The inverse q-numerical range problem and connections to the Davis-Wielandt shell and the pseudospectra of a matrix

Numerical ranges and related sets provide insights into the behavior of algorithms involving matrices, and inverse numerical range problems attempt to enhance these insights. We generalize the inverse numerical range problem, as proposed by Uhlig, to the inverse q-numerical range problem, and propose an algorithm for solving the problem that relies on convexity. To determine an approximation to the boundary of the q-numerical range, as needed by our algorithm, we must approximate the top of the Davis-Wielandt shell, a generalization of the numerical range. We found that the Davis-Wielandt shell is in a sense conjugate to the extreme singular values of the resolvent of a matrix. Knowing the Davis-Wielandt shell allows for the approximation of the q-numerical range, the pseudospectra and the Davis-Wielandt shell for any allowed Möbius transformation of a matrix. We provide some examples illustrating these connections, as well as how to solve the inverse q-numerical range problem.