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Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization

Intervenant(s) : Michael Overton (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 modest 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.
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