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

in which A is a large sparse symmetric positive definite matrix, λ is an eigenvalue and u is a corresponding eigenvector. The evaluation of one or more smallest eigenpairs has much practical interest for describing the characteristics of physical phenomena. For example, smallest eigenvalues characterize the base frequences of vibrating mechanical structures. Typically, the matrix A is a discretization matrix, arising as a result of finite-difference, finite element or finite volume discretization of elliptic boundary value problems with self-adjoint differential operators on a mesh with a characteristic meshsize h, which has n real positive eigenvalues