Abstract: The computation of accurate eigenvalues (and eigenvectors) plays an important role in many scientific problems. For symmetric tridiagonal matrices with eigenvalues of very different orders of magnitude, the number of correct digits in the computed approximations for the eigenvalues of smaller size depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, one can not be sure about the number of correct digits in the approximations computed. To solve this problem, we propose a modified bisection algorithm which produces an interval that is guaranteed to contain the desired eigenvalue. This interval has a very small relative gap when the eigenvalue is defined well.
We stress out that accurate computation of the eigenvalues is crucial to guarantee good orthogonality of eigenvectors computed with inverse iteration (this is the basis of the popular multiple relatively robust representation MRRR algorithm).

Brief Biography of the Speaker:
Rui Ralha is Associate Professor at the Department of Mathematics and Applications, School of Sciences of the University of Minho. He got his first degree in Mathematics (University of Coimbra, 1981) and a PhD degree at the Faculty of Electronics and Computer Science of the University of Southampton (UK) in 1990. He is a member of the Centre of Mathematics of the University of Minho (CMAT) where he has been coordinating the research group Comapp (Computational Mathematics and Applications). His main research interests are in the areas of numerical linear algebra and parallel computing.