A New Algorithm For Solving Matrix Eigenvalue Problem - Nonlinear Algorithm

Eigenvalue problem looks as a simple one. In feet, it is not. Eigenvalue problem's elementary thoeries have been acknowledged by people, but when calculating eigenvalue speedly and effectively, we offen confront with many challenged questions.This paper presents a new method for solving eigenpairs on the basis of the pioneering worker that we call it nonlinear algorithm for convenience. The method converts eigenvalue problem into nonlinear equations one. In order to improve Newton iteration convergence rate, it introduces homotopy ideas and utilizing homotopy curves traces eigenvalue. Since it solves the problem of Newton iteration starting value, the method can be executed effectively. The paper compares nonlinear algorithm with Jacobi and QR. We can see that the method is very good from these examples. In addition, the algorithm has the advantage of working in parallel.However, because Newton iteration is not converges in the case of image eigenvalue, the method can only evalue matrix's real eigenvalues and eigenvectors.The paper discusses nonlinear algorithm systematically. The third chapter analyses the convergevence and stability of the algorithm. The forth chapter analyses the parallel of the algorithm and how to realize. The fifth chapter presents some numerial example, from which we canfind more traits.The sixth chapter of the paper compare nonlinear algorithm with Jacobi and QR.From these we find that the caculatmg quantity is O(n4). But,because it can parallel running completely and generally n^ 10 when calculating in engineer,the algorithm is feasible completely today when computer technique is developed largely.