Numerical investigation of Crouzeix's conjecture

Crouzeix's conjecture states that for all polynomials p and matrices A, the inequality ‖p(A)‖≤2‖p‖W(A) holds, where the quantity on the left is the 2-norm of the matrix p(A) and the norm on the right is the maximum modulus of the polynomial p on W(A), the field of values of A. We report on some extensive numerical experiments investigating the conjecture via nonsmooth minimization of the Crouzeix ratio f≡‖p‖W(A)/‖p(A)‖, using Chebfun to evaluate this quantity accurately and efficiently and the BFGS method to search for its minimal value, which is 0.5 if Crouzeix's conjecture is true. Almost all of our optimization searches deliver final polynomial-matrix pairs that are very close to nonsmooth stationary points of f with stationary value 0.5 (for which W(A) is a disk) or smooth stationary points of f with stationary value 1 (for which W(A) has a corner). Our observations have led us to some additional conjectures as well as some new theorems. We hope that these give insight into Crouzeix's conjecture, which is strongly supported by our results.