A Fast QR Algorithm for Companion Matrices

  • Shiv Chandrasekaran
  • Ming Gu
  • Jianlin Xia
  • Jiang Zhu
Part of the Operator Theory: Advances and Applications book series (OT, volume 179)


It has been shown in [4, 5, 6, 31] that the Hessenberg iterates of a companion matrix under the QR iterations have low off-diagonal rank structures. Such invariant rank structures were exploited therein to design fast QR iteration algorithms for finding eigenvalues of companion matrices. These algorithms require only O(n) storage and run in O(n2) time where n is the dimensiosn of the matrix. In this paper, we propose a new O(n2) complexity QR algorithm for real companion matrices by representing the matrices in the iterations in their sequentially semi-separable (SSS) forms [9, 10]. The bulge chasing is done on the SSS form QR factors of the Hessenberg iterates. Both double shift and single shift versions are provided. Deflation and balancing are also discussed. Numerical results are presented to illustrate both high efficiency and numerical robustness of the new QR algorithm.


Companion matrices sequentially semi-separable matrices structured QR iterations structured bulge chasing Givens rotation swaps 

Mathematics Subject Classification (2000)

65F15 65H17 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Shiv Chandrasekaran
    • 1
  • Ming Gu
    • 2
  • Jianlin Xia
    • 3
  • Jiang Zhu
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of California at Santa BarbaraUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyUSA
  3. 3.Department of MathematicsUniversity of California at Los AngelesUSA

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