A new preconditioner of the tensor splitting iterative method for solving multi-linear systems with $$\mathcal {M}$$ M
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A new preconditioner of the tensor splitting iterative method for solving multi-linear systems with M-tensors Lu-Bin Cui1
· Xiao-Qing Zhang1 · Shi-Liang Wu2
Received: 18 February 2020 / Revised: 7 May 2020 / Accepted: 14 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we propose a new preconditioner of the tensor splitting iterative method for solving multi-linear systems with M-tensors. We theoretically show that the spectral radii of the preconditioner iterative tensor decrease as the parameters in the new preconditioners increase, if the preconditioned tensor is a strong M-tensor. Based on this, we give the comparison for spectral radii of preconditioned iterative tensors. Numerical examples are given to show our theoretical results and the efficiency of our new preconditioner. We also show the efficiency of our preconditioner used to solving higher order Markov chain. Keywords Multi-linear system · Preconditioning · Strong M-tensor · Iterative method · Higher order Markov chain Mathematics Subject Classification 15A18 · 15A69
Communicated by Jinyun Yuan. This document is the results of the research project funded by National Natural Science Foundations of China (Nos. 11571095, 11601134, 11961082, 17HASTIT012) and 2019 Scientific Research Project for Postgraduates of Henan Normal University(No.YL201920).
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Lu-Bin Cui [email protected] Xiao-Qing Zhang [email protected] Shi-Liang Wu [email protected]
1
School of Mathematics and Information Science, Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China
2
School of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, People’s Republic of China 0123456789().: V,-vol
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1 Introduction Recently, the research of solving the multi-linear system has attracted considerable attention. It arises from some engineering and scientific computing (Cui et al. 2015; Qi and Luo 2017; Wei and Ding 2016), such as image processing (Cui et al. 2016; Yang et al. 2019, 2020), data mining (Li and Ng 2015), tensor complementarity problems (Ding and Wei 2016; Luo et al. 2017), and numerical partial differential equations (Ding and Wei 2016). Considering the following multi-linear system: Axm−1 = b, (1.1) where A ∈ R[m,n] is an order m dimension n tensor, x, b ∈ Rn are n-dimensional vectors, and the tensor-vector product is a vector which is defined by: (Axm−1 )i =
n
aii2 ,...,im xi2 · · · xim , i = 1, . . . , n,
(1.2)
i 2 ,...,i m =1
where xi is the ith component of x. We can see that the multi-linear system is composed of a series of non-linear equations. There have been many theoretical analysis and algorithms for solving (1.1) (see Cui et al. 2019; Ding and Wei 2016; Li et al. 2018; Li and Ng 2015; Liu et al. 2018, 2020; Wang et al. 2019; Xie et al. 2018; Zhang 2020; Zhang et al. 2020). In Ding and Wei (2016) and Liu et al. (2020), the authors used th
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