The Algorithm And Theory Research On Symmetric Tensor Eigenvalue Problem

In past decade,eigenvalues of symmetric tensor have become increasingly important in the area of numerical multilinear algebra.Many authors have solved the problem of symmetric tensor eigenvalue problems by translating them into an optimization problem.In this thesis,we first introduce the shifted power method(SS-HOPM)for computing symmetric tensor Z-eigenvalues and adaptive shifted power method(GEAP)for computing generalized tensor eigenproblems.Then,two new methods are presented which can be recognized as acceleration convergence method from the power method of matrix eigenvalues.The one is to increase the eigenvalue at every iteration to speed up the convergence process,which is called an improved generalized eigenproblem adaptive power method(IGEAP).The convergence of the method is proved.The other is based on the Aitken extrapolation technique when the number of iteration is greater than six and is a multiple of,called generalized eigenproblem adaptive power method based on Aitken extrapolation(AGEAP).The AGEAP method is proved that the sequence{}can converge to attracting fixed point_*.Numerical examples of the three methods are given.Further comparison and analysis of these methods are made.The results of these examples show that the AGEAP method and the IGEAP method are superior to the GEAP method,and the AGEAP and the IGEAP method have their own advantages.