Tight Bounds on the Radius of Nonsingularity

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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gärtner, B., Matoušek, J.: Approximation Algorithms and Semidefinite Programming. Springer, Heidelberg (2012)CrossRefMATHGoogle Scholar
  8. 8.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)CrossRefMATHGoogle Scholar
  9. 9.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  10. 10.
    Packard, A., Doyle, J.C.: The complex structured singular value. Automatica 29, 71–109 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Stein, G., Doyle, J.C.: Beyond singular values and loop shapes. J. Guidance Control Dyn. 14(1), 5–16 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

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