Bounding the Solutions of Sylvester-like Absolute Value Equation

Abstract

We develop several methods (including two direct methods and an iterative method) for computing an enclosure of the solutions of the so-called Sylvester-like absolute value equations. The proposed direct methods are modifications of the Bauer-Skeel and Hansen-Bliek-Rohn bounds, which were introduced for outer approximation of the solutions of the standard and generalized absolute value equations. These approaches have the advantage of considerably reducing computational costs, in contrast to simple Kroneckerization, i.e., the direct application of the aforementioned bounds to the Kronecker form of the Sylvester-like absolute value equations. We also propose an iterative approach, which refines some initial bounds and produces highly efficient enclosures for the solutions. Moreover, the iterative method can be terminated at any time and provides a numerically guaranteed distance to the unique solution.