Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T00:10:28.657Z Has data issue: false hasContentIssue false
  • English
  • Français

THE HYPERBOLIC QUADRATIC EIGENVALUE PROBLEM

Part of: Numerical linear algebra Error analysis and interval analysis Basic linear algebra

Published online by Cambridge University Press:  13 August 2015

XIN LIANG  and
REN-CANG LI
XIN LIANG
Affiliation:
Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany; [email protected]
REN-CANG LI
Affiliation:
Department of Mathematics, University of Texas at Arlington, PO Box 19408, Arlington, TX 76019, USA; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant–Fischer type min–max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt–Lidskii type min–max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt–Lidskii–Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh–Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches—steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem—to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.

MSC classification

Primary: 15A42: Inequalities involving eigenvalues and eigenvectors 65F15: Eigenvalues, eigenvectors
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 65F08: Preconditioners for iterative methods 65F50: Sparse matrices 65G99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e13
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Al-Ammari, M. and Tisseur, F., ‘Hermitian matrix polymomials with real eigenvalues of definite type. Part I. Classification’, Linear Algebra Appl. 436(10) (2012), 3954–3973.Google Scholar
Amir-Moéz, A. R., ‘Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations’, Duke Math. J. 23 (1956), 463–476.Google Scholar
Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S. and Sorensen, D., LAPACK Users’ Guide, 3rd edn (SIAM, Philadelphia, 1999).Google Scholar
Bai, Z., Li, R.-C. and Su, Y., Lecture notes on matrix eigenvalue computations. Prepared for 2009 Summer School on Numerical Linear Algebra, Chinese Academy of Science, July 2009.Google Scholar
Barkwell, L. and Lancaster, P., ‘Overdamped and gyroscopic vibrating systems’, J. Appl. Mech. 59(1) (1992), 176–181.Google Scholar
Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C. and Tisseur, F., ‘NLEVP: a collection of nonlinear eigenvalue problems’, ACM Trans. Math. Software 39(2) (2013), 7:1–7:28.CrossRefGoogle Scholar
Bhatia, R., Matrix Analysis, Graduate Texts in Mathematics, 169 (Springer, New York, 1996).Google Scholar
Bhatia, R., Kittaneh, F. and Li, R.-C., ‘Some inequalities for commutators and an application to spectral variation. II’, Linear Multilinear Algebra 43(1–3) (1997), 207–220.Google Scholar
Bhatia, R. and Li, R.-C., ‘On perturbations of matrix pencils with real spectra. II’, Math. Comp. 65(214) (1996), 637–645.Google Scholar
Chu, M. T. and Xu, S., ‘Spectral decomposition of real symmetric quadratic 𝜆-matrices and its applications’, Math. Comp. 78 (2009), 293–313.Google Scholar
Davis, G., ‘Numerical solution of a quadratic matrix equation’, SIAM J. Sci. Statist. Comput. 2(2) (1981), 164–175.Google Scholar
Demmel, J., Applied Numerical Linear Algebra (SIAM, Philadelphia, PA, 1997).Google Scholar
Duffin, R., ‘A minimax theory for overdamped networks’, Indiana Univ. Math. J. 4 (1955), 221–233.Google Scholar
Dzeng, D. C. and Lin, W. W., ‘Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems’, J. Aust. Math. Soc. B 32 (1991), 437–456, 4.Google Scholar
Faddeev, D. K. and Faddeeva, V. N., Computational Methods of Linear Algebra, Undergraduate Mathematics Books (W.H. Freeman & Co. Ltd, San Francisco, 1963), Translated by R. C. Williams.Google Scholar
Fan, K., ‘On a theorem of Weyl concerning eigenvalues of linear transformations. I’, Proc. Natl. Acad. Sci. USA 35(11) (1949), 652–655.Google Scholar
Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials Classics in Applied Mathematics, 58, (SIAM, Philadelphia, 2009).Google Scholar
Golub, G. H. and Van Loan, C. F., Matrix Computations, 3rd edn (Johns Hopkins University Press, Baltimore, MD, 1996).Google Scholar
