We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...

We investigate the behavior of the Lanczos process when it is used to find all the eigenvalues of large sparse symmetric matrices. We study the convergence of classical Lanczos (i.e., without reorthogonalization) to the point where there is a cluster of Ritz values around each eigenvalue of the input matrix A. At that point, convergence to all the eigenvalues can be ascertained if A has no mult...

We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues—even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available dur...

We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted approximation process. An approximation to the subspace to be deflated is successively refined in the course ...

The open profile thin elastic wavy shell natural oscillations problem is considered. proposed method for determining the numerical values of lowest frequencies and corresponding forms vibrations shells with a complicated shape based on Rayleigh-Ritz energy method. results calculation rigidly pinched lower contour are presented.

Numerical computation of optimum values for nonlinear parameters in a Rayleigh-Ritz variational trial function is considerably more difBcult than numerical computation of optimum values for linear parameters. Thus, an analytic understanding of the mechanisms that determine these optimum values can be quite useful. Uniform asymptotic expansions can be used to explore these mechanisms for the non...

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...