This paper considers the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients. The solution of such problems will, in general, be rough, and it is well known that the usual (Ritz or displacement) finite element method will be inaccurate in general. The purpose of the paper is to help clarify the issue of whether the use of mixed variational...

An alternative to the classical Ritz method for approximate optimization is investigated. In the extended Ritz method, sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n-tuples of functions from a given basis. This alternative scheme, called variable-basis approximation, includes functions computable by trigonometric polynomials with f...

Ritz Values and Arnoldi Convergence for Nonsymmetric Matrices by Russell Carden The restarted Arnoldi method, useful for determining a few desired eigenvalues of a matrix, employs shifts to refine eigenvalue estimates. In the symmetric case, using selected Ritz values as shifts produces convergence due to interlacing. For nonsymmetric matrices the behavior of Ritz values is insufficiently under...

The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to precond...

The harmonic block Arnoldi method can be used to find interior eigenpairs of large matrices. Given a target point or shift to which the needed interior eigenvalues are close, the desired interior eigenpairs are the eigenvalues nearest and the associated eigenvectors. However, it has been shown that the harmonic Ritz vectors may converge erratically and even may fail to do so. To do a better job...

The Rayleigh-Ritz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in t...

This paper deals with the computation of the eigenvalues of a three-parameter Sturm-Liouville problem in the form of ordinary differential equation using Rayleigh-Ritz Method, a method which is based on the principle of variational methods. This method has been effective in computing the eigenvalues of self-adjoint problems. The resulting equations obtained in applying Rayleigh-Ritz method on t...

Generalized block Lanczos methods for large unsymmetric eigenproblems are presented, which contain the block Arnoldi method, and the block Arnoldi algorithms are developed. The convergence of this class of methods is analyzed when the matrix A is diagonalizable. Upper bounds for the distances between normalized eigenvectors and a block Krylov subspace are derived, and a priori theoretical error...

The article touches upon the problem of natural oscillations a thin elastic wavy shell an open profile. proposed method for determining numerical values lowest frequencies and corresponding forms shells complicated shape is based on Rayleigh-Ritz energy method.
 results calculation this with hinge-fixed attachment along lower contour generatrix are presented.