Investigation of Crouzeix's Conjecture via Nonsmooth Optimization

Michael Overton (CS, NYU)

M. Crouzeix’s 2004 conjecture concerns the relationship between $\|p\|_{W(A)}$,
the norm of a polynomial $p$ on $W(A)$, the field of values of a matrix $A$,
and $\|p(A)\|_2$, the operator norm of the matrix $p(A)$. We use nonsmooth
optimization to investigate the conjecture numerically, using the BFGS
(Broyden-Fletcher-Goldfarb-Shanno) method to search for local minimizers of the
Crouzeix ratio

$\|p\|_{W(A)} / \|p(A)\|_2$ and Chebfun to compute the
boundary of the field of values. The conjecture states that the globally
minimal value of the Crouzeix ratio is 1/2. We present numerical results that
lead to some theorems and further conjectures about globally and locally
minimal values of the Crouzeix ratio when varying only $A$ (of given order,
with $p$ fixed) or varying only p (of given degree, with $A$ fixed), as well as
locally minimal values of the ratio when minimizing over all $p$ and $A$. All
the computations strongly support the truth of Crouzeix’s conjecture.

This is joint work with Anne Greenbaum and Adrian Lewis.