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 under- lying 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.
The distances considered here are the distance to uncontrollability and the distance to instability. A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results. Two methods are developed to estimate the distance to uncontrollability, giving both a lower bound and an upper bound.