This paper proposes an algorithm based on the Shultz iterative method and divided differences to find the roots of nonlinear equations. Using the new method, we present a rapid and powerful algorithm to compute an approximate inverse of an invertible tensor. Analysis of the convergence error shows that the convergence order of the method is a linear combination of the Fibonacci sequence and also is rapid and powerful in finding and keeping sparsity of the obtained approximate inverse of the sparse tensors. The algorithm is extended for computing the Moore-Penrose inverse of a tensor. As an application, we use the iterates obtained by the algorithm as a preconditioner for the tensorized Krylov subspace method, e.g., LSQR based tensor form to solve the multilinear system $\mathcal{A}{★}_N\mathcal{X}=\mathcal{B}$. Several examples are also provided to show the efficiency of the proposed method. Finally, some concluding remarks are given.

本文提出了一种基于Shultz迭代法和除法求非线性方程根的算法。使用新方法，我们提出了一种快速而强大的算法来计算可逆张量的近似逆。对收敛误差的分析表明，该方法的收敛阶次是斐波那契数列的线性组合，而且在求得稀疏张量近似逆的稀疏性和保持稀疏性方面快速而有力。该算法被扩展用于计算张量的 Moore-Penrose 逆。作为应用，我们使用算法获得的迭代作为张量化 Krylov 子空间方法的预处理器，例如基于 LSQR 的张量形式来求解多线性系统$\mathcal{一种}{★}_N\mathcal{X}=\mathcal{乙}$. 还提供了几个例子来显示所提出方法的效率。最后，给出了一些结论性意见。