We describe the development of a method for the efficient computation of the smallest singular values and corresponding vectors for large sparse matrices [4]. The method combines state-of-the-art techniques that make it a useful computational tool appropriate for large scale computations. The method relies upon Lanczos bidiagonalization (LBD) with partial reorthogonalization [6], enhanced with ...

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...

In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit, after which the current vectors are reorthogonalized. A theorem of Simon 4] shows that the Rayleigh quotient | i.e., the tridiagonal matrix produced by the Lanczos recursion | contains fully accurate approximations to the Ritz values in spite...

Eigenvalue iterative methods, such as Arnoldi and Jacobi-Davidson, are typically used with restarting. This has signiicant performance shortcomings, since important components of the invariant subspace may be discarded. One way of saving more information at restart is the idea of \thick" restarting which keeps more Ritz vectors than needed. Our previously proposed dynamic thick restarting choos...

In this paper for a given prescribed Ritz values that satisfy in the some special conditions, we find a symmetric nonnegative matrix, such that the given set be its Ritz values.

Abstract In this paper we construct Ritz-type projectors with boundary interpolation properties in finite dimensional subspaces of the usual Sobolev space and provide a priori error estimates for them. The abstract analysis is exemplified by considering spline spaces equip corresponding explicit constants. This complements our results recently obtained based on classical Ritz (Numer Math 144(4)...

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...

Ω fvdxdy. There have been a lot of work related to the finite element approximation to problem (1) in the framework of both semidiscrete and fully discrete schemes. For example, some superconvergence properties of conforming linear finite element (FE) were obtained by V. Thomeé with necessary help of Ritz projection and Lin Qun with a new analysis form, i.e., an analysis for the “short side” in...