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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 31, 2014

Complexity issues for the symmetric interval eigenvalue problem

  • Milan Hladík EMAIL logo
From the journal Open Mathematics

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.

References

[1] R. E. Moore, R. B. Kearfott, and M. J. Cloud, Introduction to interval analysis. Philadelphia, PA: SIAM, 2009. 10.1137/1.9780898717716Search in Google Scholar

[2] A. Neumaier, Interval methods for systems of equations. Cambridge: Cambridge University Press, 1990. 10.1017/CBO9780511526473Search in Google Scholar

[3] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational complexity and feasibility of data processing and interval computations. Kluwer, 1998. 10.1007/978-1-4757-2793-7Search in 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. Search in 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. Search in 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. 10.1109/9.126593Search in Google 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. Search in Google 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. Search in 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. 10.1007/s10898-013-0099-1Search in Google 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. Search in Google 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. 10.1007/s11155-006-4875-1Search in Google Scholar

[12] L. V. Kolev, “Determining the positive definiteness margin of interval matrices,” Reliab. Comput., vol. 13, no. 6, pp. 445–466, 2007. 10.1007/s11155-007-9046-5Search in Google 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. Search in 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. Search in 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. Search in Google 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. 10.1109/9.481618Search in Google 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. 10.1016/j.amc.2007.05.051Search in Google 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. 10.1063/1.3498650Search in Google Scholar

[19] S. M. Rump, “Verification methods: Rigorous results using floating-point arithmetic,” Acta Numer., vol. 19, pp. 287–449, 2010. 10.1017/S096249291000005XSearch in Google 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. Search in 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. 10.1007/BF01211741Search in Google Scholar

[22] J. Rohn, “Interval matrices: Singularity and real eigenvalues,” SIAM J. Matrix Anal. Appl., vol. 14, no. 1, pp. 82–91, 1993. 10.1137/0614007Search in 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. 10.12988/ams.2013.13019Search in Google Scholar

[24] I. C. F. Ipsen, “Relative perturbation results for matrix eigenvalues and singular values,” Acta Numer., vol. 7, pp. 151–201, 1998. 10.1017/S0962492900002828Search in Google Scholar

[25] J. Rohn, “Computing the norm kAk1;1 is NP-hard,” Linear Multilinear Algebra, vol. 47, no. 3, pp. 195–204, 2000. 10.1080/03081080008818644Search in Google Scholar

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

Received: 2013-9-30
Accepted: 2014-11-26
Published Online: 2014-12-31

© 2015 Milan Hladík*

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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