Comments on finite termination of the generalized Newton method for absolute value equations

2024/01/22 by Chun‐Hua Guo, Guo, Chun-Hua
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2401.12407

openalex publication_date 2024/01/22 · openalex created_date 2024/01/25 · openalex updated_date 2026/08/01

Abstract

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) Ax-|x|=b. The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever ρ(|A^-1|)<1/3. We also present new results for the case where A-I is a nonsingular M-matrix or an irreducible singular M-matrix. When A-I is an irreducible singular M-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.

Related

• The Unique Solvability Conditions for the Generalized Absolute Value Equations
• Sufficient conditions for the unique solvability of absolute value matrix equations
• On the global convergence of the inexact semi-smooth Newton method for absolute value equation
• Convergence results for some piecewise linear solvers
• On the unique solution of the generalized absolute value equation