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A Fast QR Algorithm for Companion Matrices

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Book cover Recent Advances in Matrix and Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 179))

Abstract

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.

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References

  1. Mario Ahues and Francoise Tisseur, A new deflation criterion for the QR algorithm, Technical Report CRPC-TR97713-S, Center for Research on Parallel Computation, January 1997.

    Google Scholar 

  2. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users’ Guide, Release 2.0, SIAM, Philadelphia, PA, USA, second edition, 1995.

    Google Scholar 

  3. D. Bailey, Software: MPFUN90 (Fortran-90 arbitrary precision package), available online at http://crd.lbl.gov/~dhbailey/mpdist/index.html

    Google Scholar 

  4. D. Bindel, S. Chandresekaran, J. Demmel, D. Garmire, and M. Gu, A fast and stable nonsymmetric eigensolver for certain structured matrices, Technical report, University of California, Berkeley, CA, 2005.

    Google Scholar 

  5. D. A. Bini, F. Daddi, and L. Gemignani, On the shifted QR iteration applied to companion matrices, Electronic Transactions on Numerical Analysis 18 (2004), 137–152.

    MATH  MathSciNet  Google Scholar 

  6. D. A. Bini, Y. Eidelman, L. Gemignani and I. Gohberg, Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices, Technical Report no.1587, Department of Mathematics, University of Pisa, 2005.

    Google Scholar 

  7. S. Chandrasekaran and M. Gu, Fast and stable algorithms for banded plus semiseparable matrices, SIAM J. Matrix Anal. Appl. 25 no. 2 (2003), 373–384.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Chandrasekaran, M. Gu, and W. Lyons, A fast and stable adaptive solver for hierarchically semi-separable representations, Technical Report UCSB Math 2004–20, U.C. Santa Barbara, 2004.

    Google Scholar 

  9. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, X. Sun, A.-J. van der Veen, and D. White, Fast stable solvers for sequentially semi-separable linear systems of equations and least squares problems, Technical report, University of California, Berkeley, CA, 2003.

    Google Scholar 

  10. S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, X. Sun, A.-J. van der Veen, and D. White, Some fast algorithms for sequentially semiseparable representations, SIAM J. Matrix Anal. Appl 27 (2005), 341–364.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Chandrasekaran, M. Gu, X. Sun, J. Xia, J. Zhu, A superfast algorithm for Toeplitz systems of linear equations, SIAM J. Mat. Anal. Appl., to appear.

    Google Scholar 

  12. T.-Y. Chen and J.W. Demmel, Balancing sparse matrices for computing eigenvalues, Lin. Alg. and Appl. 309 (2000), 261–287.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Van Dooren and P. Dewilde, The eigenstructure of an aribtrary polynomial matrix: Computational aspects, Lin. Alg. and Appl. 50 (1983), 545–579.

    Article  MATH  Google Scholar 

  14. Y. Eidelman and I. Gohberg, On a new class of structured matrices, Integral Equations Operator Theory 34 (1999), 293–324.

    Article  MATH  MathSciNet  Google Scholar 

  15. Y. Eidelman, I. Gohberg and V. Olshevsky, The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order, Lin. Alg. and Appl. 404 (2005), 305–324.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Edelman and H. Murakami, Polynomial roots from companion matrix eigenvalues, Mathematics of Computation 64 (1995), 763–776.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Golub and C. V. Loan, Matrix Computations, The John Hopkins University Press, 1989.

    Google Scholar 

  18. J. G. F. Francis, The QR transformation. II, Comput. J. 4 (1961/1962), 332–345.

    Article  MathSciNet  Google Scholar 

  19. V. N. Kublanovskaya, On some algorithms for the solution of the complete eigenvalue problem, U.S.S.R. Comput. Math. and Math. Phys. 3 (1961), 637–657.

    Google Scholar 

  20. C. Moler, Roots — of polynomials, that is, The Mathworks Newsletter 5 (1991), 8–9.

    Google Scholar 

  21. V. Pan, On computations with dense structured matrices, Math. Comp. 55 (1990), 179–190.

    Article  MATH  MathSciNet  Google Scholar 

  22. B. Parlett, The symmetric eigenvalue problems, SIAM, 1997.

    Google Scholar 

  23. B. Parlett, The QR algorithm, Computing in Science and Engineering 2 (2000), 38–42. Special Issue: Top 10 Algorithms of the Century.

    Google Scholar 

  24. G. Sitton, C. Burrus, J. Fox, and S. Treitel, Factoring very-high-degree polynomials. IEEE Signal Processing Mag. 20 no. 6 (2003), 27–42.

    Article  Google Scholar 

  25. M. Stewart, An error analysis of a unitary Hessenberg QR algorithm, Tech. Rep. TR-CS-98-11, Department of Computer Science, Australian National University, Canberra 0200 ACT, Australia, 1998.

    Google Scholar 

  26. F. Tisseur, Backward stability of the QR algorithm, TR 239, UMR 5585 Lyon Saint-Etienne, October 1996.

    Google Scholar 

  27. K.-C. Toh and L. N. Trefethen. Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math. 68 (1994), 403–425.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Van Barel and A. Bultheel, Discrete Linearized Least Squares Rational Approximation on the Unit Circle, J. Comput. Appl. Math. 50 (1994), 545–563.

    Article  MATH  MathSciNet  Google Scholar 

  29. J. H. Wilkinson, The algebraic eigenvalue problem, Oxford University Press, London, 1965.

    MATH  Google Scholar 

  30. J. Xia, Fast Direct Solvers for Structured Linear Systems of Equations, Ph.D. Thesis, University of California, Berkeley, 2006.

    Google Scholar 

  31. J. Zhu, Structured Eigenvalue Problems and Quadratic Eigenvalue Problems, Ph.D. Thesis, University of California, Berkeley, 2005.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of California at Santa Barbara, USA

    Shiv Chandrasekaran

  2. Department of Mathematics, University of California at Berkeley, USA

    Ming Gu & Jiang Zhu

  3. Department of Mathematics, University of California at Los Angeles, USA

    Jianlin Xia

Authors
  1. Shiv Chandrasekaran
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  2. Ming Gu
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  3. Jianlin Xia
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  4. Jiang Zhu
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Joseph A. Ball

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Yuli Eidelman

  3. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, California, 92093-0112, USA

    J. William Helton

  4. Department of Mathematics, University of Connecticut, 196 Auditorium Road, U-9, Storrs, CT, 06269, USA

    Vadim Olshevsky

  5. Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA, 22904-4137, USA

    James Rovnyak

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Chandrasekaran, S., Gu, M., Xia, J., Zhu, J. (2007). A Fast QR Algorithm for Companion Matrices. In: Ball, J.A., Eidelman, Y., Helton, J.W., Olshevsky, V., Rovnyak, J. (eds) Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 179. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8539-2_7

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