# Applied Math Seminar

**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.