Fast and backward stable computation of eigenvalues and eigenvectors of matrix polynomials

Jared Aurentz, Thomas Mach, Leonardo Robol, Raf Vandebril, David S. Watkins

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost. The algorithm is a variant of Francis's implicitly shifted QR algorithm applied on the companion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of k matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stability after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well. Numerical experiments illustrate stability and efficiency of the proposed methods.

Original languageEnglish
Pages (from-to)313-347
Number of pages35
JournalMathematics of Computation
Issue number315
Publication statusPublished - Jan 1 2018


  • Core chasing algorithm
  • Eigenvalues
  • Eigenvectors
  • Matrix polynomial
  • Product eigenvalue problem

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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