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

Davidson method and Newton method are two effective methods for computing the extreme eigenvalues of symmetric matrices. In the paper we study the relationship between Davidson method and Newton method and emphasize on studying the inexact Newton method.The inexact Newton method is generalized and improved. Firstly, it is generalized to the block case. Secondly, we absorb the preconditioning idea of Davidson method and preconditioned Lanczos method and propose preconditioned inexact Newton method and preconditioned block inexact Newton method. In order to accelerate the eigenvalues' convergence rate, we apply Chebyshev iteration to Davidson method, block Davidson method, block inexact Newton method and preconditioned block inexact Newton method.All kinds of methods are compared in the numerical experiments. The numerical results show that the block inexact Newton method, the preconditioned inexact Newton method, the preconditioned block inexact Newton method and the Chebyshev acceleration presented in the paper are effective.