ABSTRACT

In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the $\mathcal{M}$-tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order $\mathcal{M}$-tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the $\mathcal{M}$-tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.

摘要

本文研究了具有爱因斯坦积的四阶张量 Riccati 方程。使用爱因斯坦积，我们引入$\mathcal{米}$-张量，谱和块张量乘法。我们还提出了四阶置换张量的表达式，并将 Riccati 方程从矩阵推广到张量情况。四阶解的存在唯一性$\mathcal{米}$-张量Riccati方程进行了研究。此外，我们提供了对混合条件数和分量条件数的扰动分析$\mathcal{米}$-张量 Riccati 方程。提出了一种求解方程的算法。数值例子展示了我们算法的有效性。