Some Research about the Property of Refined Approximate Eigenvector Problems
|Keywords||Eigenvalue problem Arnoldi method Refined Arnoldi method Refined Ritz vectors|
This paper studies the fine approximation of the nature of the feature vectors , including solving symmetric eigenvalue problem between the refined vector orthogonal refined Arnoldi method and how to solve the matrix multiple eigenvalues ??. The text is divided into three chapters . The first chapter describes the source of the problem of large-scale matrix , the basic method to solve this kind of problem and the development of the discipline dynamic , and outlines the main work of this paper . The second chapter studies the refined Ritz vector orthogonal . In finite precision , how refined Arnoldi method seeking a set of orthogonal symmetric matrix extent can reach the the machine precision approximate eigenvectors group of . This chapter first gives a new expression of the refined Ritz vectors , the expression that theoretically different characteristics approximate value , generally can not be guaranteed refined Arnoldi method to determine the refined Ritz vector orthogonal group . Further mining orthogonal method can get a set of orthogonal degree of machine precision can reach the standard orthogonal approximate eigenvectors group . Finally , the numerical results to verify the accuracy of the conclusions obtained after orthogonal approximation of the amount of residue is unchanged . The third chapter studies how refined Arnoldi method gives the eigenvalues ??approximation and estimate its multiplicity . This chapter first study refined approximation eigenvectors of nature , the nature that refined Arnoldi method can not be determined directly repeated characteristic roots of multiplicity . Further , the chapter presents a refined Arnoldi method to determine the multiplicity of the eigenvalue algorithm, numerical case shows the feasibility of the algorithm .