On the convergence theory of double $K$-weak splittings of type II

summary:Recently, Wang (2017) has introduced the $K$-nonnegative double splitting using the notion of matrices that leave a cone $K\subseteq \mathbb {R}^{n}$ invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for $K$-weak regular and $K$-nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes of matrices. We then obtain the comparison results for two double splittings of a $K$-monotone matrix. Most of these results are completely new even for $K= \mathbb {R}^{n}_+$. The convergence behavior is discussed by performing numerical experiments for different matrices derived from the discretized Poisson equation