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2015 | 13 | 1 |
Tytuł artykułu

Complexity issues for the symmetric interval eigenvalue problem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
otrzymano
2013-09-30
zaakceptowano
2014-11-26
online
2014-12-31
Twórcy
  • Charles University, Faculty of Mathematics and Physics, Department of Applied
    Mathematics, Malostranské nám. 25, 11800, Prague, Czech Republic, milan.hladik@matfyz.cz
Bibliografia
  • [1] R. E. Moore, R. B. Kearfott, and M. J. Cloud, Introduction to interval analysis. Philadelphia, PA: SIAM, 2009.
  • [2] A. Neumaier, Interval methods for systems of equations. Cambridge: Cambridge University Press, 1990.
  • [3] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational complexity and feasibility of data processing and intervalcomputations. Kluwer, 1998.
  • [4] J. Rohn, “Checking properties of interval matrices,” Technical Report 686, Institute of Computer Science, Academy of Sciences ofthe Czech Republic, Prague, 1996.
  • [5] J. Rohn, “A handbook of results on interval linear problems,” Technical Report 1163, Institute of Computer Science, Academy ofSciences of the Czech Republic, Prague, 2012.
  • [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.[Crossref]
  • [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.[WoS][Crossref]
  • [8] Y. Becis-Aubry and N. Ramdani, “State-bounding estimation for nonlinear models with multiple measurements,” in AmericanControl Conference (ACC 2012), (Montréal, Canada), pp. 1883–1888, IEEE Computer Society, 2012.
  • [9] M. S. Darup, M. Kastsian, S. Mross, and M. Mönnigmann, “Efficient computation of spectral bounds for hessian matrices onhyperrectangles for global optimization,” J. Glob. Optim., pp. 1–22, 2013. DOI: 10.1007/s10898-013-0099-1.[WoS][Crossref]
  • [10] M. Hladík, D. Daney, and E. Tsigaridas, “Bounds on real eigenvalues and singular values of interval matrices,” SIAM J. MatrixAnal. Appl., vol. 31, no. 4, pp. 2116–2129, 2010.[Crossref][WoS]
  • [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.
  • [12] L. V. Kolev, “Determining the positive definiteness margin of interval matrices,” Reliab. Comput., vol. 13, no. 6, pp. 445–466, 2007.[WoS]
  • [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.
  • [14] O. Beaumont, “An algorithm for symmetric interval eigenvalue problem,” Tech. Rep. IRISA-PI-00-1314, Institut de recherche eninformatique et systèmes aléatoires, Rennes, France, 2000.
  • [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.[WoS]
  • [16] J. Rohn, “An algorithm for checking stability of symmetric interval matrices,” IEEE Trans. Autom. Control, vol. 41, no. 1,pp. 133–136, 1996.[Crossref]
  • [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.[WoS]
  • [18] S. Miyajima, T. Ogita, S. Rump, and S. Oishi, “Fast verification for all eigenpairs in symmetric positive definite generalizedeigenvalue problems,” Reliab. Comput., vol. 14, pp. 24–45, 2010.
  • [19] S. M. Rump, “Verification methods: Rigorous results using floating-point arithmetic,” Acta Numer., vol. 19, pp. 287–449, 2010.[WoS][Crossref]
  • [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.
  • [21] A. Nemirovskii, “Several NP-hard problems arising in robust stability analysis,” Math. Control Signals Syst., vol. 6, no. 2,pp. 99–105, 1993.
  • [22] J. Rohn, “Interval matrices: Singularity and real eigenvalues,” SIAM J. Matrix Anal. Appl., vol. 14, no. 1, pp. 82–91, 1993.
  • [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.
  • [24] I. C. F. Ipsen, “Relative perturbation results for matrix eigenvalues and singular values,” Acta Numer., vol. 7, pp. 151–201, 1998.[Crossref]
  • [25] J. Rohn, “Computing the norm kAk1;1 is NP-hard,” Linear Multilinear Algebra, vol. 47, no. 3, pp. 195–204, 2000.
  • [26] G. H. Golub and C. F. Van Loan, Matrix computations. Baltimore: Johns Hopkins University Press, 3rd ed., 1996.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_math-2015-0015
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