Two Algorithms For Computing Eigenvalues Of Symmetric Tensors

Finding the eigenvalues of a symmetric tensor is an important topic in recent years.The dissertation mainly studies algorithms on eigenvalues problems of symmetric tensors.Firstly,a Newton sequential subspace projection method(NSSPM)is proposed to calcu-late the extreme Z-eigenvalues and corresponding eigenvectors of a real symmetric tensor.It is based on sequential subspace projection method(SSPM).SSPM algorithm solves the problem by constructing a two-dimensional subspace and then projecting the original optimization prob-lem into the subspace.The two-dimensional subspace is formed by the current iteration point and the gradient direction,and in this dissertation it is replaced by Newton direction.Com-pared with SSPM,the results of computing the extreme Z-eigenvalue of a symmetric tensor are improved for some tensors.The second problem is to solve all H-eigenvalues of a symmetric tensor.In the dissertation it is transformed into solving an equivalent system of nonlinear equations,and then solved by quasi-Newton method to generate iterative sequence.The numerical results show the method is promising.The dissertation is divided into three parts.The first chapter introduces the concepts of tensors,eigenvalues of tensor,methods of computing eigenvalues of symmetric tensors and their applications.The second chapter reviews SSPM of calculating the smallest/largest Z-eigenvalue and then introduces NSSPM and show the results of numerical tests.The third chapter introduces the quasi-newton method for computing H-eigenvalues of symmetric tensors,and the results of numerical experiments are presented.