Evaluation Evaluation of Small Elements of the Eigenvectors of Certain Symmetric Tridiagonal Matrices with High Relative Accuracy

of the eigenvectors of symmetric tridiagonal matrices is one of the most basic tasks in numerical linear algebra. It is a widely known fact that, in the case of well separated eigenvalues, the eigenvectors can be evaluated with high relative accuracy. Nevertheless, in general, each coordinate of the eigenvector is evaluated with only high absolute accuracy. In particular, those coordinates whose magnitude is below the machine precision are not expected to be evaluated to any correct digit at all. In this paper, we address the problem of evaluating small (e.g. 10 −50) coordinates of the eigenvectors of certain symmetric tridiagonal matrices with high relative accuracy. We propose a numerical algorithm to solve this problem, and carry out error analysis. Also, we discuss some applications in which this task is necessary. Our results are illustrated via several numerical experiments.