| Title:
|
On the convergence theory of double $K$-weak splittings of type II (English) |
| Author:
|
Shekhar, Vaibhav |
| Author:
|
Mishra, Nachiketa |
| Author:
|
Mishra, Debasisha |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
341-369 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Recently, Wang (2017) has introduced the $K$-nonnegative double splitting using the notion of matrices that leave a cone $K\subseteq \mathbb {R}^{n}$ invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for $K$-weak regular and $K$-nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes of matrices. We then obtain the comparison results for two double splittings of a $K$-monotone matrix. Most of these results are completely new even for $K= \mathbb {R}^{n}_+$. The convergence behavior is discussed by performing numerical experiments for different matrices derived from the discretized Poisson equation. (English) |
| Keyword:
|
linear system |
| Keyword:
|
iterative method |
| Keyword:
|
$K$-nonnegativity |
| Keyword:
|
double splitting |
| Keyword:
|
convergence theorem |
| Keyword:
|
comparison theorem |
| MSC:
|
15A06 |
| MSC:
|
15A09 |
| MSC:
|
15B48 |
| MSC:
|
65F10 |
| idZBL:
|
Zbl 07547199 |
| idMR:
|
MR4409310 |
| DOI:
|
10.21136/AM.2021.0270-20 |
| . |
| Date available:
|
2022-04-14T13:36:52Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150319 |
| . |
| Reference:
|
[1] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences.Classics in Applied Mathematics 9. SIAM, Philadelphia (1994). Zbl 0815.15016, MR 1298430, 10.1137/1.9781611971262 |
| Reference:
|
[2] Climent, J.-J., Perea, C.: Some comparison theorems for weak nonnegative splittings of bounded operators.Linear Algebra Appl. 275-276 (1998), 77-106. Zbl 0936.65063, MR 1628383, 10.1016/S0024-3795(97)10065-9 |
| Reference:
|
[3] Climent, J.-J., Perea, C.: Comparison theorems for weak nonnegative splittings of $K$-monotone matrices.Electron. J. Linear Algebra 5 (1999), 24-38. Zbl 0919.65023, MR 1668923, 10.13001/1081-3810.1029 |
| Reference:
|
[4] Climent, J.-J., Perea, C.: Comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices.Linear Algebra Appl. 302-303 (1999), 355-366. Zbl 0948.65030, MR 1733540, 10.1016/S0024-3795(99)00158-5 |
| Reference:
|
[5] Collatz, L.: Functional Analysis and Numerical Mathematics.Academic Press, New York (1966). Zbl 0148.39002, MR 0205126 |
| Reference:
|
[6] Golub, G. H., Loan, C. F. Van: Matrix Computations.The John Hopkins University Press, Baltimore (1996). Zbl 0865.65009, MR 1417720 |
| Reference:
|
[7] Golub, G. H., Varga, R. S.: Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods I.Numer. Math. 3 (1961), 147-156. Zbl 0099.10903, MR 145678, 10.1007/BF01386013 |
| Reference:
|
[8] Hou, G.: Comparison theorems for double splittings of $K$-monotone matrices.Appl. Math. Comput. 244 (2014), 382-389. Zbl 1335.15015, MR 3250585, 10.1016/j.amc.2014.06.101 |
| Reference:
|
[9] Marek, I., Szyld, D. B.: Comparison theorems for weak splittings of bounded operators.Numer. Math. 58 (1990), 389-397. Zbl 0694.65023, MR 1077585, 10.1007/BF01385632 |
| Reference:
|
[10] Miao, S.-X., Zheng, B.: A note on double splittings of different monotone matrices.Calcolo 46 (2009), 261-266. Zbl 1185.65058, MR 2563785, 10.1007/s10092-009-0011-z |
| Reference:
|
[11] Nandi, A. K., Shekhar, V., Mishra, N., Mishra, D.: Alternating stationary iterative methods based on double splittings.Comput. Math. Appl. 89 (2021), 87-98. Zbl 7336184, MR 4233331, 10.1016/j.camwa.2021.02.015 |
| Reference:
|
[12] Saad, Y.: Iterative Methods for Sparse Linear Systems.SIAM, Philadelphia (2003). Zbl 1031.65046, MR 1990645, 10.1137/1.9780898718003 |
| Reference:
|
[13] Shekhar, V., Giri, C. K., Mishra, D.: A note on double weak splittings of type II.(to appear) in Linear Multilinear Algebra. 10.1080/03081087.2020.1795057 |
| Reference:
|
[14] Shen, S.-Q., Huang, T.-Z.: Convergence and comparison theorems for double splittings of matrices.Comput. Math. Appl. 51 (2006), 1751-1760. Zbl 1134.65341, MR 2245703, 10.1016/j.camwa.2006.02.006 |
| Reference:
|
[15] Shen, S.-Q., Huang, T.-Z., Shao, J.-L.: Convergence and comparison results for double splittings of Hermitian positive definite matrices.Calcolo 44 (2007), 127-135. Zbl 1150.65008, MR 2352718, 10.1007/s10092-007-0132-1 |
| Reference:
|
[16] Song, Y.: Comparison theorems for splittings of matrices.Numer. Math. 92 (2002), 563-591. Zbl 1012.65028, MR 1930390, 10.1007/s002110100333 |
| Reference:
|
[17] Song, J., Song, Y.: Convergence for nonnegative double splittings of matrices.Calcolo 48 (2011), 245-260. Zbl 1232.65056, MR 2827007, 10.1007/s10092-010-0037-2 |
| Reference:
|
[18] Wang, C.: Comparison results for $K$-nonnegative double splittings of $K$-monotone matrices.Calcolo 54 (2017), 1293-1303. Zbl 1385.65028, MR 3735816, 10.1007/s10092-017-0230-7 |
| Reference:
|
[19] Woźnicki, Z. I.: Estimation of the optimum relaxation factors in partial factorization iterative methods.SIAM J. Matrix Anal. Appl. 14 (1993), 59-73. Zbl 0767.65025, MR 1199544, 10.1137/0614005 |
| . |
