求置换因子循环矩阵的极小多项式和逆的算法

本文引入了任意域上置换因子循环矩阵,利用多项式环的理想的Gr(?)bner基的算法给出了任意域上置换因子循环矩阵的极小多项式和公共极小多项式的算法,同时给出了这类矩阵逆矩阵的两种算法最后,利用Schur补给出了任意域上具有置换因子循环矩阵块的分块矩阵逆的一个算法,在有理数域或模素数剩余类域上,这一算法可由代数系统软件CoCoA4．0实现。     

求鳞状因子循环矩阵的逆阵及广义逆阵的快速算法

利用多项式快速算法，给出了求鳞状因子循环矩阵的逆阵、自反g-逆、群逆及Moore-Penrose逆的快速算法。该算法避免了一般快速算法中，要计算大量的三角函数等可能带来误差及影响效率的问题。该算法仅用到鳞状因子循环矩阵的第一行元素及对角阵D中的常数d1，d2，…，dn进行计算，在计算机上实现时只有舍入误差。特别地，在有理数域上用计算机求得的结果是精确的。     

实方阵的Moore-Penrose广义逆的MCG算法探究

证明了矩阵Moore-Penrose逆的唯一性以及建立了求矩阵Moore-Penrose逆的算法。首先将求矩阵的Moore-Penrose逆转为求解含有三个矩阵变量的矩阵方程组，其次建立求该矩阵方程组的修正共轭梯度算法(MCG算法)，给出了MCG算法的性质和收敛性证明，对于任意给定的初始矩阵该算法能在有限步迭代计算后得到矩阵的Moore-Penrose逆。最后给出数值算例，证明MCG算法在求解矩阵Moore-Penrose逆中具有很高的计算效率。     

g-r循环矩阵求逆的快速算法

利用FFT,给出g-r循环矩阵求逆的快速算法,计算复杂性为O(n log2 n) (g 2)n。     

部分逆M矩阵2-弦图的完备问题

本文采用图论的方法对任意阶部分逆M矩阵,当其对应的图为2-弦图时,研究了其逆M矩阵的完备问题。给出了完备定理以及具体完备的算法。     

Hankel矩阵及其逆矩阵的快速三角分解算法的改进

本文改进了求Hankel矩阵及其逆矩阵三角分解的Chun-Kailath快速算法,减少了该算法的计算量,提高了精度。     

鳞状因子循环矩阵方程解的条件与求解的快速算法

利用多项式快速算法,给出了鳞状因子循环矩阵方程AX=b可解的条件与求解的快速算法.当鳞状因子循环矩阵非奇异时,该快速算法求出线性方程组的唯一解;当鳞状因子循环矩阵奇异时,该快速算法求出线性方程组的特解与通解.该快速算法仅用到鳞状因子循环矩阵的第一行元素及对角矩阵中的对角上的常数进行计算.在计算机上实现时只有舍入误差.特别地,在有理数域上用计算机求得的结果是精确的.     

双回路图的部分逆M矩阵完备

逆M矩阵是一类非常重要的非负矩阵,在生物学、物理学等很多领域中都得到了广泛的应用．利用图论理论研究逆M矩阵的完备问题是逆M矩阵研究领域的一个重要方向．双回路图是由两条简单有向回路在任意多个顶点处相交所构建的有向图．本文对此类图形的逆M矩阵完备问题进行了研究,给出了此类图形所对应的部分矩阵在两种不同情况下具有逆M矩阵完备的充分必要条件：当双回路图中的顶点均已知时,每一条回路的回路积均小于其对角元素的乘积;当双回路图中可包含未知顶点时,每一条回路至少包含一个未知顶点．同时,本文给出了具体的完备算法,并通过数值算例验证了算法的有效性．     

厄米特矩阵一种求逆算法的程序说明和性态分析

本文是前文的继续,就厄米特矩阵一种新的求逆算法在计算机上的实现进行了分析,着重就程序作了说明,对本算法的速度和精度作了讨论,同时指出,本算法的前半部分也是求厄米特矩阵特征根的有效方法.     

范德蒙类矩阵之逆阵的一种三角分解

给出了范德蒙类矩阵之逆矩阵的一种三角分解算法，所需计算量为O(n^2).     

Inverse diffraction problems for buried objects in the layered elastic media: the anti-plane problem

The authors study direct and inverse scattering problems for buried objects in the stratified layered elastic media. It is assumed that a source load is applied on the upper boundary surface, to generate a certain wave field inside the medium. Then we build the solutions to the direct problems through specially constructed Green’s functions for every considered case. Using the known measured amplitude of the upper boundary oscillations over a certain finite-length interval, the inverse problems in a one-mode frequency range are solved. Some numerical examples demonstrate the stability of the proposed algorithm for the formulated inverse problems.     

两种解振动反问题算法的同一性及应用

根据矩阵最佳逼近与加权残值理论，把求解振动反问题时所使用的矩阵逼近法和极值化算法统一为不同范数定义下的最小二乘问题，这对部分振频和／或振型给定情况下振动反问题的求解提供了一个有效工具。结合某大型飞机机翼颤振吹风模型的动力设计，本文给出了工程应用的具体数值例子。大量的算例结果表明：与矩阵逼近法具有同一性的极值化方法计算精度和计算效率很高，可直接用于工程实际。     

Inverse mass matrix via the method of localized lagrange multipliers

An efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces and constructs the free‐free inverse mass matrix using the standard FEM procedure. The various boundary conditions are realized by the the method of localized Lagrange multipliers. In particular, the present paper constructs the kernel inverse matrix by using the standard FEM elemental mass matrices. It is shown that the accuracy of the present inverse mass matrix is almost identical to that of a conventional consistent mass matrix or a combination of lumped and consistent mass matrices. Numerical experiments with the proposed inverse mass matrix are conducted to validate its effectiveness when applied to vibration analysis of bars, beams, and plain stress problems.     

对称矩阵反问题的总体最小二乘解

最小二乘法是近年来求解对称矩阵反问题的一种常用方法,但因系数矩阵常常存在误差,方法本身具有很大的局限性.鉴于此,本文提出并讨论了对称矩阵反问题的总体最小二乘解,给出了解的一般表达式;证明了最佳逼近问题解的存在唯一性,给出了其具体表达式及数值算法,并将数值结果应用于求解对称矩阵反问题.     

An Application of DHW Algorithm for the Solution of Inverse Heat Conduction Problem

A damped heat wave (DHW) algorithm is applied for the temperature distribution calculation in a solution of a linear inverse heat conduction problem (IHCP). A nonlinear least squares algorithm is used for calculation of the unknown boundary heat flux history in a one-dimensional medium. The solution is based on the assumption that the temperature measurements are available, at least, at one point of the medium over the whole time domain. Sample calculations, for a comparison between exact heat sources and estimated ones, are made to confirm the validity of the proposed method. The close agreement between the exact and estimated values calculated for both exact and noisy data shows the potential of the proposed method for finding a relatively accurate heat source distribution in a one-dimensional homogeneous finite medium. The proposed method of solving inverse heat conduction problems is very simple and easy to implement.Paper presented at the Seventeenth European Conference on Thermophysical Properties, September 5–8, 2005, Bratislava, Slovak Republic.M. L?ffler: Deceased