An inexact framework of the Newton-based matrix splitting iterative method for the generalized absolute value equation
Dongmei Yu, Cairong Chen, Deren Han
TL;DR
This paper introduces an inexact framework (INMS) for the Newton-based matrix splitting (NMS) method to solve the generalised absolute value equation (GAVE) $Ax - B|x| - b = 0$. By allowing inexact solves of the linear subproblems with a residual tolerance $\theta$, the authors establish global linear convergence under a concrete bound involving $\| (\Omega+M)^{-1} \|$ and problem data, and show that INMS subsumes several Newton-type schemes as special cases. Numerical experiments demonstrate that INMS variants outperform their exact counterparts in CPU time while maintaining comparable accuracy, highlighting practical scalability for large-scale problems. The results suggest that exploiting inexact linear solves within the NMS framework yields both theoretical and empirical benefits for solving GAVE and its AVE special case.
Abstract
An inexact framework of the Newton-based matrix splitting (INMS) iterative method is developed to solve the generalized absolute value equation, whose exact version was proposed by Zhou, Wu and Li [H.-Y. Zhou, S.-L. Wu and C.-X. Li, \textit{J. Comput. Appl. Math.}, 394 (2021), 113578]. Global linear convergence of the INMS iterative method is investigated in detail. Some numerical results are given to show the superiority of the INMS iterative method.