Direct solution of piecewise linear systems

TL;DR

This paper introduces a modified Gaussian elimination method called signed Gaussian elimination for solving certain classes of absolute value equations efficiently, which are generally NP-hard but become tractable under specific conditions.

Contribution

It presents a new direct solution method for specific classes of absolute value equations, extending Gaussian elimination techniques to handle piecewise linear systems efficiently.

Findings

The signed Gaussian elimination algorithm matches classical Gaussian elimination complexity for dense matrices.

For tridiagonal systems, the method's cost is comparable to sorting n numbers.

The paper establishes the sharpness of the restrictions on matrix S for the method's applicability.

Abstract

Let $S$ be a real $n\times n$ matrix, $z,\hat c\in \mathbb R^n$, and $| z|$ the componentwise modulus of $z$. Then the piecewise linear equation system $$z-S| z| = \hat c$$ is called an \textit{absolute value equation} (AVE). It has been proven to be equivalent to the general \textit{linear complementarity problem}, which means that it is NP hard in general. We will show that for several system classes the AVE essentially retains the good natured solvability properties of regular linear systems. I.e., it can be solved directly by a slightly modified Gaussian elimination that we call the signed Gaussian elimination. For dense matrices $S$ this algorithm has the same operations count as the classical Gaussian elimination with symmetric pivoting. For tridiagonal systems in $n$ variables its computational cost is roughly that of sorting $n$ floating point numbers. The sharpness of the proposed restrictions on $S$ will be established.