Comparison results for proper multisplittings of rectangular matrices
For researchers working on iterative methods for rectangular linear systems, this provides a theoretical framework to compare and select faster-converging matrix splittings, though the results are incremental and domain-specific.
This paper addresses the slow convergence of iterative methods for least squares solutions of rectangular linear systems and proposes comparison theorems for proper multisplittings to accelerate convergence. The authors derive convergence and comparison results for proper weak regular and nonnegative splittings, enabling faster selection of optimal splittings.
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.