For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses signiicantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In re...

We present an approach for the acceleration of the restarted Arnoldi iteration for the computation of a number of eigenvalues of the standard eigenproblem Ax = λx. This study applies the Chebyshev polynomial to the restarted Arnoldi iteration and proves that it computes necessary eigenvalues with far less complexity than the QR method. We also discuss the dependence of the convergence rate of t...

We investigate the generalized second-order Arnoldi (GSOAR) method, a generalization of the SOAR method proposed by Bai and Su [SIAM J. Matrix Anal. Appl., 26 (2005): 640–659.], and the Refined GSOAR (RGSOAR) method for the quadratic eigenvalue problem (QEP). The two methods use the GSOAR procedure to generate an orthonormal basis of a given generalized second-order Krylov subspace, and with su...

Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. The algorithm GMRES(m) restarts every m iterations. It is generally thought the information of previous GMRES cycles is lost at the time of a restart, so that each cycle contributes to the global convergence individually. However, this is not the full story. In this paper, we shed light on the rela...

This paper presents a new preconditioning technique for the restarted GMRES algorithm. It is based on an invariant subspace approximation which is updated at each cycle. Numerical examples show that this deea-tion technique gives a more robust scheme than the restarted algorithm, at a low cost of operations and memory.

We show in this text how the idea of the Implicitly Restarted Arnoldi method can be generalised to the non-symmetric Lanczos algorithm, using the two-sided Gram-Schmidt process or using a Lanczos tridiagonalisation. The implicitly restarted Lanczos method can be combined with an implicit lter. It can also be used in case of breakdown and ooers an alternative for look-ahead.

We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a twoelectron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. Th...

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 ...