Path-Following Method to Determine the Field of Values of a Matrix with High Accuracy

Abstract

We describe a novel and efficient algorithm for calculating the field of values boundary, $\partial\textrm{W}(\cdot)$, of an arbitrary complex square matrix: the boundary is described by a system of ordinary differential equations which are solved using Runge--Kutta (Dormand--Prince) numerical integration to obtain control points with derivatives then finally Hermite interpolation is applied to produce a dense output. The algorithm computes $\partial\textrm{W}(\cdot)$ both efficiently and with low error. Formal error bounds are proven for specific classes of matrix. Furthermore, we summarise the existing state of the art and make comparisons with the new algorithm. Finally, numerical experiments are performed to quantify the cost-error trade-off between the new algorithm and existing algorithms.