Some Theories And Algorithms For Solving Large Unsymmetric Matrix Eigenproblems

This thesis presents several theories and algorithms for large unsymmetric matrix eigenproblems. It consists of four parts.Chapter one gives the background of large unsymmetric matrix eigenproblems and basic numerical algorithms for solving them. We review the state of the art of this subject. Finallly, we describe the work of this thesis.Chapter two investigates a variant on harmonic Arnoldi method. It is well known that the m-step Arnoldi process constructs an orthonomal basis of the Krylov subspace K_m+1(A,v₁) in practice as well as the restricted matrix H_m,, that is, the (m +1)th basis v_m+1 is already available. However, conventional harmonic Arnoldi method uses harmonic Ritz vectors as approximate eigenvectors. Thus the (m +1)th basis v_m+1 is wasted and it contributes nothing to the wanted eigenvectors. The modified harmonic Arnoldi methods retains harmonic Ritz value as the approximate eigenvalue, while the approximate eigenvectors are formed by an ingenious linear combination of the original harmonic Ritz vectors and the (m +1)th basis vector v_m+1. These new vectors are linear combinations of the harmonic Ritz vectors and v_m+1, such that the residual norms of the modified approximate eigenpairs are minimal in some degree. Theoretical analysis shows the efficiency of the new method. Finally, numerical results also confirm the efficiency of the modified harmonic method.In Chapter three we study the modified block harmonic Arnoldi method. Suppose that the upper Hessenberg matrix H_m which is obtained in the m-step Arnoldi process is irreducible, i. e. the offdiagonal elements are nonzero. Thenthe multiple harmonic Ritz values of A have only one harmonic Ritz vector associated with them, so harmonic Arnoldi method is inefficient for multiple eigenvalue problems. Moreover, harmonic Arnoldi method is also inefficient in computing clustered eigenvalues. In order to calculate the clustered eigenvalues that lie in the interior of the spectrum, block harmonic Arnoldi methods is presented. Comparing with the standard block Arnoldi method, modified block harmonic Arnoldi method utilizes harmonic Ritz values as the approximation of the required eigenvalues, while in the formation of approximate eigenvectors, it makes full use of the basis information and choose a new vector which is linear conbinations of harmonic Ritz vector and block basis Vm+i-called modified harmonic Ritz vector-to generate better approximation. Theoretical analysis shows the efficiency of the new modified method.In Chapter four we discuss the eigenproblems of large scale arrowhead matrices. An arrowhead matrix has such a property which is zeros except for its main diagonal and the zth row and the zth column. An algorithm for computing all the eigenvalues and eigenvectors of such matrices is presented. Taking advantage of the matrix structure, we convert the eigenproblems to a polynomial equation problems of the degree at most n. In addition, we point out that the computation for one simple eigenvalue is completely independent of others. Our algorithms can be implemented parallely. Rounding error analysis shows the stability of the new method.