A simple algorithm for the inverse field of values problem

The field of values of a matrix is the closed convex subset of the complex plane comprising all Rayleigh quotients, a set of interest in the stability analysis of dynamical systems and convergence theory of matrix iterations, among other applications. Recently, Uhlig proposed the inverse field of values problem: given a point in the field of values, determine a vector for which this point is the corresponding Rayleigh quotient. Uhlig also devised a sophisticated algorithm involving random vectors and the boundaries of ellipses for solving the inverse field of values problem. We propose a simpler deterministic algorithm that must converge (in exact arithmetic) and for most points yields an exact result in only a few iterations. The algorithm builds upon the fact that the inverse field of values problem can be solved exactly in the two-dimensional case. We also resolve a conjecture posed by Uhlig concerning the number of linearly independent vectors that generate a point in the field of values and propose a more challenging inverse field of values problem that is of interest in eigenvalue computations.