Fast and flexible preconditioners for solving multilinear systems

TL;DR

This paper introduces fast, flexible preconditioners for multilinear systems involving -tensors, demonstrating their theoretical convergence benefits and practical efficiency through numerical experiments.

Contribution

It proposes new preconditioners for multilinear systems and proves their effectiveness in accelerating iterative methods.

Findings

Preconditioners improve convergence speed of iterative methods.

Theoretical convergence theorems support the effectiveness of the preconditioners.

Numerical examples confirm the practical efficiency of the proposed methods.

Abstract

This paper investigates a type of fast and flexible preconditioners to solve multilinear system $\mathcal{A}\textbf{x}^{m-1}=\textbf{b}$ with $\mathcal{M}$-tensor $\mathcal{A}$ and obtains some important convergent theorems about preconditioned Jacobi, Gauss-Seidel and SOR type iterative methods. The main results theoretically prove that the preconditioners can accelerate the convergence of iterations. Numerical examples are presented to reverify the efficiency of the proposed preconditioned methods.