Evaluation of small elements of the eigenvectors of certain symmetric tridiagonal matrices with high relative accuracy

Evaluation 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 with any accuracy whatsoever. It turns out that, under certain conditions, frequently ecountered in applications, small (e.g. $10^{-50}$) coordinates of eigenvectors of symmetric tridiagonal matrices can be evaluated with high $relative$ accuracy. In this paper, we investigate such conditions, carry out the analysis, and describe the resulting numerical schemes. While our schemes can be viewed as a modification of already existing (and well known) numerical algorithms, the related error analysis appears to be new. Our results are illustrated via several numerical examples.