Abstract
A bisection method is used to compute lower and upper bounds on the distance from a quadratic matrix polynomial to the set of quadratic matrix polynomials having an eigenvalue on the imaginary axis. Each bisection step requires to check whether an even quadratic matrix polynomial has a purely imaginary eigenvalue. First, an upper bound is obtained using Frobenius-type linearizations. It takes into account rounding errors but does not use the even structure. Then, lower and upper bounds are obtained by reducing the quadratic matrix polynomial to a linear palindromic pencil. The bounds obtained this way also take into account rounding errors. Numerical illustrations are presented.
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Malyshev, A., Sadkane, M. Computing the distance to continuous-time instability of quadratic matrix polynomials. Numer. Math. 145, 149–165 (2020). https://doi.org/10.1007/s00211-020-01108-0
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DOI: https://doi.org/10.1007/s00211-020-01108-0