Inexact Methods For Nonlinear Equations And Inverse Eigenvalue Problems

The convergence analysis of inexact methods and the solutions of inverse eigenvalue problems are given.The local convergence of the inexact method and the semi-local convergence of the inexact Newton method are analyzed and moreover,some algorithms for solving inverse eigenvalue problems are presented and the convergence analysis are provided.The main work done in this dissertation is organized as follows.In Chapter 1,we shall study the convergence of inexact methods for solving nonlinear operator equations.Under a type of residual controls and the H(o|¨)lder condition of the first derivative,a local convergence analysis for the inexact method is presented.The radius of the convergence ball is estimated,and the linear and/or superlinear convergence property is proved.Some applications to special cases are provided which include the inexact Newton method and the case when the H(o|¨)lder condition is not satisfied.Furthermore,under another kind of residual controls and the constant Lipschitz condition,a Kantorovich-type theorem for the inexact Newton method is established.Implicit and explicit convergence criteria are included in our results.Some applications to special cases,which include the residual controls of Guo and Newton's method are given.In Chapter 2,we will study the solutions of inverse eigenvalue problems.Motivated by Moser's method and Ulm's method,we propose some algorithms for solving the IEP and present the convergence analysis of these algorithms.We propose a Moser-like method for solving the IEP where the approximate eigenvectors are obtained by the one step inverse power method.Approximations to the inverses of the Jacobian matrices are provided and hence the approximate Jacobian equations are avoided to solve in outer iterations.Under the assumption that the given eigenvalues are distinct,we show that the Moser-like method converges quadratically.Numerical experiments are given and comparisons with the inexact Newton-like method are made.Furthermore,we propose a deformed Cayley transform method for solving the IEP where the approximate eigenvectors are obtained by applying Cayley transforms and matrix exponentials.Under the assumption that the given eigenvalues are distinct,we also show that this method converges quadratically.Numerical experiments are given and comparisons with the inexact Cayley transform method are made.