Abstract
In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N.
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Anderson, E., Bai, Z., Bischof, C., Blackford, S., Dongarra, J., Demmel, J., Du Croz, J., Greenbaum, A., Hammarling, S.: LAPACK Users’ Guide. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)
Bini, D.A., Gemignani, L., Pan, V.Y.: Inverse power and Durand–Kerner iterations for univariate polynomial root-finding. Comput. Math. Appl. 47(2–3), 447–459 (2004)
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)
Chandrasekaran, S., Gu, M.: Fast and stable eigendecomposition of symmetric banded plus semi-separable matrices. Linear Algebra Appl. 313(1–3), 107–114 (2000)
Eidelman, Y., Gemignani, L., Gohberg, I.: On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms. Linear Algebra Appl. 420, 86–101 (2007)
Eidelman, Y., Gohberg, I., Olshevsky, V.: The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order. Linear Algebra Appl. 404, 305–324 (2005)
Fasino, D., Mastronardi, N., Van Barel, M.: Fast and stable algorithms for reducing diagonal plus semiseparable matrices to tridiagonal and bidiagonal form. In Fast algorithms for structured matrices: theory and applications (South Hadley, MA, 2001), Contemp. Math.,vol. 323, pp. 105–118. Amer. Math. Soc., Providence, RI (2003)
Francis, J.G.F.: The QR transformation: a unitary analogue to the LR transformation. I. Comput. J. 4, 265–271 (1961/1962)
Francis, J.G.F.: The QR transformation. II. Comput. J. 4, 332–345 (1961/1962)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD (1996)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd ed. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002)
Juang, J., Lin, W.W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20(1), 228–243 (1999) (electronic)
Lu, L., Ng, M.K.: Effects of a parameter on a nonsymmetric algebraic Riccati equation. Appl. Math. Comput. 172(2), 753–761 (2006)
Tisseur, F.: Backward stability of the QR algorithm. Technical Report 239, UMR 5585 Lyon Saint-Etienne (1996)
Vandebril, R., Van Barel, M., Mastronardi, N.: An implicit QR algorithm for symmetric semiseparable matrices. Numer. Linear Algebra Appl. 12(7), 625–658 (2005)
Watkins, D.S.: Fundamentals of matrix computations, 2nd edn. Pure and Applied Mathematics (New York). Wiley-Interscience (John Wiley & Sons), New York (2002)
Watkins, D.S., Elsner, L.: Chasing algorithms for the eigenvalue problem. SIAM J. Matrix Anal. Appl. 12(2), 374–384 (1991)
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Eidelman, Y., Gemignani, L. & Gohberg, I. Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations. Numer Algor 47, 253–273 (2008). https://doi.org/10.1007/s11075-008-9172-0
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DOI: https://doi.org/10.1007/s11075-008-9172-0