Proper Weak Regular Splitting and its Application to Convergence of Alternating Iterations

TL;DR

This paper revisits weak regular splittings for rectangular matrices and introduces an alternating iterative method using Moore-Penrose inverse, with convergence analysis and comparison showing faster convergence.

Contribution

It extends the theory of weak regular splittings and proposes a new alternating iterative method with proven convergence for solving rectangular linear systems.

Findings

Proposed method converges faster than existing schemes.

Extended convergence theory for the new alternating iterative method.

Comparison results support improved efficiency of the method.

Abstract

The theory of matrix splitting is a useful tool for finding solution of rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which insures faster convergence rate of the proposed alternating iterative scheme.