An algorithm for computing the numerical radius

Received on 23 October 1995. Revised on 15 July 1996. This paper is concerned with the calculation of the numericalradius of a matrix, an important quantity in the analysis ofconvergence of iterative processes. An algorithm is developedwhich enables the numerical radius to be obtained to a givenprecision, using a process which successively refines lowerand upper bounds. It uses an iteration procedure analogous tothe power method for computing the largest modulus eigenvalueof a Hermitian matrix. In contrast to that method, convergenceis possible here to a local maximum of the underlying optimizationproblem which is not global, so that only a lower bound is provided.This is used in conjunction with a technique based on the solutionof a generalized cigenvalue problem to provide an upper bound.Numerical results illustrate the performance of the method.