Inexact Iterative WYD Method for Eigenvalue Problems

In this paper, we propose a new inexact iterative subspace projection method for real symmetric positive definite generalized eigenvalue problems based on the WYD (an abbreviation of the initials of the proposers: Wilson, Yuan, and Dickens [1]) method. Firstly an analysis of the convergence condition of the approximate eigenvectors is given when ${\bf Ar}_{k+1}=\lambda{\bf Br}_k$ is inexactly solved by the Krylov subspace methods. Then the inexact iterative WYD (IIWYD) method is constructed, which utilizes the approximate Ritz subspace generated by the WYD method with an inexact solver as the main search space. The IIWYD method improves the quality of the search space during the iterative process, significantly reduces the number of iteration steps, and improves the overall computational efficiency and stability. The results of numerical experiments show that the IIWYD method is more efficient and stable compared to the locally optimal block preconditioned conjugate gradient (LOBPCG) method and the Jacobi-Davidson (JD) method. In addition, we also discuss the effects of the refined strategy and the conjugate strategy in our method.