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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. [ABSTRACT FROM AUTHOR]

• Convergence theory of two-stage iterative scheme for K-weak regular splittings of both types in the proper cone setting is established. • Numerical implementation has been discussed on next generation matrix and PageRank matrix. Monotone matrices play a key role in the convergence theory of regular splittings and different types of weak regular splittings. If monotonicity fails, then it is difficult to guarantee the convergence of the above-mentioned classes of matrices. In such a case, K -monotonicity is sufficient for the convergence of K -regular and K -weak regular splittings, where K is a proper cone in R n. However, the convergence theory of a two-stage iteration scheme in a general proper cone setting is a gap in the literature. Especially, the same study for weak regular splittings of type II (even if in standard proper cone setting, i.e., K = R + n), is open. To this end, we propose convergence theory of two-stage iterative scheme for K -weak regular splittings of both types in the proper cone setting. We provide some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a K -regular splitting and then establish some comparison theorems. We also study K -monotone convergence theory of the stationary two-stage iterative method in case of a K -weak regular splitting of type II. The most interesting and important part of this work is on M -matrices appearing in the Covid-19 pandemic model. Finally, numerical computations are performed using the proposed technique to compute the next generation matrix involved in the pandemic model. The computation of large PageRank matrices shows that the two-stage Gauss-Seidel method performs better than the Gauss-Seidel methods. [ABSTRACT FROM AUTHOR]