Variable Step-Size Newton-Raphson Iterative Algorithm for Solving Multi-linear Systems with $\mathcal{M}$ -tensors

Abstract

In this paper, the major work is to investigate how the effect of step-size on the convergence performance of the Newton-Raphson iterative algorithm for solving multi-linear systems with $\mathcal{M}$ -tensors. Via utilizing the discrete-time Lyapunov theory, it is revealed that the range of the step-size is between 0 and 2. Only in this way can ensure the convergence of Newton-Raphson iterative algorithm. Additionally, when $\gamma=1$ , that is, the traditional Newton-Raphson iterative algorithm possessing the fastest convergence speed for solving multi-linear systems with $\mathcal{M}$ -tensors. Simultaneously, the corresponding numerical examples about the multi-linear systems with $\mathcal{M}$ -tensors with 3-order 10-dimensional synthesized by the variable step-size Newton-Raphson iterative algorithm are performed, whose results substantiate and support above the mentioned conclusions.