关于几类矩阵的注记

特殊矩阵在矩阵分析里起着核心的作用.运用Cramer法则和Lagrange插值公式,处理循环矩阵,Vandermonde矩阵,Hilbert矩阵,Cauchy矩阵的一些基本问题:给出Ramakrishnan的矩阵分解定理的一种推广,计算Vandermonde矩阵,Hilbert矩阵,Cauchy矩阵的行列式,当它们可逆...     

跳行范德蒙矩阵的三角分解

跳行范德蒙矩阵是一种重要的矩阵,在函数插值等方面有着重要的应用.根据跳行范德蒙矩阵的特殊结构,将跳行范德蒙矩阵分解为一系列下三角矩阵与一系列上三角矩阵的乘积.进一步给出了其逆矩阵分解为一系列上三角矩阵与一系列下三角矩阵的乘积的表达式.     

Cramer法则在数值计算中的若干应用

用Cramer法则给出了Lagrange插值公式和Newton插值公式的简洁证明,同时得到了Vandermonde矩阵的逆矩阵的LU分解.     

n维特殊伸缩矩阵的构造与n维广义插值细分函数向量

本文构造了一种特殊的n维特殊伸缩矩阵,且定义了n维正交广义插值多小波.基于这种特殊的伸缩矩阵,讨论n维正交广义插值多小波的构造算法. 并且最后给出了算例.     

二元矩阵有理插值函数的构造

一般构造矩阵值有理函数的方法是利用连分式给出的,其算法的可行性不易预知,且计算量大.本文对于二元矩阵值有理插值的计算,通过引入多个参数,定义一对二元多项式:代数多项式和矩阵多项式,利用两多项式相等的充分必要条件通过求解线性方程组确定参数,并由此给出了矩阵值有理插值公式.该公式简单,具有广阔的应用前景.     

矩形网格上一类二元有理插值问题

本文首先利用Ｖａｎｄｅｒｍｏｎｄｅ矩阵得到矩形网格上二元多项式插值公式,然后利用该公式建立一类二元有理插值问题的存在性判别准则及有理插值函数的表现公式,并给出数值例子     

三对角矩阵逆的估计

1引言 三对角矩阵出现在很多应用中,例如,在求解常系数微分方程的比值问题,三次样条插值等应用中都会遇到三对角矩阵.因此这类矩阵非常重要,而且也有很多学者致力于这类矩阵的研究.在一些应用中,比如估计条件数和构造稀疏近似逆预条件子,需要计算三对角矩阵的逆,或者估计其逆元素的界.文献[1-7]给出了关于三对角矩阵逆的一些很好的结果,但是,这些结果大都建立在矩阵对角占优的条件之下,这限制了他们的应用.在本文中,我们给出一种一般三对角矩阵逆元素的估计办法.     

带重点的广义Cauchy矩阵及其相关问题

本文详细讨论了带重点的广义Cauchy矩阵的位移结构、快速求逆公式、可逆性判别条件及其与矩阵有理切插值问题的关系,并给出了求解此类矩阵的线性方程组的快速算法,推广了文[1]和[2]的结果.     

GF(Pn)上的插值理论及其在信息隐藏中的应用

给出了信息隐藏的一种数学描述,得到了GF(Pn)上的Newton插值公式,讨论了Newton插值公式和范德蒙逆矩阵在信息隐藏中的应用,给出了算法和算例.     

矩形网格上一类二元有理插值问题

本首先利用Vandermonde矩阵得到矩形网格上二元多项式插值公式．然后利用该公式建立一类二元有理插值问题的存在性判别准则及有理插值函数的表现公式,并给出数值例于。     

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Matrix Szeg? biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szeg? polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szeg? matrix and the associated Szeg? recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szeg? polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.     

The precise consistency consensus matrix in a local AHP-group decision making context

In matrix theory, Fu and Markham showed using majorization technique that if a Hermitian matrix satisfies certain conditions, then the matrix must be block-diagonal. In this paper, we extend this result to the setting of simple Euclidean Jordan algebras by using the Cauchy interlacing theorem and the Schur complement Cauchy interlacing theorem.     

A Lanczos‐type algorithm for the QR factorization of regular Cauchy matrices

We present a fast algorithm for computing the QR factorization of Cauchy matrices with real nodes. The algorithm works for almost any input matrix, does not require squaring the matrix, and fully exploits the displacement structure of Cauchy matrices. We prove that, if the determinant of a certain semiseparable matrix is non‐zero, a three term recurrence relation among the rows or columns of the factors exists. Copyright © 2002 John Wiley & Sons, Ltd.     

A Quantile Goodness-of-Fit Test for Cauchy Distribution, Based on Extreme Order Statistics

A test statistic for testing goodness-of-fit of the Cauchy distribution is presented. It is a quadratic form of the first and of the last order statistic and its matrix is the inverse of the asymptotic covariance matrix of the quantile difference statistic. The distribution of the presented test statistic does not depend on the parameter of the sampled Cauchy distribution. The paper contains critical constants for this test statistic, obtained from 50,000 simulations for each sample size considered. Simulations show that the presented test statistic is for testing goodness-of-fit of the Cauchy distributions more powerful than the Anderson-Darling, Kolmogorov-Smirnov or the von Mises test statistic.     

Factorizations of Cauchy matrices

The development of fast algorithms for the solution of linear systems of equations with a Cauchy matrix has recently received considerable attention. Several of these algorithms factor a Cauchy matrix or its inverse into triangular and possibly diagonal matrices. The numerical properties of the factorization methods depend on the selection of pivots. This note presents elementary derivations of some factorization methods and describes a new strategy for searching both rows and columns for suitable pivots.     

Displacement structure approach to Cauchy and Cauchy–Vandermonde matrices: inversion formulas and fast algorithms

The confluent Cauchy and Cauchy–Vandermonde matrices are considered, which were studied earlier by various authors in different ways. In this paper, we use another way called displacement structure approach to deal with matrices of this kind. We show that the Cauchy and Cauchy–Vandermonde matrices satisfy some special type of matrix equations. This leads quite naturally to the inversion formulas and fast algorithms for matrices of this kind.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Matrix versions of the Cauchy and Kantorovich inequalities

Summary A version of Cauchy's inequality is obtained which relates two matrices by an inequality in the sense of the Loewner ordering. In that ordering a symmetric idempotent matrix is dominated by the identity matrix and this fact yields a simple proof.A consequence of this matrix Cauchy inequality leads to a matrix version of the Kantorovich inequality, again in the sense of Loewner.     

线性泛函微分方程关于部分变元稳定性的充要条件

本文在引进具有界滞量线性泛函微分方程的Cauchy矩阵的截断矩阵与截断解算子概念的基础上 ,讨论了这类方程关于部分变元各种稳定性的充要条件 .本文包含此类方程相应的关于全体变元稳定性的经典结论为其特例 .     

Discrete second-order Ablowitz–Kaup–Newell–Segur equation and its modified form

Theoretical and Mathematical Physics - By introducing shift relations satisfied by a matrix $$\boldsymbol{r}$$ , we propose a generalized Cauchy matrix scheme and construct a discrete second-order...