On a semidefinite programming characterizations of the numerical radius and its dual norm

We give a semidefinite programming characterization of the dual norm of numerical radius for matrices. This characterization yields a new proof of semidefinite characterization of the numerical radius for matrices, which follows from Ando's characterization. We show that the computation of the numerical radius and its dual norm within $\varepsilon$ precision are polynomially time computable in the data and $|\log \varepsilon |$ using the short step, primal interior point method.