Golub, G. H. and Ye, Q., ‘An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems’, SIAM J. Sci. Comput. 24(1) (2002), 312–334.Google Scholar
Greenbaum, A., Iterative Methods for Solving Linear Systems (SIAM, Philadelphia, 1997).CrossRefGoogle Scholar
Guo, C.-H., ‘Numerical solution of a quadratic eigenvalue problem’, Linear Algebra Appl. 385(0) (2004), 391–406.Google Scholar
Guo, C.-H., Higham, N. J. and Tisseur, F., ‘Detecting and solving hyperbolic quadratic eigenvalue problems’, SIAM J. Matrix Anal. Appl. 30(4) (2009), 1593–1613.Google Scholar
Guo, C.-H. and Lancaster, P., ‘Algorithms for hyperbolic quadratic eigenvalue problems’, Math. Comp. 74 (2005), 1777–1791.Google Scholar
Higham, N. J. and Kim, H.-M., ‘Numerical analysis of a quadratic matrix equation’, IMA J. Numer. Anal. 20(4) (2000), 499–519.Google Scholar
Higham, N. J., Li, R.-C. and Tisseur, F., ‘Backward error of polynomial eigenproblems solved by linearization’, SIAM J. Matrix Anal. Appl. 29(4) (2007), 1218–1241.Google Scholar
Higham, N. J., Mackey, D. and Tisseur, F., ‘Definite matrix polynomials and their linearization by definite pencils’, SIAM J. Matrix Anal. Appl. 31(2) (2009), 478–502.Google Scholar
Higham, N. J., Tisseur, F. and Van Dooren, P. M., ‘Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems’, Linear Algebra Appl. 351–352 (2002), 455–474.Google Scholar
Knyazev, A. V., ‘Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method’, SIAM J. Sci. Comput. 23(2) (2001), 517–541.Google Scholar
Knyazev, A. V. and Neymeyr, K., ‘A geometric theory for preconditioned inverse iteration III: a short and sharp convergence estimate for generalized eigenvalue problems’, Linear Algebra Appl. 358(1–3) (2003), 95–114.CrossRefGoogle Scholar
Kovač-Striko, J. and Veselić, K., ‘Trace minimization and definiteness of symmetric pencils’, Linear Algebra Appl. 216 (1995), 139–158.Google Scholar
Lancaster, P., ‘Inverse spectral problems for semisimple damped vibrating systems’, SIAM J. Matrix Anal. Appl. 29(1) (2007), 279–301.Google Scholar
Lancaster, P. and Tisseur, F., ‘Hermitian quadratic matrix polynomials: solvents and inverse problems’, Linear Algebra Appl. 436(10) (2012), 4017–4026.Google Scholar
Li, R.-C., ‘On perturbations of matrix pencils with real spectra’, Math. Comp. 62 (1994), 231–265.CrossRefGoogle Scholar
Li, R.-C., ‘On perturbations of matrix pencils with real spectra, a revisit’, Math. Comp. 72 (2003), 715–728.Google Scholar
Li, R.-C., ‘On Meinardus’ examples for the conjugate gradient method’, Math. Comp. 77(261) (2008), 335–352. Electronically published on September 17, 2007.Google Scholar
Li, R.-C., Rayleigh quotient based optimization methods for eigenvalue problems. Technical Report 2014-04, Department of Mathematics, University of Texas at Arlington, January 2014. Lecture summary for 2013 G. Golub SIAM Summer School; to appear in Ser. Contemp. Appl. Math.Google Scholar
Li, R.-C., Lin, W.-W. and Wang, C.-S., ‘Structured backward error for palindromic polynomial eigenvalue problems’, Numer. Math. 116(1) (2010), 95–122.Google Scholar
Liang, X. and Li, R.-C., ‘Extensions of Wielandt’s min–max principles for positive semi-definite pencils’, Linear Multilinear Algebra 62(8) (2014), 1032–1048.Google Scholar
Liang, X. and Li, R.-C., The hyperbolic quadratic eigenvalue problem. Technical Report 2014-01, Department of Mathematics, University of Texas at Arlington, January 2014. Available at http://www.uta.edu/math/preprint/.Google Scholar
Liang, X., Li, R.-C. and Bai, Z., ‘Trace minimization principles for positive semi-definite pencils’, Linear Algebra Appl. 438 (2013), 3085–3106.Google Scholar
Lidskii, V. B., ‘The proper values of the sum and product of symmetric matrices’, Dokl. Akad. Nauk SSSR 75 (1950), 769–772. (in Russian). Translation by C. Benster available from the National Translation Center of the Library of Congress.Google Scholar
Longsine, D. E. and McCormick, S. F., ‘Simultaneous Rayleigh-quotient minimization methods for Ax =𝜆 Bx’, Linear Algebra Appl. 34 (1980), 195–234.Google Scholar
Mackey, D. S., Mackey, N., Mehl, C. and Mehrmann, V., ‘Structured polynomial eigenvalue problems: good vibrations from good linearizations’, SIAM J. Matrix Anal. Appl. 28(4) (2006), 1029–1051.Google Scholar
