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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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910888826503168 |
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| author | Shekhar, Vaibhav Mishra, Nachiketa Mishra, Debasisha |
| author_facet | Shekhar, Vaibhav Mishra, Nachiketa Mishra, Debasisha |
| contents | 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 $\mathbb{R}^n$. 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., $K=\mathbb{R}^n_+$), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_06047 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Convergence of two-stage iterative scheme for $K$-weak regular splittings of type II with application to Covid-19 pandemic model Shekhar, Vaibhav Mishra, Nachiketa Mishra, Debasisha Numerical Analysis 65F10, 65F50 G.1.3 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 $\mathbb{R}^n$. 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., $K=\mathbb{R}^n_+$), 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. |
| title | Convergence of two-stage iterative scheme for $K$-weak regular splittings of type II with application to Covid-19 pandemic model |
| topic | Numerical Analysis 65F10, 65F50 G.1.3 |
| url | https://arxiv.org/abs/2007.06047 |