Visit at Cerfacs 2 Research Report

1 Activity Report The two-month visit at CERFACS has been a pleasure and an exciting experience, for the opportunity to share ideas with the researchers working there, on a large variety of elds. My staying coincided with two important workshops, \Direct Methods" and \Eigenvalues and beyond" in the ILAY frame. They were both extremely valuable and it was a great opportunity for me to meet people from diierent countries working on challenging problems. Part of the work presented at the Eigenvalues workshop was carried out during my staying, and the paper which will be submitted to BIT for the special issue devoted to the conference was also ultimate in this period. It is my pleasure to say that during my staying I have started a collaboration with Dr. Mario Arioli and Dr. D. Ruiz, within the European project HCM # ER-BCHRXCT930420, which will be continued in the next months. A short report on the state of the work follows. In 1], Arioli and Ruiz have proposed an inner-outer method for computing a basis of the invariant subspace of a symmetric positive deenite matrix A corresponding to its smallest eigenvalues. Then, a deeation technique can make use of the generated basis for improving the convergence of an iterative method applied on the linear system Ax = b. The method can be applied when A is characterized by strongly clustered and very small eigenvalues, which may excessively slow down the convergence of the iterative process. Known deeation techniques consist of removing from the system the eigencomponents with smallest eigenvalues 3]. It is clear that this requires the availability of, or an accurate approximation to, the corresponding invariant subspace. In some applications, this can be derived from the structure of the problem, in general, however, a basis for the (approximate) invariant subspace needs to be generated. We would like to stress that the goodness of the approximation can also be evaluated in terms of orthogonality with respect to the rest of the spectrum. Our interest is in exploring the results in 1], where a simultaneous inverse iteration procedure was used for computing a suuciently accurate basis of the looked after invariant subspace; see 4]. For large matrices, the inverse iteration can be seen as an inner-outer scheme. The outer loop replaces X k with a block X k+1 in the subspace spanned by 1