Reliable eigenvalues of symmetric tridiagonals

Abstract

For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a 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‖₂ 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.