Iterative Methods For Computing Z-Eigenvalues Of Symmetric Tensor Group

Z-eigenvalues of symmetric tensor groups have important applications in hypergraph matching,hypergraph clustering,target tracking and data mining.This dissertation mainly studies the solution of Z-eigenvalue and Z-eigenvector of symmetric tensor group,and the obtained results are as follows:Firstly,the Z-eigenvalue problem of symmetric tensor group is transformed into the minimum problem of nonlinear function.When the cosine of the angle between Newton direction and the negative gradient direction of nonlinear function is less than a fixed value,the descent direction of Newton method is improved,and an improved Newton method for solving Z-eigenvalue of symmetric tensor group is proposed.It is proved that the improved Newton method is globally superlinear convergent.Secondly,based on Shifted Symmetric High Order Power Method(SS-HOPM)for solving the Z-eigenvalues of symmetric tensor group,the improved Newton method is applied to preprocess the initial values of SS-HOPM,and Newton Preconditioning SS-HOPM(NPSS-HOPM)for solving the Z-eigenvalues of symmetric tensor group is obtained.Finally,numerical examples show that the improved Newton method can calculate more Z-eigenvalues and Z-eigenvectors in a shorter time than SS-HOPM,that NPSS-HOPM can calculate more Z-eigenvalues and Z-eigenvectors than the improved Newton method with almost no increase in time,and that when different displacements are taken,the convergence speed of NPSS-HOPM is less affected than that of SS-HOPM.