Numerical Methods For Nonlinear Eigenvalue Problems And Their Applications

The nonlinear eigenvalue problems(NEPs)arise in a number of various applications,such as numerical simulation of quantum dots,electromagnetic field computation,vibration analysis of viscoelastic structures,stability analysis of time-delay systems and so on.Studies on these problems have important theoretical significance and application value.In this dissertation,we consider some numerical methods for solving the NEPs and concern the stability of n-dimensional time-delay systems by using contour integral method.The main contributions are as follows.We propose a successive m th(7)m ?2(8)approximation method for solving the NEPs,and analyze the local convergence of the successive m th approximation method.To solve the large-scale polynomial eigenvalue problems(PEPs)caused by the successive m th approximation method,we present a partially orthogonal projection method with the successive m th approximation for solving the NEPs by using the partially orthogonal projection method for the PEPs.In order to reduce the computational cost of Newton method for solving the NEPs based on singular value decomposition(SVD),we consider approximating them by using inverse iteration instead of computing accurately the smallest singular value and the associated left and right singular vectors at each step,present a modification of the Newton method,and establish the locally quadratic convergence of the modified Newton method.The numerical method of determining the algebraic multiplicity of the eigenvalues of the NEPs in a given region is studied.We give an extension of the argument principle,and propose a numerical method for determining the algebraic multiplicity of the eigenvalues of the NEPs in a given region.To compute all eigenvalues of the PEPs in an open half plane,we transform a PEP in the open half plane to another PEP in the open unit disc by fractional linear mapping,establish the relation between all the eigenvalues of the two problems,and develop the contour integral method for computing all eigenvalues of the PEPs in the open half plane.For n-dimensional linear time-delay systems,we propose a numerical method for computing the rightmost eigenvalues of these systems by using the contour integral method,and present a novel method to study the local asymptotic stability of these systems at the positive equilibrium.For n-dimensional nonlinear time-delay Lotka-Volterra system,a numerical method for determining the local asymptotic stability of the system at positive equilibrium is developed.Numerical results show that the proposed numerical methods for solving the NEPs are efficient.