Error bounds and a condition number for the absolute value equations

TL;DR

This paper develops error bounds and introduces a condition number for absolute value equations, enhancing understanding of their stability and computational properties, especially in relation to the linear complementarity problem.

Contribution

It provides new error bounds and a condition number for absolute value equations, including analysis for various matrix norms and special matrix classes.

Findings

Derived explicit error bounds for absolute value equations.

Introduced and analyzed a condition number for stability assessment.

Explored computational complexity and bounds for the condition number.

Abstract

Absolute value equations, due to their relation to the linear complementarity problem, have been intensively studied recently. In this paper, we present error bounds for absolute value equations. Along with the error bounds, we introduce an appropriate condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, its computational complexity, its bounds and also exact values for special classes of matrices. We consider also matrices that appear based on the transformation from the linear complementarity problem.