Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the late...

This work examines various statistical distributions in connection with random Vandermonde matrices and their extension to d-dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be O(logN) and O(logN/ log logN) respectively where N is the dimension of the matrix, generalizing the results in [21]. We further study the behavior of the mini...

In this paper we extend the concept of ∆-confluence to ∆k-confluence by allowing more generalized junctions, called ∆k-junctions. We present an algorithm for recognizing graphs that are ∆k-confluent. We then generalize ∆k-confluence to Ok-confluence by allowing non-intersecting chords within a junction, resulting in Ok-junctions. We present an algorithm for recognizing graphs that are Ok-conflu...

Suppose that K ⊂ IR is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c1, c2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, Fn ⊂ K, c1 n ≤ min b∈Fn b 6=a dist(a, b) ≤ c2 n , ∀a ∈ Fn where dist(a, b) is a natural distance on K that will be described in the text. §

We present a combinatorial method for solving a certain system of polynomial equations of Vandermonde type in 2N variables by reducing it to the problem of solving two special linear systems of size N and rooting a single univariate polynomial of degree N . Over C, all solutions can be found with fixed precision using, up to polylogarithmic factors, O(N2) bitwise operations in the worst case. F...

We analyse a generalisation of the Quicksort algorithm, where k uniformly at random chosen pivots are used for partitioning an array of n distinct keys. Specifically, the expected cost of this scheme is obtained, under the assumption of linearity of the cost needed for the partition process. The integration constants of the expected cost are computed using Vandermonde matrices.

We consider a special class of the generalized Vandermonde matrices and obtain an LU factorization for its member by giving closed-form formulae of the entries of L and U . Moreover, we express the matrices L and U as products of 1-banded (bidiagonal) matrices. Our result is applied to give the closed-form formula of the inverse of the considered matrix.

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.