On the Nearest Singular Matrix Pencil

Abstract

Given a regular matrix pencil A + µE, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil A + ∆A + µ(E + ∆E), one approach for general singular pencils and another one such that A + ∆A and E + ∆E have a common left/right null vector. For the latter case the nearest singular pencil is shown to differ from the original pencil by rank-one matrices ∆A and ∆E. In both cases we consider also the situation where only A is perturbed. The nearest singular pencil is approached by a two-level iteration, where a gradient flow is driven to an equilibrium point in the inner iteration and the outer level uses a fast iteration for the distance parameter. This approach extends also to structured matrices A and E.

Computing lower rank approximations of matrix polynomials

“…All experiments are done using quad precision floating point arithmetic, with about 35 decimal digits of accuracy. We compare some degree one examples to the recent results of (Guglielmi et al, 2017).…”

“…In this section we consider Examples 2.10, 2.11 and 2.12 from Guglielmi et al (2017), where we compare our results to real perturbations. Note that complex perturbations are a straight-forward generalization of the theory presented here, and can be re-formulated as a problem over R.…”

Computing the nearest stable matrix pairs

“…Because we require a stable pair to be regular, it is also complementary to the distance to the nearest singular pencil, which is a long-standing open problem. [14][15][16] The nearest stable matrix pair problem occurs in system identification, where one needs to identify a stable matrix pair depending on observations, similar to the nearest stable matrix problem. 12,13 For example, when a real-world problem is approximated by a system model (1), the stability of the physical system may not be preserved, that is, the approximation process (e.g., finite element or finite difference models, linearization, or model order reduction) may make the stable system unstable.…”

“…An important consequence of the proof of Theorem 4 is an explicit construction of the DH characterization of a matrix pair (E, A), as follows: (a) solve the system (13) (if the system does not admit a solution, the pair is not regular, of index at most one, and asymptotically stable), and (b) use (15) to construct (J, R, Q).…”

A new bound for the distance to singularity of a regular matrix pencil

“…A matrix pencil A(s) = sE − A with E, A ∈ C n×n is called regular if det A(s) is not the zero polynomial, otherwise it is called singular. In the numerical treatment of matrix pencils it turns out that regular pencils which are close to singular pencils are difficult to handle [5]. In general, it is not even possible to compute canonical forms because rank decisions turn out to be impossible.…”