Convergence of two-stage iterative scheme for K-weak regular splittings of type II with application to Covid-19 pandemic model
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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 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.
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