范德蒙德行列式与多项式的不动点

以范德蒙德行列式为工具,得到关于多项式不动点的两个新结论,以及线性代数中某已知结论的一个新证明,另外提供实例说明多项式不动点的应用．     

一般广义Vandermonde行列式的直接计算公式

Vandermonde行列式构造独特,是高等代数中一个典型的行列式,在数值计算,数值逼近等领域有着广泛的应用.通过对已得到的几类广义Vandermonde行列式的结果进行变形分析,推广得到了最一般的广义Vandermonde行列式的直接计算公式.     

构造辅助函数计算准Vandermonde行列式

通过构造线性方程组和一元高次方程,利用线性方程组的解与一元高次方程根与系数的关系推导出第一类准Vandermonde行列式的值.通过构造辅助函数计算一个特殊的第一类准Vandermonde行列式,并把这种方法推广于两类特殊第二类准Vandermonde行列式的计算.     

On vandermonde systems solution in the symmetric case

Vandermonde matrices defined by pairs of symmetric points with respect to the origin are analysed. Direct methods for the determinant computation and for the linear system solution, both for primal and dual case, are proposed that reduce the computational complexity by a factor four and two respectively with respect to the known algorithms of ordern ². Moreover perturbation bounds for the new LUD factorization error are found depending on a condition number of an halved dimension Vandermonde matrix.

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Sunto Si considerano matrici di Vandermonde definite da coppie di punti simmetrici rispetto all'origine. Sono proposti metodi diretti per il calcolo del determinante e per la soluzione dei sistemi lineari, sia nel caso primario che duale, che riducono rispettivamente di un fattore quattro e due la complessità computazionale degli algoritmi noti di ordinen ². Inoltre si determinano limitazioni dell'errore della nuova fattorizzazione LUD dipendenti dall'indice di condizionamento di una matrice di Vandermonde di dimensioni dimezzate.

     

一类广义Vandermonde矩阵的可逆条件及求逆公式

利用广义Vandermonde行列式的显式表示式,给出了广义Vandermonde矩阵可逆的充要条件及求逆公式.     

Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants

The concepts of Vandermonde determinant and confluent Vandermonde determinant are extended to the multidimensional setting by relating them to multivariate interpolation problems. With an approach different from that of other recent papers on this subject, the values of these determinants are computed, recovering and extending the results of those papers.Partially supported by Research Grant PS900121 DGICYT.     

On some generalizations of the Vandermonde matrix and their relations with the Euler beta-function

A multiple Vandermonde matrix which, besides the powers of variable, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. for the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler betafunction.     

Computing Fekete and Lebesgue points: Simplex, square, disk

We have computed point sets with maximal absolute value of the Vandermonde determinant (Fekete points) or minimal Lebesgue constant (Lebesgue points) on three basic bidimensional compact sets: the simplex, the square, and the disk. Using routines of the Matlab Optimization Toolbox, we have obtained some of the best bivariate interpolation sets known so far.     

广义Vandermonde行列式的计算公式

给出两类广义Vandermonde行列式的计算公式以及各类广义Vandermonde行列式的计算方法.     

Divided differences and a unified approach to evaluating determinants involving generalized factorials

By employing divided differences, a unified approach to the evaluation of some determinant involving generalized factorials is proposed. Previous generalizations of the Vandermonde determinant and the Cauchy determinant due to Chu-Claudio, Chu-Wang-Zhang and Johnson are included as special cases of our unified treatment.     

一元切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,分别针对iα=2,iα=3(i=1,2,…,s)的情形给出切触有理插值问题有解的条件及解的表达式.     

范德蒙行列式的新证明及其应用

首先给出范德蒙行列式的一个新的证明,与以前的证明相比,本文给出的证明更具有通用性．其次介绍范德蒙行列式在n阶行列式计算和在向量组线性相关性证明中的两个应用．     

增次广义Vandermonde矩阵

定义了增次广义Vandermonde矩阵,并利用反证法或行列式推得它们的秩和某些逆.     

两类Stirling数的行列式表示

通过计算两个广义的范德蒙(Vandermonde)行列式,得到了第一类无符号Stirling数和第二类Stirling数的一种新的表示方法:用行列式来表示.     

An application of the Vandermonde determinant

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this method is much simpler.     

Cayley-Hamilton定理的一个证明

从Vandermonde行列式出发,给出了Cayley-Hamilton定理的一个新的证明,也给出了有限维向量空间的一些结论的新的证明.     

A note on the determinant of a functional confluent vandermonde matrix and controllability

A factorization expression for the determinant of the confluent Vandermonde matrix is extended to a functional form. With the exponential function, simpler necessary and sufficient conditions for controllability of sampled input systems follow.     

一类无穷级数的求和

应用微分方程的理论和线性代数中的克莱姆法则,范德蒙行列式,得到一类无穷级数∑x<'mn>/(mn)!的求和公式,其中m是任意给定的正整数.     

Determinants of Random Matrices and Jack Polynomials of Rectangular Shape

We consider an N -dimensional real integral, indexed by a parameter that specifies the power of a Vandermonde determinant. For two particular values of the parameter, this integral arises from matrix integrals, over real symmetric and complex Hermitian   N × N   matrices. When it is normalized, it gives the expectation of an arbitrary power of the determinant. The results are given as finite summations, using terminating hypergeometric series. We relate the integral to a specific coefficient in the Jack polynomial indexed by a partition of rectangular shape, and present data for this coefficient in terms of the parameter α.     

Binomial Determinant Evaluations

By means of matrix decomposition and the partial fraction method, we establish several determinant evaluation formulas, which can be considered as generalizations of the Vandermonde and Cauchy determinants. Received February 24, 2005