Efficient Splitting Methods for Solving Tensor Absolute Value Equation

Jing Ning,Yajun Xie,Jie Yao

doi:10.3390/sym14020387

Jing Ning, Yajun Xie + Show 1 more

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https://doi.org/10.3390/sym14020387

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Abstract

The tensor absolute value equation is a class of interesting structured multilinear systems. In this article, from the perspective of pure numerical algebra, we first consider a tensor-type successive over-relaxation method (SOR) (called TSOR) and tensor-type accelerated over-relaxation method (AOR) (called TAOR) for solving tensor absolute value equations. Furthermore, one type of preconditioned tensor splitting method is also applied for solving structured multilinear systems. Numerical experiments adequately demonstrate the efficiency of the presented methods.

Highlights

In this paper, we focus on the following tensor absolute value equation (TAVE): Axm−1 − |x|[m−1] = b, (1)
Tensor splitting iteration schemes type accelerated overrelaxation method (TAOR) and type successive overrelaxation method (TSOR) are proposed for solving the tensor absolute value equation (TAVE), which can be regarded as generalizations of accelerated over-relaxation method (AOR) and SAOR for linear systems
Some efficient preconditioned techniques are provided to improve the efficiency of solution-finding of TAVE

Summary

Introduction

We focus on the following tensor absolute value equation (TAVE): Axm−1 − |x|[m−1] = b,. Han [23] investigated a homotopy method by the Euler–Newton prediction–correction technique to solve multilinear systems with nonsymmetric M-tensors, which demonstrated a better result than the Newton method in the sense of convergence performances. Some comparison results for splitting iteration for solving multilinear systems were investigated widely in [25,26]. Du et al presented the tensor absolute value equation (TAVE) [27] and introduced the Levenberg–Marquardt method for solving this (1) from the point of an optimization theory.

Preliminaries

Numerical Experiments

Conclusions