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The tensor splitting with application to solve multi-linear systems. (English) Zbl 1376.65040

Summary: In this paper, firstly, we introduce the variant tensor splittings, and present some equivalent conditions for a strong \(\mathcal{M}\)-tensor based on the tensor splitting. Secondly, the existence and uniqueness conditions of the solution for multi-linear systems are given. Thirdly, we propose some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong \(\mathcal{M}\)-tensor. As an application, a tensor splitting algorithm for solving the multi-linear model of higher order Markov chains is proposed. Numerical examples are given to demonstrate the efficiency of the proposed algorithms.

MSC:

65F10 Iterative numerical methods for linear systems
15A69 Multilinear algebra, tensor calculus
65C40 Numerical analysis or methods applied to Markov chains
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