Computing the distance to continuous-time instability of quadratic matrix polynomials

A bisection method is used to compute lower and upper bounds on the distance froma quadratic matrix polynomial to the set of quadratic matrix polynomials having aneigenvalue on the imaginary axis. Each bisection step requires to check whether aneven quadratic matrix polynomial has a purely imaginary eigenvalue. First, an upperbound is obtained using Frobenius-type linearizations. It takes into account roundingerrors but does not use the even structure. Then, lower and upper bounds are obtainedby reducing the quadratic matrix polynomial to a linear palindromic pencil. The boundsobtained this way also take into account rounding errors. Numerical illustrations arepresented