An Implicitly Restarted Block Arnoldi Method In A Vector-wise Fashion

Many scientific applications lead to large-scale eigenvalue problems, where typically only a few eigenvalues are of interest. For such problems Krylov methods are well suited. One of the Krylov methods called Arnoldi method is a good iterative method for approximating a few eigenvalues of large matrices. Sorensen devised an approach for the single-vector implicitly restarted Arnoldi method [24]. Later, the implicitly restarted technique was generalized to block Arnoldi by Lehoucq and Maschhoff [10]. The resulting method has good convergence properties, but Lehoucq and Maschhoff must take special care not to disturb the Hessenberg structure. In paper [6] which is about the model reduction problem, Freund introduces an algorithm which implements the block Arnoldi method in a vector-wise fashion, as opposed to the block-wise construction. The vector-wise construction is preferable to the block-wise construction because it greatly simplifies both the detection of necessary deflation and the actual deflation itself.Our main work is the development of an implicitly restarted block Arnoldi algorithm in a vector-wise fashion to solve the large-scale eigenproblems. We point out that the implicitly restarted technique can be combined with the vector-wise block Arnoldi method (Theorem 3.2.2), and the method in vector-wise fashion converges much faster than the method in block-wise fashion (Theorem 3.2.4), since the later one must preserve the Hessenberg structure while the former one doesn't have this limit any more. Lastly, our numerical experiments shows that our algorithm has the advantages of both the blockmethod and the vector-wise fashion, it is efficient for solving the multiple and/or clustered real eigenvalues, and it converges rather fast.In Chapter 1, we give an overview of the background of eigenvalue problems and the research history of solving eigenvalue problems in Section 1. In Section 2, we introduce the block Arnoldi process and the advantages and disadvantages between block and non-block methods. In Section 3, we explain the implicitly restarted block Arnoldi method. In Chapter 2, we give some notations and the vector-wise block Arnoldi process which is introduced in [6]. We also give an example for better understanding of the process. In Chapter 3 we first prove that the implicitly restarted technique can be combined with the vector-wise block Arnoldi method, then we develop an implicitly restarted block Arnoldi algorithm in a vector-wise fashion to solve the large-scale eigenproblems. Lastly, we present the superiority of our vector-wise implicitly restarted block Arnoldi method for the computation of eigenvalues, using both the theory and numerical experiments.