Numerical range of a matrix: some effective criteria

AbstractAlgorithms are presented which decide, for a given complex number w and a given complex n×n matrix S, whether w is in the numerical range W(S) of S, whether w is a boundary point of W(S), whether w is an extreme point of W(S), whether w is a bare point of W(S), and whether w is a vertex of W(S). Further algorithms decide whether W(S) intersects a given line (or a given ray), whether W(S) is included in a given open half plane (or a given closed half plane), and, for a given real number r, whether the numerical radius ρs of S is > r, whether ρs=r, and whether ρs≥r. A simple effective criterion for H-stability is also given: a nonsingular H-semistable matrix S is H-stable iff the nullity of (S+S∗)S-1(S+S∗) is twice the nullity of S+S∗. The computations involved in all these algorithms are elementary (rational operations, the max operation on pairs of real numbers, the degree of a nonzero polynomial, and the number of sign variations in the coefficients of a nonzero real polynomial), must be carried out exactly, and give exact (i.e., 100% reliable) results. Examples are worked out to illustrate the application of some of the algorithms