Mackey, D. S., Mackey, N., Mehl, C. and Mehrmann, V., ‘Vector spaces of linearizations for matrix polynomials’, SIAM J. Matrix Anal. Appl. 28(4) (2006), 971–1004.CrossRefGoogle Scholar
Markus, A. S., Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, 71 (American Mathematical Society, Providence, RI, 1988).Google Scholar
Mirsky, L., ‘Symmetric gauge functions and unitarily invariant norms’, Quart. J. Math. 11 (1960), 50–59.Google Scholar
Moler, C. B. and Stewart, G. W., ‘An algorithm for generalized matrix eigenvalue problems’, SIAM J. Numer. Anal. 10(2) (1973), 241–256.Google Scholar
Nocedal, J. and Wright, S., Numerical Optimization, 2nd edn (Springer, New York, 2006).Google Scholar
Ovtchinnikov, E. E., ‘Sharp convergence estimates for the preconditioned steepest descent method for Hermitian eigenvalue problems’, SIAM J. Numer. Anal. 43(6) (2006), 2668–2689.Google Scholar
Parlett, B. N., The Symmetric Eigenvalue Problem (SIAM, Philadelphia, 1998).Google Scholar
Polyak, B. T., Introduction to Optimization (Optimization Software, New York, 1987).Google Scholar
Rogers, E. H., ‘A mimmax theory for overdamped systems’, Arch. Ration. Mech. Anal. 16 (1964), 89–96.Google Scholar
Rogers, E. H., ‘Variational properties of nonlinear spectra’, Indiana Univ. Math. J. 18 (1969), 479–490.Google Scholar
Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd edn (SIAM, Philadelphia, 2003).Google Scholar
Samokish, B., ‘The steepest descent method for an eigenvalue problem with semi-bounded operators’, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (1958), 105–114. (in Russian).Google Scholar
Stewart, G. W., ‘Perturbation bounds for the definite generalized eigenvalue problem’, Linear Algebra Appl. 23 (1979), 69–86.Google Scholar
Stewart, G. W., Matrix Algorithms, Vol II: Eigensystems (SIAM, Philadelphia, 2001).CrossRefGoogle Scholar
Stewart, G. W. and Sun, J.-G., Matrix Perturbation Theory (Academic Press, Boston, 1990).Google Scholar
Sun, J.-G., ‘A note on Stewart’s theorem for definite matrix pairs’, Linear Algebra Appl. 48 (1982), 331–339.Google Scholar
Sun, J.-G., ‘Perturbation bounds for eigenspaces of a definite matrix pair’, Numer. Math. 41 (1983), 321–343.Google Scholar
Sun, W. and Yuan, Y.-X., Optimization Theory and Methods—Nonlinear Programming (Springer, New York, 2006).Google Scholar
Takahashi, I., ‘A note on the conjugate gradient method’, Inform. Process. Japan 5 (1965), 45–49.Google Scholar
Tisseur, F., ‘Backward error and condition of polynomial eigenvalue problems’, Linear Algebra Appl. 309(1–3) (2000), 339–361.Google Scholar
Tisseur, F. and Meerbergen, K., ‘The quadratic eigenvalue problem’, SIAM Rev. 43(2) (2001), 235–386.Google Scholar
Veselić, K., ‘A Jacobi eigenreduction algorithm for definite matrix pairs’, Numer. Math. 64 (1993), 241–269.Google Scholar
Veselić, K., ‘Note on interlacing for hyperbolic quadratic pencils’, inRecent Advances in Operator Theory in Hilbert and Krein Spaces, (eds. Behrndt, J., Förster, K.-H. and Trunk, C.) Operator Theory: Advances and Applications, 198 (Birkhäuser, Boston, 2010), 305–307.Google Scholar
Veselić, K., Damped Oscillations of Linear Systems, Lecture Notes in Mathematics, 2023 (Springer, Berlin, 2011).Google Scholar
Voss, H., ‘A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter’, Numer. Linear Algebra Appl. 16(11–12) (2009), 899–913.Google Scholar
Voss, H. and Werner, B., ‘A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems’, Math. Methods Appl. Sci. 4 (1982), 415–424.Google Scholar
Wei, S. and Kao, I., ‘Vibration analysis of wire and frequency response in the modern wiresaw manufacturing process’, J. Sound Vib. 231(5) (2000), 1383–1395.Google Scholar
Weyl, H., ‘Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)’, Math. Ann. 71 (1912), 441–479.Google Scholar
Wielandt, H., ‘An extremum property of sums of eigenvalues’, Proc. Amer. Math. Soc. 6 (1955), 106–110.Google Scholar
Yang, H., ‘Conjugate gradient methods for the Rayleigh quotient minimization of generalized eigenvalue problems’, Computing 51 (1993), 79–94.Google Scholar