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The theory of splitting is a useful tool for finding solution of a system of linear equations. Many woks are going on for singular system of linear equations. In this article, we have introduced a new splitting called indexproper nonnegative splitting for singular square matrices. Several convergence and comparison results are also established. We then apply the same theory to double splitting.

Climent and Perea [Journal of Computational and Applied Mathematics 58:43–48, 2003; MR2013603] proposed first the convergence theory of two-stage iterative scheme for solving real rectangular linear systems. In this article, we revisit the same theory. The first main result provides some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a proper weak regular splitting. We then establish a few comparison results. Out of these, many are even new in nonsingular matrix setting. Further, we study the monotone convergence theory of the two-stage iterative method. Besides these, we also prove the uniqueness of a proper splitting of a rectangular matrix under certain assumptions.

In the original version of the book, the following correction has been incorporated: In Chapter 10, the second author's affiliation has been changed from "Depart-