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Erschienen in:
2014 | OriginalPaper | Buchkapitel
On the Computation of the Determinant of a Generalized Vandermonde Matrix
verfasst von : Takuya Kitamoto
Erschienen in: Computer Algebra in Scientific Computing
Verlag: Springer International Publishing
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“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]).