In this paper, the author gives a simpler alternative approach to the LU factorization and 1-banded factorization of the Vandermondematrix, and obtains explicit formulas of the triangular factors and 1-bandedmatrices by using symmetric functions. © 2004 Elsevier B.V. All rights reserved.

Pλ(y; θ), (1) where δ := (n− 1, n− 2, . . . , 1, 0) and λ are partitions, aδ = ∏ 1≤i<j≤n(yi − yj) is the Vandermonde determinant and u is a free parameter. Under a general result, S. Sahi proves [8, Theorem 5.2] the existence of a unique polynomial P ∗ μ(y; θ), now known as shifted Jack polynomials, satisfying a certain vanishing condition. In the special case θ = 1, Okounkov and Olshanski [6,7...

In this paper we compare the numerical properties of the well-known fast O(n 2) Traub and Bjj orck-Pereyra algorithms, which both use the special structure of a Vandermonde matrix to rapidly compute the entries of its inverse. The results of numerical experiments suggest that the Parker variant of what we shall call the Parker-Traub algorithm allows one not only fast O(n 2) inversion of a Vande...

In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensorsw...

According to the assumption that si = sj for i = j, the matrix (5) has an inverse and the system (2) and (4) can be solved. We denote by V the Vandermonde determinant of the matrix (5) and by Vk the Vandermonde determinant of order (n − 1) of the variables s1, . . . , sk−1, sk+1, . . . , sn. We also denote by Φ r the fundamental symmetric function of the r-th order of (n − 1) variables s1, . . ...

The Gaussian matrix model is known to deform the $q,t$-matrix model. We consider further deformation elliptic $q,t$ by properly deforming density as well Vandermonde factor. Properties of an associated basis symmetric functions that provide property $\sim {\rm char}$ in deformed case are discussed.

Multipoint polynomial evaluation and interpolation are fundamental for modern algebraic and numerical computing. The known algorithms solve both problems over any field by using O(N log N) arithmetic operations for the input of size N , but the cost grows to quadratic for numerical solution. Our study results in numerically stable algorithms that use O(uN log N) arithmetic time for approximate ...