic equations with complex matrices. A static stiffness matrix is also derived and the associated consistent mass and geometric stiffness matrices are given. Modal masses are evaluated. A FORTRAN program and a numerical example are included.     

Time-dependent linear systems derivable from a variational principle

This paper presents a study of time-varying linear systems of second order ordinary differential equations, which can be derived from a Lagrangian after multiplication by a suitable matrix. It concerns a generalization of previous studies on systems with constant coefficients. After a simplification of the Helmholtz conditions, it is shown that the problem is reduced to a purely algebraic one, provided one can solve a matrix differential equation which produces the transformation to canonical form of the given system. This further leads to a theoretical characterization of all systems admitting a multiplier. Various algebraic relations are derived, involving constant matrices only, which can help to detect, prior to any integration procedure, whether or not: a multiplier exists. They are referred to as the generalized commutativity conditions. The first of these, which is sufficient for the existence of a Lagrangian, is shown to allow also a simple construction of a quadratic first integral, and to have some other interesting features. The paper ends with an example.     

Generalized Bezout Identity

We describe a new approach of the generalized Bezout identity for linear time-varying ordinary differential control systems. We also explain when and how it can be extended to linear partial differential control systems. We show that it only depends on the algebraic nature of the differential module determined by the equations of the system. This formulation shows that the generalized Bezout identity is equivalent to the splitting of an exact differential sequence formed by the control system and its parametrization. This point of view gives a new algebraic and geometric interpretation of the entries of the generalized Bezout identity. Received: June 6, 1997; revised version: December 10, 1997