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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

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ISSN
2391-5455
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Volume 13, Issue 1 (Dec 2014)

Issues

Complexity issues for the symmetric interval eigenvalue problem

Milan Hladík
  • Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 11800, Prague, Czech Republic
  • Email:
Published Online: 2014-12-31 | DOI: https://doi.org/10.1515/math-2015-0015

Abstract

We study the problem of computing the maximal and minimal possible eigenvalues of a symmetric matrix when the matrix entries vary within compact intervals. In particular, we focus on computational complexity of determining these extremal eigenvalues with some approximation error. Besides the classical absolute and relative approximation errors, which turn out not to be suitable for this problem, we adapt a less known one related to the relative error, and also propose a novel approximation error. We show in which error factors the problem is polynomially solvable and in which factors it becomes NP-hard.

Keywords : Interval matrix; Interval analysis; Eigenvalue; Eigenvalue bounds; NP-hardness

References

  • [1] R. E. Moore, R. B. Kearfott, and M. J. Cloud, Introduction to interval analysis. Philadelphia, PA: SIAM, 2009. Google Scholar

  • [2] A. Neumaier, Interval methods for systems of equations. Cambridge: Cambridge University Press, 1990. Google Scholar

  • [3] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational complexity and feasibility of data processing and interval computations. Kluwer, 1998. Google Scholar

  • [4] J. Rohn, “Checking properties of interval matrices,” Technical Report 686, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 1996. Google Scholar

  • [5] J. Rohn, “A handbook of results on interval linear problems,” Technical Report 1163, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 2012. Google Scholar

  • [6] D. Hertz, “The extreme eigenvalues and stability of real symmetric interval matrices,” IEEE Trans. Autom. Control, vol. 37, no. 4, pp. 532–535, 1992. CrossrefGoogle Scholar

  • [7] M. Hladík, D. Daney, and E. P. Tsigaridas, “Characterizing and approximating eigenvalue sets of symmetric interval matrices,” Comput. Math. Appl., vol. 62, no. 8, pp. 3152–3163, 2011. Web of ScienceCrossrefGoogle Scholar

  • [8] Y. Becis-Aubry and N. Ramdani, “State-bounding estimation for nonlinear models with multiple measurements,” in American Control Conference (ACC 2012), (Montréal, Canada), pp. 1883–1888, IEEE Computer Society, 2012. Google Scholar

  • [9] M. S. Darup, M. Kastsian, S. Mross, and M. Mönnigmann, “Efficient computation of spectral bounds for hessian matrices on hyperrectangles for global optimization,” J. Glob. Optim., pp. 1–22, 2013. DOI: 10.1007/s10898-013-0099-1. Web of ScienceCrossrefGoogle Scholar

  • [10] M. Hladík, D. Daney, and E. Tsigaridas, “Bounds on real eigenvalues and singular values of interval matrices,” SIAM J. Matrix Anal. Appl., vol. 31, no. 4, pp. 2116–2129, 2010. Web of ScienceCrossrefGoogle Scholar

  • [11] L. V. Kolev, “Outer interval solution of the eigenvalue problem under general form parametric dependencies,” Reliab. Comput., vol. 12, no. 2, pp. 121–140, 2006. Google Scholar

  • [12] L. V. Kolev, “Determining the positive definiteness margin of interval matrices,” Reliab. Comput., vol. 13, no. 6, pp. 445–466, 2007. Web of ScienceGoogle Scholar

  • [13] M.-H. Matcovschi and O. Pastravanu, “A generalized Hertz-type approach to the eigenvalue bounds of complex interval matrices,” in IEEE 51st Annual Conference on Decision and Control (CDC 2012), (Hawaii, USA), pp. 2195–2200, IEEE Computer Society, 2012. Google Scholar

  • [14] O. Beaumont, “An algorithm for symmetric interval eigenvalue problem,” Tech. Rep. IRISA-PI-00-1314, Institut de recherche en informatique et systèmes aléatoires, Rennes, France, 2000. Google Scholar

  • [15] M. Hladík, D. Daney, and E. P. Tsigaridas, “A filtering method for the interval eigenvalue problem,” Appl. Math. Comput., vol. 217, no. 12, pp. 5236–5242, 2011. Web of ScienceGoogle Scholar

  • [16] J. Rohn, “An algorithm for checking stability of symmetric interval matrices,” IEEE Trans. Autom. Control, vol. 41, no. 1, pp. 133–136, 1996. CrossrefGoogle Scholar

  • [17] Q. Yuan, Z. He, and H. Leng, “An evolution strategy method for computing eigenvalue bounds of interval matrices,” Appl. Math. Comput., vol. 196, no. 1, pp. 257–265, 2008. Web of ScienceGoogle Scholar

  • [18] S. Miyajima, T. Ogita, S. Rump, and S. Oishi, “Fast verification for all eigenpairs in symmetric positive definite generalized eigenvalue problems,” Reliab. Comput., vol. 14, pp. 24–45, 2010. Google Scholar

  • [19] S. M. Rump, “Verification methods: Rigorous results using floating-point arithmetic,” Acta Numer., vol. 19, pp. 287–449, 2010. Web of ScienceCrossrefGoogle Scholar

  • [20] J. Rohn, “Checking positive definiteness or stability of symmetric interval matrices is NP-hard,” Commentat. Math. Univ. Carol., vol. 35, no. 4, pp. 795–797, 1994. Google Scholar

  • [21] A. Nemirovskii, “Several NP-hard problems arising in robust stability analysis,” Math. Control Signals Syst., vol. 6, no. 2, pp. 99–105, 1993. Google Scholar

  • [22] J. Rohn, “Interval matrices: Singularity and real eigenvalues,” SIAM J. Matrix Anal. Appl., vol. 14, no. 1, pp. 82–91, 1993. Google Scholar

  • [23] V. Kreinovich, “How to define relative approximation error of an interval estimate: A proposal,” Appl. Math. Sci., vol. 7, no. 5, pp. 211–216, 2013. Google Scholar

  • [24] I. C. F. Ipsen, “Relative perturbation results for matrix eigenvalues and singular values,” Acta Numer., vol. 7, pp. 151–201, 1998. CrossrefGoogle Scholar

  • [25] J. Rohn, “Computing the norm kAk1;1 is NP-hard,” Linear Multilinear Algebra, vol. 47, no. 3, pp. 195–204, 2000. Google Scholar

  • [26] G. H. Golub and C. F. Van Loan, Matrix computations. Baltimore: Johns Hopkins University Press, 3rd ed., 1996. Google Scholar

About the article

Received: 2013-09-30

Accepted: 2014-11-26

Published Online: 2014-12-31


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0015.

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© 2015 Milan Hladík*. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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