For Solving Large-scale Non-symmetric Matrix Eigenvalue Problem Two Refined Version Of The Lanczos Method

Large unsymmetric eigenproblems arise in many applications in scientific-and engineering computing. Theoretical analysis and numerical experiments show that the classical .orthogonal and the oblique projection methods have potential danger in finding eigenvectors of the matrix involved. That is, Ritz vectors may not converge to the desired eigenvectors even if the corresponding Ritz values converge. In order to circumvent the flaw, Jia suggested to use certain refined vectors to approximate the desired eigenvectors. The resulting refined projection methods avoid the above danger and converge faster. It is important to combine the refined strategy with many other techniques or methods. More robust and reliable algorithms can be derived from the combination. This thesis consists of the following parts.1.The quasi-refined biorthogonalization Lanczos method is proposed according to the refined projection strategy and the quasi-refined idea. Moreover, we discuss the relationship between the residual norm of the quasi-refined approximate eigenpair and that of the refined eigenpair, and the quasi-refined biorthogonalization Lanczos algorithm is given. Some numerical experiments are carried on, and the results illustrate that the newalgorithm is superior to the classic one. 2.We propose the semi-refined biorthogonalization Lanczos method which is based on the refined projection idea. Furthermore, the relationship between the residual norm of the semi-refined approximate eigenpair and that of the approximate refined eigenpair is discussed. The semi-refined biorthogonalization Lanczos algorithm is given. We carry on some numerical experiments, and the results show that the new algorithm is more efficient than its conventional counterpart.