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Tight Bounds on the Radius of Nonsingularity

Part of the Lecture Notes in Computer Science book series (LNTCS,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

D. Hartman—Supported by grants 13-17187S and 13-10660S of the Czech Science Foundation.

M. Hladík—Supported by grant 13-10660S of the Czech Science Foundation.

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Hartman, D., Hladík, M. (2016). Tight Bounds on the Radius of Nonsingularity. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds) Scientific Computing, Computer Arithmetic, and Validated Numerics. SCAN 2015. Lecture Notes in Computer Science(), vol 9553. Springer, Cham. https://doi.org/10.1007/978-3-319-31769-4_9

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  • DOI: https://doi.org/10.1007/978-3-319-31769-4_9

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