The tensor splitting methods for solving tensor absolute value equation
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The tensor splitting methods for solving tensor absolute value equation Fan Bu1 · Chang-Feng Ma1 Received: 20 March 2019 / Revised: 9 May 2020 / Accepted: 14 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Recently, D.-D. Liu et al. (2018) presented the tensor splitting methods for solving multilinear systems and S.-Q. Du et al. (2018) generalized tensor absolute value equations. In this paper, we verify the existence of solutions of tensor absolute value equations and propose the tensor splitting methods for solving this class of equation. Furthermore, the convergence analysis of the tensor splitting method is also studied under suitable conditions. Finally, numerical examples show that our algorithm is an efficient iterative method. Keywords Tensor absolute value equation · Tensor splitting · Strong M-tensor Mathematics Subject Classification 15A69 · 15A72 · 53A45
1 Introduction In recent years, tensors have become a hot research field and have wide applications in engineering and scientific computing. Many scholars extended methods of matrix equations to the aspect of solving the classes of tensor systems. Thereinto the following multi-linear systems have attracted much attention: Axm−1 = b,
(1.1)
where A is an m-order n-dimensional tensor in R n×n×...×n , x and right-hand b are both vectors in R n . As for Axm−1 , the ith entry is given below: (Axm−1 )i =
n i 2 =1
···
n
aii2 ···im xi2 · · · xim (i = 1, 2, . . . , m).
(1.2)
i n =1
Communicated by Jinyun Yuan.
B 1
Chang-Feng Ma [email protected] College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350117, People’s Republic of China 0123456789().: V,-vol
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Fan Bu and C.-F. Ma
In this paper, we consider the following tensor absolute value equation (TAVE) Axm−1 − |x|[m−1] = b,
(1.3)
where |x|[m−1] is a vector in R n with |x|[m−1] = (|x1 |m−1 , . . . , |xn |m−1 ) . Many theoretical analysis and algorithms for solving Eq. (1.1) have been proposed X. Wang et al. (2019); M. Che et al. (2016, 2019); W.-Y. Ding and Y. Wei (2016); D.-D. Liu et al. (2018); W. Li et al. (2018); C.-Q. Lv and C.-F. Ma (2018); D.-H. Li et al. (2017); L. Han (2017); X.-T. Li and M.K. Ng (2015). W.-Y. Ding and Y. Wei (2016) studied some special structured multi-linear systems, particularly the coefficient tensors which are M-tensors. In addition, strong M-tensor is also called nonsingular M-tensor. They proved that system (1.1) has unique solution when coefficient tensor is nonsingular tensor and the right hand is positive. Besides, they also extended Jacobi method, the Gauss–Seidel method and the Newton method to solve nonsingular M-equations. D.-D. Liu et al. (2018) proposed some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong M-tensors as well. W. Li et al. (2018) compared spectral radius of two different iterative tensors. Based on D.-D. Liu et al. (2018), they presented the preconditioned tensor splitting method and gave th
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