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Implicit QR algorithms for palindromic and even eigenvalue problems

by D. Kressner and Ch. Schroeder and D. S. D.S. Watkins

(Report number 2008-18)

Abstract
In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques.

Keywords: palindromic eigenvalue problem, implicit QR algorithm, bulge chasing, bulge exchange, (even eigenvalue problem, convergence theory)

@Techreport{KSD08_382,
  author = {D. Kressner and Ch. Schroeder and D. S. D.S. Watkins},
  title = {Implicit QR algorithms for palindromic and even eigenvalue problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-18},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-18.pdf },
  year = {2008}
}

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