The Estimations Of The Eigenpairs Of Tensors And Related Algorithms

The eigenvalues of tensors have many applications in automatical control,spectral hypergraph theory,quantum entanglement,magnet-ic resonance imaging and high-order Markov chains.Computing the eigenpairs(i.e.,eigenvalues and eigenvectors)of a higher order tensor is equivalent to finding nontrivial solutions of a system of homogeneous or inhomogeneous polynomial equations in several variables.However,the task in the computation of eigenvalues and eigenvectors of a tensor will be much more difficult when its order and dimension are very large.Hence,It is very important to study the estimates and numerical algorithms of the eigenpairs of tensors.This thesis mainly studies the estimates of the Z-eigenpairs for irreducible nonnegative tensors and some effective algo-rithms for calculating generalized eigenpairs and Z-eigenpairs of(weak)symmetric tensors.The details are given as follows:Firstly,some new lower and upper bounds for the Z-eigenvector and Z-spectral radius of an irreducible(weakly symmetric)nonnegative tensors are provided,which mainly generalize and improve the ones in[Linear Algebra Appl.483:182-199,2015].Besides,some upper bounds for Z1-eigenvalues of general tensors are specifically presented.Some numerical examples are given to show that our bounds are better than the existing ones,and are attainable.Secondly,we are concerned with computing Z-eigenpairs of sym-metric tensors.We show that computing Z-eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear sys-tem of equations,and propose a modified normalized Newton method(MNNM)for it.Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions,which great-ly improves the Newton correction method and the orthogonal Newton correction method with quadratic convergence rate in[SIAM J.Matrix Anal.Appl.,39:1071-1094,2018].As an application of the MNNM method,we calculate the US-eigenpairs of a complex-valued symmetric tensor arising from the computation of quantum entanglement in quan-tum physics.Some numerical experiments are performed to illustrate the efficiency and effectiveness of our proposed method.Thirdly,we propose a spectral residual method(SREIG)for com-puting generalized eigenpairs of weakly symmetric tensors.Under appro-priate conditions,the globally convergence of this method is established.The performed numerical examples illustrate that compared with the spectral gradient projection method in[Comput.Optim.Appl.66:285-307,2017],our method is more efficient and can find more(even all)generalized eigenvalues of weakly symmetric tensors by running it mul-tiple times with different starting points.Finally,in order to improve the convergence rate of the above S-REIG method,based on the MNNM method mentioned previously,we transform equivalently computing generalized eigenpairs of a symmet-ric tensor into finding the nonzero solutions of a nonlinear system of equations,and propose a normalized Newton method(NNMEIG)for it.The local and quadratic convergence of this method is established un-der the generalized-y-Newton-stability condition that can guarantee the nonsingularity of the Jacobian matrix of this nonlinear system of equa-tions.However,in general,the nonsingularity of the Jacobian matrix is too strong.Hence we present a modified Levenberg-Marquardt method(MLMEIG)for computing generalized eigenpairs of symmetric tensors.Using a local error bound condition that is weaker than the nonsingu-larity of the Jacobian matrix,we prove that the MLMEIG method is locally and quadratically convergent.Numerical examples are given to illustrate that our proposed methods are more effective and efficient than the SREIG method and the other existing ones.