"Vandermonde" matrix is a matrix whose (i,j)th entry is in the form of x_i~j. 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(x_i) where f_j(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 f_j (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 f_j (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 H_2 norm of a system achievable by a static feedback controller (for details, see [18],[19]).
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