Classical orthogonal polynomials and Leverrier-Faddeev algorithm for the matrix pencilssE−A

Javier Hernández,Francisco Marcellán

doi:10.1155/ijmms/2006/74507

Abstract

In this contribution we present an extension of the Leverrier-Faddeev algorithm for the simultaneous computation of the determinant and the adjoint matrixB(s)of a pencilsE−AwhereEis a singular matrix butdet⁡(sE−A)≢0. Using a previous result by the authors we expressB(s)anddet⁡(sE−A)in terms of classical orthogonal polynomials.

Highlights

IntroductionThe computation of (sE − A)−1 can be carried out by using the Cramer rule, which requires the evaluation of n2 determinants of (n − 1) × (n − 1) polynomial matrices
Time-invariant, multivariable singular system described in the state space as follows: Ex = Ax + Bu, (1.1)
The computation of−1 can be carried out by using the Cramer rule, which requires the evaluation of n2 determinants of (n − 1) × (n − 1) polynomial matrices

Summary

Introduction

The computation of (sE − A)−1 can be carried out by using the Cramer rule, which requires the evaluation of n2 determinants of (n − 1) × (n − 1) polynomial matrices. This is not a practical procedure for large n. We will describe an extension of the classical Leverrier-Faddeev algorithm using families of classical orthogonal polynomials following our previous contribution [2] when instead of a singular matrix E we used In. Here we generalize a recent result [6] based on the Chebyshev polynomials, a very.

Leverrier-Faddeev algorithm and classical orthogonal polynomials

Regular pencils