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The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke–Lewis–Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n 4) operations on average, which is an improvement… (More)

This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T , we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplic-ities. We consider backward errors that only allow constant perturbations, which do… (More)

The eigenvalues of a Hermitian matrix function that depends on one parameter analytically can be ordered so that each eigenvalue is an analytic function of the parameter. Ordering these analytic eigenvalues from the largest to the smallest yields continuous and piece-wise analytic functions. For multi-variate Hermitian matrix functions that depend on d… (More)

The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix… (More)

We consider the 2-norm distance τr(A, B) from a linear time-invariant dynamical system (A, B) of order n to the nearest system (A + ∆A * , B + ∆B *) whose reachable subspace is of dimension r < n. We first present a characterization to test whether the reachable sub-space of the system has dimension r, which resembles and can be considered as a… (More)

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