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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)
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 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)
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)
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)
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions, and of practical interest because of wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the(More)
We are concerned with the computation of the H∞-norm for H∞-functions of the form H(s) = C(s)D(s) −1 B(s), where the middle factor is the inverse of an analytic matrix-valued function, and C(s), B(s) are analytic functions mapping to short-and-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay(More)