The Error and Perturbation Bounds of the General Absolute Value Equations

To our knowledge, the error and perturbation bounds of the general absolute value equations (AVE) are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error and perturbation bounds of these two AVEs: Ax-B|x|=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax-B|x|=b$$\end{document} and Ax-|Bx|=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax-|Bx|=b$$\end{document}. Some useful error bounds and perturbation bounds of the above two AVEs are provided. Without limiting the matrix type, some computable estimates for the relevant upper bounds are given. By applying the absolute value equations, a new approach for some existing perturbation bounds of the linear complementarity problem (LCP) in (SIAM J. Optim., 18 (2007) 1250-1265) is provided. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds.