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

It turns out that, under certain conditions, frequently encountered 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.