Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems L. Bergamaschi & A. Martínez a Department of Civil, Environmental and Architectural Engineering, University of Padua, via Trieste 63, 35100 Padova, Italy b Department of Mathematics, University of Padua, via Trieste 63, 35100 Padova, Italy Accepted author version posted online: 14 Apr 2014.Published online...

The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions [21]. This paper reports on new methods for finding eigenvalues of very large matrices by a synthesis of evolutionary computation, parallel programming, and e...

We consider the class of the Orthogonal Projection Methods (OPM) to solve iteratively large and generalised eigenvalue problems. An OPM is a method that projects a large eigenvalue problem on a smaller subspace. In this subspace, an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method. We show that many iterative eigenvalue solvers, such...

We propose a restarted Arnoldi’s method with Faber polynomials and discuss its use for computing the rightmost eigenvalues of large non hermitian matrices. We illustrate, with the help of some practical test problems, the benefit obtained from the Faber acceleration by comparing this method with the Chebyshev based acceleration. A comparison with the implicitly restarted Arnoldi method is also ...

We consider the development and implementation of eigen-solvers on distributed memory parallel arrays of vector processors and show that the concomitant requirements for vectorization and paralleliza-tion lead both to novel algorithms and novel implementation techniques. Performance results are given for several large-scale applications and some performance comparisons made with LAPACK and ScaL...

In this presentation we review iterative projection methods for sparse nonlinear eigenvalue problems which have proven to be very efficient. Here the eigenvalue problem is projected to a subspace V of small dimension which yields approximate eigenpairs. If an error tolerance is not met then the search space V is expanded in an iterative way with the aim that some of the eigenvalues of the reduc...

The simple Lanczos process is very eeective for nding a few extreme eigenvalues of a large symmetric matrix. The main task in each iteration step consists in evaluating a matrix-vector product. It is shown in this paper how to apply a fast wavelet-based product in order to speed up computations. Some numerical results are given for the simple case of the Harmonic Oscillator.

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...

This paper studies a number of Newton methods and use them to define new secondary linear systems of equations for the Davidson eigenvalue method. The new secondary equations avoid some common pitfalls of the existing ones such as the correction equation and the Jacobi-Davidson preconditioning. We will also demonstrate that the new schemes can be used efficiently in test problems.

Iterative methods for solving large, sparse, symmetric eigenvalue problems often encounter convergence diiculties because of ill-conditioning. The Generalized Davidson method is a well known technique which uses eigenvalue preconditioning to surmount these diiculties. Preconditioning the eigenvalue problem entails more subtleties than for linear systems. In addition, the use of an accurate conv...