Gradient iterations for the Rayleigh quotient are elemental methods for computing the smallest eigenvalues of a pair of symmetric and positive definite matrices. A considerable convergence acceleration can be achieved by preconditioning and by computing Rayleigh-Ritz approximations from subspaces of increasing dimensions. An example of the resulting Krylov subspace eigensolvers is the generaliz...

A convergence analysis for the nonsymmetric Lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive approximation bounds. In particular, the analysis implies the classical results for the symmetric Lanczos algorithm.

In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and a singular source. This class of problems covers general point sources, line the combination point-line sources has broad range practical applications. The proposed approach is decomposing true solution into part that known analytically using fundamental Lapla...

We show that a combination of the Generalized Davidson method and harmonic Ritz values (called harmonic Davidson) is well-suited for solving large interior eigenvalue problems using a plane wave basis. The algorithm enables us to calculate impurity and band edge states for systems of 100,000 atoms in about one day on 32 cores. We demonstrate the capabilities of the method by calculating the ele...

In order to solve the boundary value problems of elliptic equations, especially with singularities and unbounded domains, the simplified hybrid-combined method, which is equivalent to the coupling method of Zienkiewicz et al. [15], is presented. This is a combination of the Ritz-Galerkin and the finite element methods. Its optimal error estimates are proved in this paper, and the solution strat...

In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction [25] for a generalized eigenvalue problem. The method applies ...