Extensions of theory of regular and weak regular splittings to singular matrices

Abstract

Matrix splittings are useful in finding a solution of linear systems of equations, iteratively. In this note, we present some more convergence and comparison results for recently introduced matrix splittings called index-proper regular and index-proper weak regular splittings. We then apply to theory of double index-proper splittings.

Three-step alternating iterations for index 1 and non-singular matrices

“…In this section, we recall first the definition of proper G-regular splittings and proper G-weak regular splittings, and then present some new results for index 1 matrices. Definition 2.1 and 2.2, [13] reduce to the following two definitions, respectively when we use the group inverse in the place of the Drazin inverse. Below is an algorithm which we have used for computing proper G-weak regular splittings in this article.…”

“…However, the iteration scheme (2) converges very slow in many practical cases. To overcome this, several comparison results are proposed in the literature (see [9], [13], [14], [15] and [34] and the references cited therein). In case of a matrix having many proper splittings, comparison results are not so useful to find the best splitting (in the sense that the iteration matrix arising from a matrix splitting has the smallest spectral radius).…”

Index-proper nonnegative splittings of matrices

“…Very recently, Baliarsingh and Jena [2] have introduced index-proper regular and index-proper weak regular splittings for singular square matrices. Jena [5] again studied the same theory. We now introduce another splitting which covers a larger class of matrices in comparison to the above two splittings.…”