Refining-Strategy And Deflation Technique Of Block Davidson Method For Solving Large Sparse Eigenproblems

Block Davidson method is very efficient for computing the extreme eigenpairs of large symmetric matrices.But it needs much memory, so restarting process is always used in it.This paper mainly studies the restarting process of the Davidson method,and proposes two new methods : refined block Davidson method and refined block Davidson method with deflation.To some eigenproblems, the corresponding Ritz vectors always converge very slowly even sometimes don't converge when Ritz values have converged. So in order to improve the convergence of Ritz vectors, firstly, refining -strategy is applied to the Davidson method in this paper. The process restarts with the refined Ritz vectors as the initial vectors, and then we get the refined block Davidson method. By analyzing the convergence of the new method, we give the theorem of it. Secondly, combined by a deflation technique and the refined block Davidson method, the refined block Davidson method with deflation is proposed.By means of numerical experiments, the block Lanczos method, the block Davidson method , the refined block Davidson method and the refined block Davidson method with deflation are compared. Though the refined block Davidson method is much better than the block Lanczos and Davidson methods , the refined block Davidson method with deflation is best. Nevertheless, the new methods all improves the convergence of block Davidson method greatly.