Abstract
The Vandermonde matrix is ubiquitous in mathematics and engineering. Both the Vandermonde matrix and its inverse are often encountered in control theory, in the derivation of numerical formulas, and in systems theory. In some cases block vandermonde matrices are used. Block Vandermonde matrices, considered in this paper, are constructed from a full set of solvents of a corresponding matrix polynomial. These solvents represent block poles and block zeros of a linear multivariable dynamical time-invariant system described in matrix fractions. Control techniques of such systems deal with the inverse or determinant of block vandermonde matrices. Methods to compute the inverse of a block vandermonde matrix have not been studied but the inversion of block matrices (or partitioned matrices) is very well studied. In this paper, properties of these matrices and iterative algorithms to compute the determinant and the inverse of a block Vandermonde matrix are given. A parallelization of these algorithms is also presented. The proposed algorithms are validated by a comparison based on algorithmic complexity.
Highlights
The Vandermonde matrix is very important and its uses include polynomial interpolation, coding theory, signal processing, etc
Block Vandermonde matrices constructed using matrix polynomials solvents are very useful in control engineering, for example in control of multi-variable dynamic systems described in matrix fractions
We used the ”tic/toc” functions of Matlab to determine the execution time of our algorithms to be compared to the execution time of Matlab functions, and the proposed inverse algorithm is found 10 times quicker, and the determinant algorithm is 14 times quicker
Summary
The Vandermonde matrix is very important and its uses include polynomial interpolation, coding theory, signal processing, etc. In [12], linear diffusion layers achieving maximal branch numbers called MDS (maximal distance separable) matrices are constructed from block Vandermonde matrices and their transposes. Block Vandermonde matrices constructed using matrix polynomials solvents are very useful in control engineering, for example in control of multi-variable dynamic systems described in matrix fractions (see [15]). It is in these particular BVM that we are interested. In this paper a new algorithm to compute the inverse and the determinant of a block Vandermonde matrix constructed from solvents are given.
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