Plenary
Lecture
Highly accurate bounds for the eigenvalues of symmetric
tridiagonal matrices
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Associate Professor Rui Manuel Silva Ralha
Department of Mathematics and Applications
University of Minho
Braga, Portugal
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.
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