Abstract
“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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Kitamoto, T. (2014). On the Computation of the Determinant of a Generalized Vandermonde Matrix. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_18
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DOI: https://doi.org/10.1007/978-3-319-10515-4_18
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