Eigenvalue Decomposition Algorithm Of Complex Tensors

With the rapid development of information age,tensor problem has become a hot research object in various fields,and the eigenvalue problem of complex tensors has important research significance in the field of quantum entanglement.This paper is mainly concerned with the calculation of US-eigenpairs of symmetric complex tensors.The US-eigenvalue can be regarded as a generalization of the Z-eigenvalue from the real number field to the complex number field,which is closely related to the problem of the optimal complex-rank one approximation of the higher order tensors.At present,there are many methods to calculate the eigenpair of complex tensors,but they all have their own advantages in the aspects of the number of eigenpairs,the maximum eigenvalues and the computational accuracy.In this paper,the feature pairs of complex tensors are calculated by means of the algorithm in the optimization theory,and the BFGS method proposed by Bai et al.is used as a comparison algorithm to give the advantages and disadvantages of each algorithm from the aspects of maximum feature value,successful rate and running time.Firstly,this paper gives the second-order complex Taylor expansion of the real value complex variable function by using the knowledge of Wirtinger calculus.Starting from the problem of equivalent nonlinear equations solved by the eigenvalues of complex tensors,the Damped-Newton method for solving nonlinear equations in the real number field is extended to the complex field,a method for calculating the complex tensors US-eigenpairs is proposed,and the convergence and speed of the algorithm are given.It can be seen from the numerical example that the maximum eigenvalue calculated by this algorithm is the same as that calculated by BFGS method,and the successful rate is improved by up to 18%.Then,this paper transforms the problem of nonlinear equations into the least square problem,makes full use of the second derivative information of the objective function,and extends the classical Levenberg-Marquardt method in the real number field to the complex number field.This algorithm not only has the advantages of gradient method and Newton method,but also solves the problem of coefficient matrix singularity in the calculation process,and also gives the convergence and speed of the algorithm.It can be seen from the results of numerical experiments that this algorithm not only guarantees the successful rate of the Damped-Newton method,but also reduces the running speed by up to 98%compared with the Damped-Newton method.Compared with the BFGS method,the operating speed was reduced by up to 89%.Finally,using the relationship between the eigenvalues of asymmetric complex tensors and the eigenvalues of embedded symmetric complex tensors,the algorithms proposed in this paper are applied to the calculation of U-eigenpairs of asymmetric complex tensors.It can also be seen from the numerical examples that the eigenvalues obtained by the algorithm are not only the same as those obtained by the BFGS method,but also improved by the Damped-Newton method by 18.2%in successful rate.Levenberg-Marquardt improved its successful rate by 18.4%while reducing its speed by 84%.The advantages of the proposed algorithm are fully proved.