Research On Some Methods Of Nonlinear Equations And Their Application To Tensor Eigenvalue Problems

Solving nonlinear equations is an important part of the optimization theory,and it has been widely applied in many fields of the national economy such as finance,trade,management,scientific research.Tensor eigenvalue problems take an important role in signal processing,medical resonance imaging,higher order Markov chains,elasticity in solid mechanical and so on.It is a prospective study to solve tensor eigenvalue problem by using the methods of nonlinear equations.This thesis mainly studies the algorithm for solving nonlinear equations and its application in tensor eigenvalue problems.Based on Chebyshev method,we propose a cubic convergent method for solving nonlinear equations.In the new method,three order tensor is replaced by the difference of Jacobian matrixes.Therefore,the new method reduces the storage and computational cost.We prove the global and cubic convergence of the algorithm.Numerical results indicate that the new method is effective and efficient for some classical test problems.We modify the cubic convergent method for solving nonlinear equations,and apply to find the largest H eigenvalue of a nonnegative irreducible tensor.Due to the particular structure of tensor,we combine Chebyshev method and the cubic method in Chapter ?.The global conver-gence and local cubic convergence are proved.Numerical results indicate that the performance of the new method is more stable and efficient.A quarticly convergent method for nonlinear equations is proposed,and applied to solving the generalized tensor eigenvalues,which is an extension of cubically convergent method.The global and quartic convergence of the algorithm are proved.We compute H eigenvalues and Z eigenvalues of tensor in numerical experiment,respectively.The data show that the new method is more stable and all Z eigenvalues are found.At last,we propose a modified CG method for the generalized eigenvalue of large scale sparse tensors.We modify the derivative-free PRP conjugate gradient method,and determine the descent property of the search direction by using Jacobian,which just needs a few extra computation,so the new method is still suitable for the solving general problems.The global convergence is established.