On comparison results for $ K $-nonnegative double splittings of different $ K $-monotone matrices

Ting Huang; Shu-Xin Miao; Ting Huang; Shu-Xin Miao

doi:10.3934/math.2021450

AIMS Mathematics

2021, Volume 6, Issue 7: 7741-7748. doi: 10.3934/math.2021450

Previous Article Next Article

Research article

On comparison results for $ K $-nonnegative double splittings of different $ K $-monotone matrices

Ting Huang ^,,
Shu-Xin Miao

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 06 March 2021 Accepted: 11 May 2021 Published: 14 May 2021
MSC : 15A09, 65F15

The comparison results for $ K $-double splittings of one $ K $-monotone matrix are established in the literatures. As comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners, we propose some comparison results for $ K $-nonnegative double splittings of different $ K $-monotone matrices in this note. The obtained results generalize the previous ones.
- $ K $-monotone matrix,
- $ K $-nonnegative double splitting,
- comparison theorem
Citation: Ting Huang, Shu-Xin Miao. On comparison results for $ K $-nonnegative double splittings of different $ K $-monotone matrices[J]. AIMS Mathematics, 2021, 6(7): 7741-7748. doi: 10.3934/math.2021450

Related Papers:

Abstract

The comparison results for $ K $-double splittings of one $ K $-monotone matrix are established in the literatures. As comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners, we propose some comparison results for $ K $-nonnegative double splittings of different $ K $-monotone matrices in this note. The obtained results generalize the previous ones.

References

[1]	A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1979.
[2]	K. Chen, Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, 2005.
[3]	J. J. Climent, C. Perea, Some comparison theorems for weak nonnegative splittings of bounded operators, Linear Algebra Appl., 275 (1998), 77-106.
[4]	J. J. Climent, C. Perea, Comparison theorems for weak nonnegative splittings of $K$-monotone matrices, The Electronic Journal of Linear Algebra, 5 (1999), 24-38.
[5]	L. Elsner, A. Frommer, R. Nabben, H. Schneider, D. B. Szyld, Conditions for strict inequality in comparisons of spectral radii of splittings of different matrices, Linear Algebra Appl., 363 (2003), 65-80. doi: 10.1016/S0024-3795(01)00535-3
[6]	C. H. Golub, R. S. Varga, Chebyshev semi-iterative methods, successive overrrelaxation iterative methods, and second order Richardson iterative methods-I, Numer. Math., 3 (1961), 147-168. doi: 10.1007/BF01386013
[7]	G.-L. Hou, Comparison theorems for double splittings of $K$-monotone matrices, Appl. Math. Comput., 244 (2014), 382-389.
[8]	G.-L. Hou, N. Li, $K$-nonnegative matrices and comparison theorems for iterative methods based on splittings, Adv. Math. (CHINA), 43 (2014), 463-479.
[9]	I. Marek, D. B. Szyld, Comparison theorems for weak splittings of bounded operators, Numer. Math., 58 (1990), 387-397. doi: 10.1007/BF01385632
[10]	S.-X. Miao, B. Zheng, A note on double splittings of different monotone matrices, Calcolo, 46 (2009), 261-266. doi: 10.1007/s10092-009-0011-z
[11]	Y. Song, Comparison theorems for nonnegative splittings of bounded operators, Mathematica Applicata, 12 (1999), 137-142.
[12]	Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn, Society for Industrial and Applied Mathematics, Philadelphia, 2003.
[13]	V. Shekhar, C. K. Giri, D. Mishra, A note on double weak splittings of type $II$, Linear and Multilinear Algebra, (2020), 1-21.
[14]	S. Q. Shen, T. Z. Huang, Convergence and comparison theorems for double splittings of matrices, Comput. Math. Appl., 51 (2006), 1751-1760. doi: 10.1016/j.camwa.2006.02.006
[15]	S. Q. Shen, T. Z. Huang, J. L. Shao, Convergence and comparison results for double splittings of Hermitian positive definite matrices, Calcolo, 44 (2007), 127-135. doi: 10.1007/s10092-007-0132-1
[16]	J. Song, Y. Song, Convergence for nonnegative double splittings of matrices, Calcolo, 48 (2011), 245-260. doi: 10.1007/s10092-010-0037-2
[17]	U. Trottenberg, C. Oosterlee, A. Schuller, Multigrid, Academic Press, San Diego, 2001.
[18]	Z. I. Woźniki, Estimation of the optimum relaxation factors in partial factorization iterative methods, SIAM J. Matrix Anal. Appl., 14 (1993), 59-73. doi: 10.1137/0614005
[19]	C. Wang, Comparison results for $K$-nonnegative double splittings of $K$-monotone matrices, Calcolo, 54 (2017), 1293-1303. doi: 10.1007/s10092-017-0230-7
[20]	C.-Y. Zhang, On convergence of double splitting methods for non-Hermitian positive semidefinite linear systems, Calcolo, 47 (2010), 103-112. doi: 10.1007/s10092-009-0015-8

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)