New bounds on the canonical angles between an invariant subspace of A and an approximating subspace by the differences between Ritz values and the targeted eigenvalues are obtained. From this result, various bounds are readily available to estimate how accurate the Ritz vectors computed from the approximating subspace may be, based on information on approximation accuracies in the Ritz values. ...

in this study, the free vibration of partially fluid-filled laminated composite circular cylindrical shell with arbitrary boundary conditions has been investigated by using rayleigh-ritz method. the analysis has been carried out with strain-displacement relations based on love’s thin shell theory and the contained fluid is assumed irrotational, incompressible and inviscid. after determining the...

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...

The relationship between the densities of ground-state wave functions (i.e., the minimizers of the Rayleigh-Ritz variation principle) and the ground-state densities in density-functional theory (i.e., the minimizers of the Hohenberg-Kohn variation principle) is studied within the framework of convex conjugation, in a generic setting covering molecular systems, solid-state systems, and more. Hav...

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A are almost linearly dependent. Furthermo...

One application of harmonic Ritz values is to approximate, with a projection method, the interior eigenvalues of a matrix A while avoiding the explicit use of the inverse A. In this context, harmonic Ritz values are commonly derived from a Petrov-Galerkin condition for the residual of a vector from the test space. In this paper, we investigate harmonic Ritz values from a slightly different pers...

Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poores...

in this paper, the stability characteristics of single-walled carbon nanotubes (swcnts) under the action of axial load are investigated. to this end, a nonlocal flügge shell model is developed to accommodate the small length scale effects. the analytical rayleigh-ritz method with beam functions is applied to the variational statement derived from the flügge-type buckling equations. molecular dy...