| With the development of tensor theories,the study of the eigenvalues of a tensor has attracted the attention of many scholars.Among them,the solution of the eigenvalues of a symmetric tensor has become a very popular research topic in recent years.Since the eigenvalues and eigenvectors of the multiple order symmetric tensors play an increasingly important role in graph matching,therefore,this paper mainly studies the problem of computing the eigenvalues and eigenvectors of the multiple order symmetric tensors.Firstly,we present the shifted higher-power method for computing the eigenvalues of the multiple order symmetric tensors.The ideal is that on the basis of the shifted symmetric higher-order power method(SS-HOPM),by constructing an auxiliary function with a shifted factor for converting the problem of computing the eigenvalues of the multiple order symmetric tensors into solving extreme points of this function,and the convergence of this algorithm is proved.Secondly,on the basis of the trust region strategy modification,we propose a Levenberg-Marquardt(LM)method to computing the eigenvalues of the multiple order symmetric tensors.The ideal of the proposed method is that the problem of solving the eigenvalues of the multiple order symmetric tensors is transformed into a nonlinear least squares problem.Then we prove the globaland local convergence of the method under a local error bound which is weaker than the non-singular condition.Finally,we verify the theoretical results of the two proposed algorithms by several numerical examples,and we use the results of the solutions of NSolve command to verify the effectiveness of the two algorithms.The numerical results show that the two algorithms are both effective,and the experimental results of the two algorithms are compared. |
