Preconditioning Techniques And Iterative Algorithms For Eigenvalue Problems

Eigenvalue problems are important,in the area of science and engineering.For solving large sparse eigenvalue problem,iteration methods are usually preferred,such as Gradient type methods,Krylov subspace methods,Davidson type methods,etc.The main content of this thesis is to study the preconditioning techniques and corresponding iterative methods for Jacobi-Davidson method and Newton type method.Firstly,for symmetric eigenvalue problems,two preconditioners based on a regularization method and shift-splitting of the saddle point matrix are given in each inner iteration step of Newton method.Then,the spectral properties of the corresponding preconditioned matrix are discussed and the convergence of the method is investigated.Finally,the efficiency of preconditioners is illustrated through numerical experiments.Restart strategy of Jacobi-Davidson method is also investigated.Secondly,for the nonsymmetric eigen-problems,a sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system.The properties of the preconditioned matrix are investigated.Numerical results are given which reveal the influence of the new proposed preconditioner.Furthermore,for the generalized eigen-problems,we prove local quadratic convergence of the inexact simplified Jacobi-Davidson method when the involved correction equation is solved by a Krylov subspace iteration.This method then shows local cubic convergence rate when the correction equation is solved to a prescribed precision proportional to the square of the norm of the current residual.Numerical experiments are given to demonstrate the theory analysis.Finally,we propose a successive quadratic approximations method combined with contour integral method for computing the rightmost characteristic roots of systems of linear time-delay differential equations.These roots are very important in the stability analysis of the time-delay systems.The effectiveness of the proposed method is illustrated by some numerical experiments.