For the eigenvalues of a symmetric tridiagonal matrix T, the most
accurate algorithms deliver approximations which are the exact
eigenvalues of another matrix whose entries differ from the
corresponding entries of T by small relative perturbations. However, for
matrices with eigenvalues of different magnitudes, the number of
correct digits in the computed approximations for eigenvalues of size
smaller than ?T?? 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, there is no simple way to
estimate well the number of correct digits in the approximations. To
remedy this, we propose a method that provides sharp bounds for the
eigenvalues of T. We present some numerical examples to illustrate the
usefulness of our method. |