A note on double splittings for rectangular matrices

The theory of matrix splittings can be applied to derive iterative solutions for rectangular linear systems of the form Ax = b. Various comparison results for different subclasses of proper splittings have been proposed in the literature to enhance the convergence rate of these iterative methods. In this article, we extend the convergence theory of double proper splittings for rectangular matrices by introducing two new subclasses: double proper weak regular splitting of type II and double proper weak splitting of type II. Additionally, we present several comparison results that can be utilized to identify a more effective splitting among various options.