Some Research About The Property Of Refined Approximate Eigenvector Problems

This thesis mainly presents the property of the refined eigenvectors,Including the orthogonality of the refined vectors when solving the symmetric eigenvalue problems and how to solving matrix double eigenvalue problems by the refined Anoldi method.It consists of three parts.Chapter one gives the background of large matrices eigenproblems and basic numerical algorithms for solving them.we review the state of the art of this subject.Finally,we describe the work of this thesis.Chapter two investigates the orthogonality of the refined Ritz vectors.How to get a group of approximate eigenvectors which could get to the machine tolerance for symmetric matrices by the refined Arnoldi method.First,we give a new expression for the refined Ritz vectors,it implies that,generally,fbr the different approximate eigenvectors,we can not promise the refined Arnoldi method and the refined Ritz vectors are orthogonal.furthermore,apply the orthogonal method again,we could get a group of standard orthogonal eigenvalues which could get to the machine tolerance.Finally,the numerical results prove the accuracy of the method,meanwhile,the new approximate residuals remain the same after the reorthogonal. In Chapter three,we investigate how to get the approximate eigenvalue and to estimate the multiply.First,we investigate the property of the approximate eigenvalue.it shows that we can't get the multiply of the given eigenvalue directly. What's more,in this chanpter,we solve this multiply by a refined Arnoldi method.the numerical result,s prove the accuracy of this method.