English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The SOR-like iteration method for solving the absolute value equations~(AVE)of finding a vector $x$ such that $Ax - |x| - b = 0$ with $\nu = \|A^{-1}\|_2 <1$ is investigated. The convergence conditions of the SOR-like iteration methodproposed by Ke and Ma ([{\em Appl. Math. Comput.}, 311:195--202, 2017]) arerevisited and a new proof is given, which exhibits some insights in determiningthe convergent region and the optimal iteration parameter. Along this line, theoptimal parameter which minimizes $\|T_\nu(\omega)\|_2$ with $$T_\nu(\omega) =\left(\begin{array}{cc} |1-\omega| & \omega^2\nu \\ |1-\omega| & |1-\omega|+\omega^2\nu \end{array}\right)$$ and the approximate optimal parameter whichminimizes $\eta_{\nu}(\omega) =\max\{|1-\omega|,\nu\omega^2\}$ are explored.The optimal and approximate optimal parameters are iteration-independent andthe bigger value of $\nu$ is, the smaller convergent region of the iterationparameter $\omega$ is. Numerical results are presented to demonstrate that theSOR-like iteration method with the optimal parameter is superior to that withthe approximate optimal parameter proposed by Guo, Wu and Li ([{\em Appl. Math.Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration methodwith the optimal parameter performs better, in terms of CPU time, than thegeneralized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009])for solving the AVE.

Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations