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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)

Two useful measures of the robust stability of the discrete-time dynamical system x k+1 = Ax k are the-pseudospectral radius and the numerical radius of A. The-pseudospectral radius of A is the largest of the moduli of the points in the-pseudospectrum of A, while the numerical radius is the largest of the moduli of the points in the field of values. We… (More)

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the… (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)

— Structured eigenvalue problems feature a prominent role in many algorithms for the computation of robust measures for the stability or controllability of a linear control system. Structures that typically arise are Hamiltonian, skew-Hamiltonian, and symplectic. The use of eigenvalue solvers that preserve such structures can enhance the reliability and… (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)

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 multiplicities. While the usual resolvent norm addresses this question for a single… (More)