Variable-step preconditioned conjugate gradient method for partial symmetric eigenvalue problems

The iterative methods for partial algebraic symmetric eigenvalue problems are considered for sparse positive definite matrices which arise in approximation of 2D and 3D boundary value problems. The approach is based on subspace iterations, Rayleigh-Ritz method, and the variable-step preconditioned conjugate gradient algorithm, including algebraic multigrid and incomplete factorization. Theorems on the properties of convergence rate are presented. The efficiency of the proposed iterative processes is demonstrated by the results of numerical experiments.