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Fast QR iterations for unitary plus low rank matrices

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Abstract

Some fast algorithms for computing the eigenvalues of a (block) companion matrix have recently appeared in the literature. In this paper we generalize the approach to encompass unitary plus low rank matrices of the form \(A=U + XY^H\) where U is a general unitary matrix. Three important cases for applications are U unitary diagonal, U unitary block Hessenberg and U unitary in block CMV form. Our extension exploits the properties of a larger matrix \(\hat{A}\) obtained by a certain embedding of the Hessenberg reduction of A suitable to maintain its structural properties. We show that \(\hat{A}\) can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first k rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR iteration. The resulting eigenvalue algorithm is fast and backward stable.

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References

  1. Ammar, G., Calvetti, D., Reichel, L.: Computing the poles of autoregressive models from the reflection coefficients. In: Proceedings of 31st Annual Allerton Conference on Communication, Control, and Computing, pp. 255–264 (1993)

  2. Ammar, G., Gragg, W., Reichel, L.: Direct and inverse unitary eigenproblems in signal processing: an overview. In: De Moor, B.L.R., Moonen, F.T., Golub, G.H. (eds.) Linear Algebra for Large Scale and Real-Time Applications. Springer, New York (1993)

    Google Scholar 

  3. Ammar, G.S., Calvetti, D., Reichel, L.: Continuation methods for the computation of zeros of Szegö polynomials. Linear Algebra Appl. 249, 125–155 (1996)

    Article  MathSciNet  Google Scholar 

  4. Ammar, G.S., Gragg, W.B., Reichel, L.: On the eigenproblem for orthogonal matrices. In: 1986 25th IEEE Conference on Decision and Control, pp. 1963–1966 (1986)

  5. Aurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of the eigenvalues and eigenvectors of matrix polynomials. Math. Comput. 88, 313–347 (2019)

    Article  MathSciNet  Google Scholar 

  6. Aurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.S.: Core-Chasing Algorithms for the Eigenvalue Problem. Fundamentals of Algorithms. SIAM, Philadelphia (2018)

    Book  Google Scholar 

  7. Aurentz, J.L., Mach, T., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of roots of polynomials. SIAM J Matrix Anal. Appl. 36(3), 942–973 (2015)

    Article  MathSciNet  Google Scholar 

  8. Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems. ACM Trans. Math. Softw. 39(2), 7:1–7:28 (2013)

    Article  MathSciNet  Google Scholar 

  9. Bevilacqua, R., Del Corso, G.M.: Structural properties of matrix unitary reduction to semiseparable form. Calcolo 41(4), 177–202 (2004)

    Article  MathSciNet  Google Scholar 

  10. Bevilacqua, R., Del Corso, G.M., Gemignani, L.: On computing efficient data-sparse representations of unitary plus low-rank matrices. Technical report (2019)

  11. Bindel, D., Chandresekaran, S., Demmel, J., Garmire, D., Gu, M.: A fast and stable nonsymmetric eigensolver for certain structured matrices. Technical report (2005)

  12. Bini, D.A., Daddi, F., Gemignani, L.: On the shifted QR iteration applied to companion matrices. Electron. Trans. Numer. Anal. 18(electronic), 137–152 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Bini, D.A., Eidelman, Y., Gemignani, L., Gohberg, I.: Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices. SIAM J. Matrix Anal. Appl. 29(2), 566–585 (2007)

    Article  MathSciNet  Google Scholar 

  14. Bini, D.A., Gemignani, L., Pan, V.Y.: Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations. Numer. Math. 100(3), 373–408 (2005)

    Article  MathSciNet  Google Scholar 

  15. Boito, P., Eidelman, Y., Gemignani, L., Gohberg, I.: Implicit QR with compression. Indagationes Mathematicae 23(4), 733–761 (2012)

    Article  MathSciNet  Google Scholar 

  16. Bunse-Gerstner, A., Elsner, L.: Schur parameter pencils for the solution of the unitary eigenproblem. Linear Algebra Appl. 154(156), 741–778 (1991)

    Article  MathSciNet  Google Scholar 

  17. Chandrasekaran, S., Gu, M., Xia, J., Zhu, J.: 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, pp. 111–143. Birkhäuser, Basel (2007)

    Chapter  Google Scholar 

  18. Del Corso, G.M., Poloni, F., Robol, L., Vandebril, R.: When is a matrix unitary or hermitian plus low rank? Numer. Linear Algebra Appl. (To appear) (2019)

  19. Del Corso, G.M., Poloni, F., Robol, L., Vandebril, R.: Factoring block Fiedler companion matrices. Springer INdAM Ser. 30, 129–155 (2019)

    Article  MathSciNet  Google Scholar 

  20. De Terán, F., Dopico, F.M., Pérez, J.: Backward stability of polynomial root-finding using Fiedler companion matrices. IMA J. Numer. Anal. 36(1), 133–173 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Edelman, A., Murakami, H.: Polynomial roots from companion matrix eigenvalues. Math. Comput. 64(210), 763–776 (1995)

