WHERE IS THE NEAREST NONREGULAR PENCIL

This is a first step toward the goal of finding a way to calculate a s mallest norm de-regularizing perturbation of a given square matrix pen cil. Minimal de-regularizing perturbations have geometric characteriza tions that include a variable projection linear least squares problem and a minimax characterization reminiscent of the Courant-Fischer theo rem. The characterizations lead to new, computationally attractive upp er and lower bounds. We give a brief survey and illustrate strengths a nd weaknesses of several upper and lower bounds some of which are well -known and some of which are new. The ultimate goal remains elusive. ( C) 1998 Elsevier Science Inc. All rights reserved.