A note on double weak splittings of type II. Issue 12 (13th August 2022)

• Record Type: Journal Article
• Authors: Shekhar, Vaibhav; Giri, Chinmay Kumar; Mishra, Debasisha
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 12(2022)
• Page Start: 2297
• Page End: 2317
• Publication Date: 2022-08-13
• Subjects: Linear system -- iterative methods -- double splitting -- non-negativity -- convergence theorem -- comparison theorem; 15A09; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1795057
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M -matrix in the Gauss–Seidel double SOR method to obtain a comparison result.