Direct solution of piecewise linear systems

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.