Leverrier's algorithm for orthogonal polynomial bases
type: Item
instance of: scholarly article
title: Leverrier's algorithm for orthogonal polynomial bases
zbMATH Open document ID: 0848.65034
DOI: 10.1016/0024-3795(94)00158-8
author: Stephen Barnett
published in: Linear Algebra and its Applications
publication date: 1996-10-07T00:00:00Z
review text: For a given \(n \times n\) matrix \(A\), the well-known Leverrier-Fadeev algorithm can be used for simultaneous computation of the adjoint \(B(\lambda)\) and determinant \(a(\lambda)\) of the characteristic matrix \(\lambda I - A\). This paper deals with the case when both \(B(\lambda)\) and \(a(\lambda)\) are expressed relative to a basis of orthogonal polynomials. By using the standard three-term recurrence relation of orthogonal polynomials, the Leverrier-Fadeev algorithm is modified and specific formulae are derived for the cases of Chebyshev, Hermite, Legendre, and Laguerre polynomials. These results may be very useful in linear control theory and elsewhere. Obviously, further numerical analysis should be investigated.
Mathematics Subject Classification ID: 65F40
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 871269
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Legendre polynomials
zbMATH Keywords: Leverrier-Fadeev algorithm
zbMATH Keywords: adjoint
zbMATH Keywords: determinant
zbMATH Keywords: basis of orthogonal polynomials
zbMATH Keywords: three-term recurrence relation
zbMATH Keywords: Laguerre polynomials
reviewed by: Liu Xinguo
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00158-8
OpenAlex ID: W1969090657
cites work: Q5558293
cites work: Q3309525
cites work: Division of generalized polynomials using the comrade matrix
cites work: Multiplication of Generalized Polynomials, with Applications to Classical Orthogonal Polynomials
cites work: Change of Basis for Products of Orthogonal Polynomials
cites work: Euclidean remainders for generalized polynomials
cites work: Leverrier’s Algorithm: A New Proof and Extensions
cites work: Q5533821
cites work: THE COLLEAGUE MATRIX, A CHEBYSHEV ANALOGUE OF THE COMPANION MATRIX
cites work: Fast projection methods for minimal design problems in linear system theory
cites work: Further remarks on the Cayley-Hamilton theorem and Leverrier's method for the matrix pencil&lt;tex&gt;(sE - A)&lt;/tex&gt;
cites work: Leverrier's algorithm for singular systems
cites work: Q5839985
description: scientific article; zbMATH DE number 871269