The Technique Of Acceleration And Preconditioning For Solving Large Symmetric Eigenvalue Problems

Block Davidson method is very efficient and favorable for large symmetric eigenvalue problems. But it is actually the combination of the preconditioning process and Rayleigh-Ritz process, and its success or not lies in the choices of the preconditi- oning matrices and the restarting vectors, so aiming at the both sides, this paper mainly studies the following work:In order to make the preconditioning matrix close to A as far as possible and easy to inverse, the third chapter of this paper brings up a new method of the choice of the preconditioning matrices.The approximate eigenvectors or Ritz vectors obtained by the block Davidson method may converge very slowly and even fail to converge even if the corresponding Ritz values do. In order to improve it, so the fourth chapter in this paper proposes four algorithms to remedy this shortcoming, furtherly accelerating the convergence of the block Davidson method. Firstly, the first section gives out a new accelerated-strategy to improve approximating eigenvectors, and also provides its theoretical analysis. According to the theory of refined-strategy, the second section exploits refined vectors to expand subspace, and revises the preconditioning equation, thereby brings up a improved refined block algorithm; In the end, combining the new accelerated-strategy and refined-strategy, we propose a refined and accelerated block Davidson method, by means of numerical experiments, the new method is more efficient than the improved refined block Davidson method and the improved block Davidson.