    Article  MathSciNet  Google Scholar 

  22. Eidelman, Y., Gohberg, I., Haimovici, I.: Separable type representations of matrices and fast algorithms. In: Eigenvalue method, Volume 235 of Operator Theory: Advances and Applications, vol. 2, Birkhäuser/Springer, Basel (2014)

  23. Fassbender, H.: On numerical methods for discrete least-squares approximation by trigonometric polynomials. Math. Comput. 66(218), 719–741 (1997)

    Article  MathSciNet  Google Scholar 

  24. Fiedler, M.: A note on companion matrices. Linear Algebra Appl. 372, 325–331 (2003)

    Article  MathSciNet  Google Scholar 

  25. Fiedler, M., Markham, T.L.: Completing a matrix when certain entries of its inverse are specified. Linear Algebra Appl. 74, 225–237 (1986)

    Article  MathSciNet  Google Scholar 

  26. Francis, J.G.F.: The QR transformation-part 2. Comput. J. 4(4), 332–345 (1962)

    Article  MathSciNet  Google Scholar 

  27. Fyodorov, Y.V., Sommers, H.-J.: Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A Math. Gen. 36(12), 3303–3347 (2003)

    Article  MathSciNet  Google Scholar 

  28. Gantmacher, F.R.: The Theory of Matrices. Number v. 1 in the Theory of Matrices. Chelsea Pub. Co. (1960)

  29. Gemignani, L.: A unitary Hessenberg QR-based algorithm via semiseparable matrices. J. Comput. Appl. Math. 184(2), 505–517 (2005)

    Article  MathSciNet  Google Scholar 

  30. Gemignani, L., Robol, L.: Fast Hessenberg reduction of some rank structured matrices. SIAM J. Matrix Anal. Appl. 38(2), 574–598 (2017)

    Article  MathSciNet  Google Scholar 

  31. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  32. Gragg, W.B.: The QR algorithm for unitary Hessenberg matrices. J. Comput. Appl. Math. 16, 1–8 (1986)

    Article  MathSciNet  Google Scholar 

  33. Gragg, W.B.: Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle. J. Comput. Appl. Math. 46(1–2), 183–198 (1993). (Computational complex analysis)

    Article  MathSciNet  Google Scholar 

  34. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)

    Book  Google Scholar 

  35. Jenkins, M.A., Traub, J.F.: Principles for testing polynomial zerofinding programs. ACM Trans. Math. Softw. 1(1), 26–34 (1975)

    Article  MathSciNet  Google Scholar 

  36. Kimura, H.: Generalized Schwarz form and lattice-ladder realizations of digital filters. IEEE Trans. Circuits Syst. 32(11), 1130–1139 (1985)

    Article  Google Scholar 

  37. Mach, T., Vandebril, R.: On deflations in extended QR algorithms. SIAM J. Matrix Anal. Appl. 35(2), 559–579 (2014)

    Article  MathSciNet  Google Scholar 

  38. Moler, C.: Fiedler Companion Matrix. Cleve’s Corner (2013)

  39. Sinap, A., Van Assche, W.: Orthogonal matrix polynomials and applications. In: Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), vol. 66, pp. 27–52 (1996)

    Article  MathSciNet  Google Scholar 

  40. Vandebril, R., Del Corso, G.M.: An implicit multishift \(QR\)-algorithm for Hermitian plus low rank matrices. SIAM J. Sci. Comput. 32(4), 2190–2212 (2010)

    Article  MathSciNet  Google Scholar 

  41. Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices, vol. I, II. Johns Hopkins University Press, Baltimore (2008)

    MATH  Google Scholar 

  42. Watkins, D.S.: The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods, 1st edn. Society for Industrial and Applied Mathematics, Philadelphia (2007)

    Book  Google Scholar 

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Acknowledgements

We would like to thank the anonymous reviewers whose helpful comments allowed to improve the quality of the presentation.

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  1. Dipartimento di Informatica, Università di Pisa, Largo Bruno Pontecorvo 3, 56127, Pisa, Italy

    Roberto Bevilacqua, Gianna M. Del Corso & Luca Gemignani

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  1. Roberto Bevilacqua
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  2. Gianna M. Del Corso
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  3. Luca Gemignani
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Correspondence to Gianna M. Del Corso.

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The research of the last two authors was partially supported by GNCS project “Analisi di matrici sparse e data-sparse: metodi numerici ed applicazioni”and by the project sponsored by University of Pisa under the Grant PRA-2017-05.

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Bevilacqua, R., Del Corso, G.M. & Gemignani, L. Fast QR iterations for unitary plus low rank matrices. Numer. Math. 144, 23–53 (2020). https://doi.org/10.1007/s00211-019-01080-4

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