A Hybrid Algorithm To Compute Z-eigenvalues Of Symmetric Tensors

A 1-order array is a vector,a 2-order array a matrix,a 3-order or more than 3-order array a tensor.As is well known,eigenvalues of matrix have many significant applications.Eigenvalues of tensors also find many applications,such as signal processing,data analysis,image analysis,higher order Markov chains and so on.In 2005,professor Qi Liqun and Lim have proposed independently the definitions of eigenvalues and eigenvectors for high order symmetric tensors,and from then on tensor's eigenvalue problems have drawn wide attention of many researchers from home and abroad.In this thesis,a hybrid algorithm is proposed to calculate extreme Z-eigenvalues of sym-metric tensors,which is based on the feasible trust region method and the sequential subspace projection method.In section 1,we introduce some methods to solve the different eigenvalues of symmetric tensor.In section 2,we firstly introduce the definitions of tensor and its eigenvalues,existing methods for computing eigenvalues of symmetric tensors and related applications.And then we propose the hybrid algorithm to compute extreme Z-eigenvalues of symmetric tensors.The main idea of hybrid algorithm is to employ feasible trust region method and sequential subspace projection method alternatively.The main idea of the feasible trust region method is to transfor-m the high-order high-dimension tensor's problems into a sequence of 2-order high-dimension subproblems in current point xk,that is to say,order reduction.The main thought of the sequen-tial subspace projection method is to transform the high-order high-dimension tensor's problems into a sequence of high-order 2-dimension subproblems in current point Xk,that is to say,di-mension reduction.The hybrid algorithm reduces order in even step and dimension in odd step,namely,reducing order and dimension alternatively.In section 3,we give the new algorithm and do numerical experiment for 5 examples.Compared with feasible trust region method and sequential subspace projection method,the hybrid algorithm finds the extreme Z-eigenvalues with higher probability,takes fewer iterations and less CPU time to some extent.