SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2014-2015
Jeudi 26 février 2015 :
Michael L. OVERTON
(Courant Institute of Mathematical Sciences, New York University)
Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization.
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 suggest 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, Adrian Lewis and Nick Trefethen.
