Malaysian Journal of Mathematical Sciences, December 2023, Vol. 17, No. 4
Richardson Iterative Method for Solving Multi-Linear System with $\mathcal{M}$-Tensor
Liang, Y., Ibrahim, A., and Omar, Z.
Corresponding Email: yanliang@m.scnu.edu.cn
Received date: 27 June 2023
Accepted date: 7 November 2023
Abstract:
In this paper, Richardson iterative method is employed to solve $\mathcal{M}$-Equation. In order to guarantee the solution can be found, convergence theorems are established and confirmed numerically. The optimal $\alpha$, which is a parameter of Richardson iterative method that can provide the best convergence rate, is also determined theoretically and numerically. Furthermore, a theorem establishing the range of initial vector for general splitting methods is extended from the range in past study. To further accelerate the convergence rate, Anderson accelerator and three preconditioners are incorporated into Richardson iterative method. Numerical results reveal that by including these accelerators, the convergence rates are enhanced. Finally, we show that Richardson iterative methods with optimal $\alpha$ perform better than the SOR type methods in past studies in terms of number of iterative steps and CPU time.
Keywords: Richardson iteration; $\mathcal{M}$-tensor; multi-linear system; Anderson acceleration; preconditioned technique
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