Convergence results for some piecewise linear solvers

TL;DR

This paper extends the convergence analysis of two solvers for absolute value equations, a class of piecewise linear systems, providing broader applicability of these methods.

Contribution

It broadens the known convergence ranges for a direct and a semi-iterative solver applied to absolute value equations.

Findings

Extended convergence ranges for the solvers.

Validated effectiveness for broader classes of matrices.

Enhanced understanding of solver behavior for piecewise linear systems.

Abstract

Let $A$ be a real $n\times n$ matrix and $z,b\in \mathbb R^n$. The piecewise linear equation system $z-A\vert z\vert = b$ is called an \textit{absolute value equation}. We consider two solvers for this problem, one direct, one semi-iterative, and extend their previously known ranges of convergence.