AN ALGORITHM FOR COMPUTING THE NUMERICAL RADIUS

This paper is concerned with the calculation of the numerical radius o f a matrix, an important quantity in the analysis of convergence of it erative processes. An algorithm is developed which enables the numeric al radius to be obtained to a given precision, using a process which s uccessively refines lower and upper bounds. It uses an iteration proce dure analogous to the power method for computing the largest modulus e igenvalue of a Hermitian matrix. In contrast to that method, convergen ce is possible here to a local maximum of the underlying optimization problem which is not global, so that only a lower bound is provided. T his is used in conjunction with a technique based on the solution of a generalized eigenvalue problem to provide an upper bound. Numerical r esults illustrate the performance of the method.