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. The other is slow, requiring computations of a large number of the smallest singular values, but it produces bounds as tight as possible and also a region containing global minimizers. Newton's method can be used to compute the global minimizers if one really wants. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number.