ON THE COMPUTATION OF THE DISTANCE TO QUADRATIC MATRIX POLYNOMIALS THAT ARE SINGULAR AT SOME POINTS ON THE UNIT CIRCLE

For a quadratic matrix polynomial, the distance to the set of quadratic matrix polynomials which have singularities on the unit circle is computed using a bisection-based algorithm. The success of the algorithm depends on the eigenvalue method used within the bisection to detect the eigenvalues near the unit circle. To this end, the QZ algorithm along with the Laub trick is employed to compute the anti-triangular Schur form of a matrix resulting from a palindromic reduction of the quadratic matrix polynomial. It is shown that despite rounding errors, the Laub trick followed, if necessary, by a simple refinement procedure makes the results reliable for the intended purpose. Several numerical illustrations are reported.