Part of the series Trends in Mathematics pp 115
On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis
 Robert M. CorlessAffiliated withOntario Research Centre for Computer Algebra and the Department of Applied Mathematics, University of Western Ontario
Abstract
Experimental observations of univariate rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable. It has recently been proved that a new rootfinding condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial (which is itself optimally small in a certain sense); and computation shows that sometimes it can be much smaller. These results extend to the matrix polynomial case, although in this case we are not computing polynomial roots but rather ‘polynomial eigenvalues’ (sometimes known as ‘latent roots’), i.e. finding the values of x where the matrix polynomial is singular. This paper gives two theorems explaining part of the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.
Keywords
Lagrange basis companion matrix polynomial eigenvalue Title
 On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis
 Book Title
 SymbolicNumeric Computation
 Pages
 pp 115
 Copyright
 2007
 DOI
 10.1007/9783764379841_1
 Print ISBN
 9783764379834
 Online ISBN
 9783764379841
 Series Title
 Trends in Mathematics
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Birkhäuser Verlag Basel/Switzerland
 Additional Links
 Topics
 Keywords

 Lagrange basis
 companion matrix
 polynomial eigenvalue
 Industry Sectors
 eBook Packages
 Editors

 Dongming Wang ^{(1)} ^{(2)}
 Lihong Zhi ^{(3)}
 Editor Affiliations

 1. School of Science, Beihang University
 2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie  CNRS
 3. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences
 Authors

 Robert M. Corless ^{(4)}
 Author Affiliations

 4. Ontario Research Centre for Computer Algebra and the Department of Applied Mathematics, University of Western Ontario, 1151 Richmond St., London, Canada
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