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...

Abstract: The inverse of a Vandermonde matrix has been used for signal processing, polynomial interpolation, curve fitting, wireless communication, and system identification. In this paper, we propose a novel fast recursive algorithm to compute the inverse of a Vandermonde matrix. The algorithm computes the inverse of a higher order Vandermonde matrix using the available lower order inverse mat...

In this paper we rst show that a con uent Vandermonde matrix may be viewed as composed of some rows of a certain block Vandermonde matrix As a result we derive a Sylvester s structure for this class of matrices that ap pears as a natural generalization of the straightforward one known for usual Vandermonde matrices Then we present some applications as an illustration of the established structur...

In this first part, analytical methods for finding moments of random Vandermonde matrices are developed. Vandermonde Matrices play an important role in signal processing and communication applications such as direction of arrival estimation, precoding or sparse sampling theory for example. Within this framework, we extend classical freeness results on random matrices with i.i.d entries and show...

The Vandermonde and con uent Vandermonde matrices are of fundamental signi cance in matrix theory. A further generalization of the Vandermonde matrix called the q-adic coe cient matrix was introduced in [V. Mani and R. E. Hartwig, Lin. Algebra Appl., to appear]. It was demonstrated there that the q-adic coe cient matrix reduces the Bezout matrix of two polynomials by congruence. This extended t...

Analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle are developed. Vandermonde Matrices play an important role in signal processing and wireless applications such as direction of arrival estimation, precoding, and sparse sampling theory, just to name a few. Within this framework, we extend classical freeness results on random matrices with indep...

The study focuses primarily on Vandermonde-like matrix systems. The idea is to express Vandermonde and Vandermonde-like matrix systems as the problems related to Krylov Matrices. The connection provides a different angle to view the Vandermondelike systems. Krylov subspace methods are strongly related to polynomial spaces, hence a nice connection can be established using LU factorization as pro...

Lower bounds on the condition number minκp(V ) of a Vandermonde matrix V are established in terms of the dimension n or n and the largest absolute value among all nodes that define the Vandermonde matrix. Many bounds here are asymptotically sharp, and compare favorably to those of the same kind due to Gautschi and Inglese (Numer. Math., 52 (1988), 241–250) who considered either all positive nod...

Multipoint polynomial evaluation and interpolation are fundamental for modern algebraic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grew to quadratic for numerical solution. We fix this discrepancy: our new numerical algorithms run in nearly linear arithmetic time. At first we restate our goals as t...