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Matrix double splitting iterations are simple in implementation while solving real non-singular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each ADS scheme induces a preconditioned system which helps in showing the convergence of the ADS schemes. We also show that the classes of matrices for which one ADS scheme is better than the other, are mutually exclusive. Numerical experiments confirm that the proposed ADS schemes have several computational advantages over the existing methods. Though the problems are considered in the rectangular matrix settings, the same problems are even new in non-singular matrix settings.

Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to deal with singular and rectangular linear systems. In this article, we introduce a new iteration scheme called three-step alternating iterations using proper splittings and group inverses to find an approximate solution of singular linear systems, iteratively. As a special case, the same findings also work for finding an approximate solution of non-singular linear systems. A preconditioned alternating iterative scheme is also proposed to relax some sufficient conditions and to obtain faster convergence as well. We then show that our scheme converges faster than the unpreconditioned one. The theoretical findings are then validated numerically.

In this paper, the concept of matrix splitting is introduced to solve a large sparse ill-posed linear system via Tikhonov?s regularization. In the regularization process, we convert the ill-posed system to a well-posed system. The convergence of such a well-posed system is discussed by using different types of matrix splittings. Comparison analysis of both systems are studied by operating certain types of weak splittings. Further, we have extended the double splitting of [Song J. and Song Y, Calcolo 48(3), 245-260, 2011] to double weak splitting of type II for nonsingular symmetric matrices. In addition to that, some more comparison results are presented with the help of such weak double splittings of type I and type II.