Mathematics in Computer Science

, Volume 1, Issue 2, pp 353–374 | Cite as

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

  • Robert M. Corless
  • Nargol Rezvani
  • Amirhossein Amiraslani
Article

Abstract.

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied.

Mathematics Subject Classification (2000).

Primary 15A42 Secondary 41A05 

Keywords.

Pseudospectra matrix polynomial conditioning Lebesgue functions Lebesgue constants 

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Copyright information

© Springer 2007

Authors and Affiliations

  • Robert M. Corless
    • 1
  • Nargol Rezvani
    • 2
  • Amirhossein Amiraslani
    • 3
  1. 1.Department Applied MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Department Computer ScienceUniversity of TorontoTorontoCanada
  3. 3.Department Mathematics & StatisticsUniversity of CalgaryCalgaryCanada

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