On finite termination of the generalized Newton method for solving absolute value equations

TL;DR

This paper analyzes the conditions under which the generalized Newton method terminates finitely when solving absolute value equations, providing theoretical bounds and numerical validation.

Contribution

It establishes finite termination conditions for GNM on AVEs with certain matrices and introduces new results on the solvability of AVEs.

Findings

GNM terminates in at most 2n+2 iterations for some matrices

New criteria for unique solvability and unsolvability of AVEs

Numerical experiments confirm theoretical results

Abstract

Motivated by the framework constructed by Brugnano and Casulli $[$SIAM J. Sci. Comput. 30: 463--472, 2008$]$, we analyze the finite termination property of the generalized Netwon method (GNM) for solving the absolute value equation (AVE). More precisely, for some special matrices, GNM is terminated in at most $2n + 2$ iterations. A new result for the unique solvability and unsolvability of the AVE is obtained. Numerical experiments are given to demonstrate the theoretical analysis.