Advertisement

Tight Bounds on the Radius of Nonsingularity

  • David HartmanEmail author
  • Milan Hladík
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9553)

Abstract

Radius of nonsingularity of a square matrix is the minimal distance to a singular matrix in the maximum norm. Computing the radius of nonsingularity is an NP-hard problem. The known estimations are not very tight; one of the best one has the relative error 6n. We propose a randomized approximation method with a constant relative error 0.7834. It is based on a semidefinite relaxation. Semidefinite relaxation gives the best known approximation algorithm for MaxCut problem, and we utilize similar principle to derive tight bounds on the radius of nonsingularity. This gives us rigorous upper and lower bounds despite randomized character of the algorithm.

Keywords

Radius of nonsingularity Bounds Semidefinite programming 

References

  1. 1.
    Poljak, S., Rohn, J.: Radius of Nonsingularity. Technical report KAM Series (88–117), Department of Applied Mathematics, Charles University, Prague (1988)Google Scholar
  2. 2.
    Poljak, S., Rohn, J.: Checking robust nonsingularity is NP-hard. Math. Control Signals Syst. 6(1), 1–9 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Rohn, J.: Checking properties of interval matrices. Technical report, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 686 (1996)Google Scholar
  4. 4.
    Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  5. 5.
    Rump, S.M.: Almost sharp bounds for the componentwise distance to the nearest singular matrix. Linear Multilinear Algebra 42(2), 93–107 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Rump, S.M.: Bounds for the componentwise distance to the nearest singular matrix. SIAM J. Matrix Anal. Appl. 18(1), 83–103 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gärtner, B., Matoušek, J.: Approximation Algorithms and Semidefinite Programming. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  9. 9.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  10. 10.
    Packard, A., Doyle, J.C.: The complex structured singular value. Automatica 29, 71–109 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stein, G., Doyle, J.C.: Beyond singular values and loop shapes. J. Guidance Control Dyn. 14(1), 5–16 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kolev, L.V.: A method for determining the regularity radius of interval matrices. Reliable Comput. 16(1), 1–26 (2011)MathSciNetGoogle Scholar
  14. 14.
    Jansson, C., Chaykin, D., Keil, C.: Rigorous error bounds for the optimal value in semidefinite programming. SIAM J. Numer. Anal. 46(1), 180–200 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Institute of Computer Science Academy of SciencesPrague 8Czech Republic

Personalised recommendations