Matrix is an important tool of mathematics,and its theory is applied to every aspect of science and engineering.As a high-order extension of the matrix,tensor is widely applied in signal and image processing,nuclear magnetic resonance and wireless communication.Research on the high order tensor algorithms for eigenvalue and complementarity problem,not only has important theoretical significance,but also has important application value.In this paper,the algorithms for solving Z-eigenvalues of higher order symmetric tensors and two methods of higher order tensor complementarity problems are mainly studied.The results are as follows:Firstly,a sequential unconstrained DFP method for solving high order tensor Z-eigenvalues is proposed,and its convergence theory also is proved.Secondly,for tensor complementarity problems,we propose smoothing Newton method and non-smooth quasi-Newton method and get numerical solutions.Next,we prove the convergence of the two methods.Finally,a large number of numerical experiments and algorithms comparison show that: Sequential Unconstrained DFP methods for solving higher-order Tensor Z-eigenvalues are very effective and more than Newton and SS-HOPM;Smoothing Newton method and non-smooth quasi-Newton method for solving tensor complementarity problem are also effective. |