On the computation of the determinant of a generalized Vandermonde matrix

Abstract

"Vandermonde" matrix is a matrix whose (i,j)th entry is in the form of xij. The matrix has a lot of applications in many fields such as signal processing and polynomial interpolations. This paper generalizes the matrix, and let its (i,j) entry be f j (xi ) where fj (x) is a polynomial of x. We present an efficient algorithm to compute the determinant of the generalized Vandermonde matrix. The algorithm is composed of two sub-algorithms: the one that depends on given polynomials fj (x) and the one that does not. The latter algorithm (the one does not depend on f j (x)) can be performed beforehand, and the former (the one that depends on fj (x)) is mainly composed of the computation of determinants of numerical matrices. Determinants of the generalized Vandermonde matrices can be used, for example, to compute the optimal H∞ and H2 norm of a system achievable by a static feedback controller (for details, see [18],[19]). © 2014 Springer International Publishing.