Title: The Inverse q-Numerical Range Problem and Connections to the Davis-Wielandt Shell and the Pseudospectra of a Matrix

Abstract: Numerical ranges and related sets provide insights into the behavior
 of iterative algorithms for solving systems of equations and computing eigenvalues.
 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 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 M\"{o}bius transformation of a matrix.
 We provide some examples illustrating these connections, as well as
 how to solve the inverse $q$-numerical range problem.