A note on the preconditioned tensor splitting iterative method for solving strong $ \mathcal{M} $-tensor systems

Qingbing Liu; Aimin Xu; Shuhua Yin; Zhe Tu; Qingbing Liu; Aimin Xu; Shuhua Yin; Zhe Tu

doi:10.3934/math.2022400

AIMS Mathematics

2022, Volume 7, Issue 4: 7177-7186. doi: 10.3934/math.2022400

Previous Article Next Article

Research article

A note on the preconditioned tensor splitting iterative method for solving strong $ \mathcal{M} $-tensor systems

Department of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Received: 19 October 2021 Revised: 26 January 2022 Accepted: 06 February 2022 Published: 09 February 2022
MSC : 15A69, 65F10

In this note, we present a new preconditioner for solving the multi-linear systems, which arise from many practical problems and are different from the traditional linear systems. Based on the analysis of the spectral radius, we give new comparison results between some preconditioned tensor splitting iterative methods. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned method.
- multi-linear systems,
- Tensor splitting,
- preconditioned method,
- spectral radius,
- comparison theorem
Citation: Qingbing Liu, Aimin Xu, Shuhua Yin, Zhe Tu. A note on the preconditioned tensor splitting iterative method for solving strong $ \mathcal{M} $-tensor systems[J]. AIMS Mathematics, 2022, 7(4): 7177-7186. doi: 10.3934/math.2022400

Related Papers:

Abstract

In this note, we present a new preconditioner for solving the multi-linear systems, which arise from many practical problems and are different from the traditional linear systems. Based on the analysis of the spectral radius, we give new comparison results between some preconditioned tensor splitting iterative methods. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned method.

References

[1]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302–1324. https://doi.org/10.1016/j.jsc.2005.05.007 doi: 10.1016/j.jsc.2005.05.007
[2]	K. Pearson, Essentially positive tensors, Int. J. Algebra., 4 (2010), 421–427.
[3]	D. Liu, W. Li, S. Vong, The tensor splitting with application to solve multi-linear systems, J. Comput. Appl. Math., 330 (2018), 75–94. https://doi.org/10.1016/j.cam.2017.08.009 doi: 10.1016/j.cam.2017.08.009
[4]	W. Li, D. Liu, S. Vong, Comparison results for splitting iterations for solving multi-linear system, Appl. Numer. Math., 134 (2018), 105–121. https://doi.org/10.1016/j.apnum.2018.07.009 doi: 10.1016/j.apnum.2018.07.009
[5]	L. Cui, M. Li, Y. Song, Preconditioned tensor splitting iterations method for solving multi-linear systems, Appl. Math. Lett., 96 (2019), 89–94. https://doi.org/10.1016/j.aml.2019.04.019 doi: 10.1016/j.aml.2019.04.019
[6]	W. Ding, L. Qi, Y. Wei, $\mathcal{M}$-tensors and nonsingular $\mathcal{M}$-tensors, Linear Algebra Appl., 439 (2013), 3264–3278. https://doi.org/10.1016/j.laa.2013.08.038 doi: 10.1016/j.laa.2013.08.038
[7]	Q. Liu, J. Huang, S. Zeng, Convergence analysis of the two preconditioned iterative methods for $\mathcal{M}$-matrix linear systems, J. Comput. Appl. Math., 281 (2015), 49–57. https://doi.org/10.1016/j.cam.2014.11.034 doi: 10.1016/j.cam.2014.11.034
[8]	K. C. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507–520. https://dx.doi.org/10.4310/CMS.2008.v6.n2.a12 doi: 10.4310/CMS.2008.v6.n2.a12
[9]	W. Ding, Y. Wei, Solving multi-linear system with $\mathcal{M}$-tensors, J. Sci. Comput., 68 (2016), 689–715. https://doi.org/10.1007/s10915-015-0156-7 doi: 10.1007/s10915-015-0156-7
[10]	L. Zhang, L. Qi, G. Zhou, $\mathcal{M}$-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437–452. https://doi.org/10.1137/130915339 doi: 10.1137/130915339
[11]	Z. Luo, L. Qi, N. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471–482. https://doi.org/10.1007/s11590-016-1013-9 doi: 10.1007/s11590-016-1013-9
[12]	M. Ng, L. Qi, G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090–1099. https://doi.org/10.1137/09074838X doi: 10.1137/09074838X
[13]	Y. Matsuno, Exact solutions for the nonlinear Klein-Gordon and Liouville equations in four-dimensional Euclidean space, J. Math. Phys., 28 (1987), 2317–2322. https://doi.org/10.1063/1.527764 doi: 10.1063/1.527764
[14]	D. Zwillinger, Handbook of differential equations, 3 Eds., Boston: Academic Press Inc, 1997.
[15]	D. Kressner, C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. Appl., 31 (2010), 1688–1714. https://doi.org/10.1137/090756843 doi: 10.1137/090756843
[16]	C. Tobler, Low-rank Tensor methods for linear systems and eigenvalue problems, Ph.D. thesis, 2012.

Reader Comments

Your name:*

Email:*
© 2022 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)