Comparison results for proper multisplittings of rectangular matrices

TL;DR

This paper introduces comparison theorems for proper multisplittings of rectangular matrices to efficiently analyze the convergence rates of iterative methods for least squares solutions.

Contribution

It develops new comparison theorems for proper multisplittings, improving the analysis of convergence speed in iterative solutions of rectangular linear systems.

Findings

Comparison theorems for proper weak regular splittings

Convergence results for proper multisplittings

Enhanced methods for analyzing iterative convergence

Abstract

The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is practically very slow in many cases. Thus, works on improving the speed of the iteration scheme have attracted great interest. In this direction, comparison of the rate of convergence of the iteration schemes produced by two matrix splittings is very useful. But, in the case of matrices having many matrix splittings, this process is time-consuming. The main goal of the current article is to provide a solution to the above issue by using proper multisplittings. To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first. We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.