Preconditioned gradient iterations for very large eigenvalue problems are efficient solvers with growing popularity. However, only for the simplest preconditioned eigensolver, namely the preconditioned gradient iteration (or preconditioned inverse iteration) with fixed step size, sharp non-asymptotic convergence estimates are known. These estimates require a properly scaled preconditioner. In t...

Iterative algorithms for large-scale eigenpair computation are mostly based subspace projections consisting of two main steps: a subspace update (SU) step that generates bases for approximate eigenspaces, followed by a Rayleigh-Ritz (RR) projection step that extracts approximate eigenpairs. A predominant methodology for the SU step makes use of Krylov subspaces that builds orthonormal bases pie...

This paper discusses a few algorithms for updating the approximate Singular Value Decomposition (SVD) in the context of information retrieval by Latent Semantic Indexing (LSI) methods. A unifying framework is considered which is based on Rayleigh-Ritz projection methods. First, a Rayleigh-Ritz approach for the SVD is discussed and it is then used to interpret the Zha-Simon algorithms [SIAM J. S...

We consider a parallel method for solving generalized eigenvalue problems that arise from molecular orbital computations. We use a moment-based method that finds several eigenvalues and their corresponding eigenvectors in a given domain, which is suitable for masterworker type parallel programming models. The computation of eigenvalues using explicit moments is sometimes numerically unstable. W...

The paper considers problem of buckling a thin elastic cylindrical shell, supported by rings various stiffness. approximate solution for clamping edge was obtained with analytical Rayleigh-Ritz method. parameters the optimal distribution mass structure between shell and stiffeners are found in order to maximize critical pressure. Frames zero eccentricity considered. solutions compared finite el...

Using  principle  of  minimum  total  potential  energy  approach  in conjunction  with  Rayleigh-Ritz  method,  the  electro-thermo- mechanical  axial  buckling  behavior  of  piezoelectric  polymeric cylindrical  shell  reinforced  with  double-walled  boron-nitride nanotube  (DWBNNT)  is  investigated. Coupling  between  electrical and mechanical  fields  are  considered  according  to  ...

The investigation of symmetry nonrestoration scenarios has led to a controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. A Rayleigh-Ritz variational approach to the problem, which might be useful in bridging the gap between perturbative and nonperturbative viewpoints, is here proposed. ...

Abstract. We investigate several generalizations of the harmonic and refined Rayleigh–Ritz method. These may be practical when one is interested in eigenvalues close to one of two targets (for instance, when the eigenproblem has Hamiltonian structure such that eigenvalues come in pairs or quadruples), or in rightmost eigenvalues close to (for instance) the imaginary axis. Our goal is to develop...

We present several variations on the harmonic Rayleigh–Ritz method. First, we introduce a relative harmonic approach for the standard, generalized, and polynomial eigenproblem. Second, a harmonic extraction method is studied for rightmost eigenvalues of generalized eigenvalue problems. Third, we propose harmonic extraction methods for large eigenvalues of generalized and polynomial eigenproblem...