基于预处理分裂迭代算法求解Mz张量方程组

Solving the Multi-Linear System with Mz-Tensor via Preconditioned Tensor Splitting Iteration

偶数阶张量的正定性和半定性由于在谱超图理论、自动控制、多项式理论、随机过程、磁共振成像等众多领域均有着广泛的应用,近年来得到了深入而广泛的研究。研究表明,M张量、B张量、H张量、希尔伯特张量以及随机张量在特定条件下可以是正定的。然而,仍有许多正定张量无法通过上述准则来确定。在本文中,我们提出了一类全新的正定张量,其非对角项相对于强M张量可以为正,我们将其称为强Mz张量。这类强Mz张量可从离散微分方程中产生,原因在于它是基于Z特征值而非传统的H特征值。由于Z特征值定义中涉及的非齐次性通常比H特征值更难处理相应的问题,因此Z特征值的计算结果远远小于h本征值的计算结果。定义在Z特征值基础上的Mz张量在结构上比已被广泛研究的M张量更复杂。对Mz张量的研究几乎是一片空白。到目前为止,还没有人提出求解强Mz方程的算法。作为本文的动机之一,我们创新性地利用张量迭代给出了求解系数张量为Mz张量的多线性系统的张量分裂迭代算法。其次,给出了相应的预处理分裂迭代算法以及理论分析。最后,通过数值算例说明了所提算法的有效性。这在图像处理以及物理领域等多维信息分析方面,乃至高阶非线性系统的稳定性研究等领域均具有广泛的应用前景。The positive definiteness and semi-definiteness of even-order tensors have received extensive and in-depth research in recent years due to their wide applications in various fields such as spectral hypergraph theory, automatic control, polynomial theory, stochastic processes, and magnetic resonance imaging. Studies have shown that M-tensors, B-tensors, H-tensors, Hilbert tensors, and random tensors can be positive definite under specific conditions. However, there are still many positive definite tensors that cannot be determined by the above criteria. In this paper, we propose a new class of positive definite tensors whose off-diagonal terms can be positive relative to strong M-tensors. We call them strong Mz-tensor. This kind of strong Mz-tensor can be generated from discrete differential equations because it is based on Z-eigenvalues rather than the traditional H-eigenvalues. Due to the non-homogeneity involved in the definition of Z-eigenvalues, which is usually more difficult to handle corresponding problems than H-eigenvalues, the computational results of Z-eigenvalues are much smaller than those of h-eigenvalues. The Mz-tensor defined on the basis of Z-eigenvalues is structurally more complex than the widely studied M-tensor. The research on Mz-tensor is almost a blank. So far, no one has proposed an algorithm for solving strong Mz equations. As one of the motivations of this paper, we innovatively use tensor iteration to give a tensor splitting iterative algorithm for solving multilinear systems with coefficient tensors being Mz-tensor. Secondly, the corresponding preconditioned splitting iterative algorithm and theoretical analysis are given. Finally, numerical examples illustrate the effectiveness of the proposed algorithm. This has broad application prospects in multi-dimensional information analysis such as image processing and the physical field, and even in the stability research of high-order nonlinear systems.