On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis

  • Robert M. Corless
Conference paper

DOI: 10.1007/978-3-7643-7984-1_1

Part of the Trends in Mathematics book series (TM)
Cite this paper as:
Corless R.M. (2007) On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis. In: Wang D., Zhi L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel

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 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Robert M. Corless
    • 1
  1. 1.Ontario Research Centre for Computer Algebra and the Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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