An algorithm for computing the numerical radius

This paper is concerned with the calculation of the numerical radius of a matrix, an important quantity in the analysis of convergence of iterative processes. An algorithm is developed which enables the numerical radius to be obtained to a given precision, using a process which successively refines lower and upper bounds. It uses an iteration procedure analogous to the power method for computing the largest modulus eigenvalue of a Hermitian matrix. In contrast to that method, convergence is possible here to a local maximum of the underlying optimization problem which is not global, so that only a lower bound is provided. This is used in conjunction with a technique based on the solution of a generalized eigenvalue problem to provide an upper bound. Numerical results illustrate the performance of the method.