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Evaluation of the eigenvectors of symmetric tridiagonal matrices is one ofthe most basic tasks in numerical linear algebra. It is a widely known factthat, in the case of well separated eigenvalues, the eigenvectors can beevaluated with high relative accuracy. Nevertheless, in general, eachcoordinate of the eigenvector is evaluated with only high $absolute$ accuracy.In particular, those coordinates whose magnitude is below the machine precisionare not expected to be evaluated with any accuracy whatsoever.  It turns out that, under certain conditions, frequently ecountered inapplications, small (e.g. $10^{-50}$) coordinates of eigenvectors of symmetrictridiagonal matrices can be evaluated with high $relative$ accuracy. In thispaper, we investigate such conditions, carry out the analysis, and describe theresulting numerical schemes. While our schemes can be viewed as a modificationof already existing (and well known) numerical algorithms, the related erroranalysis appears to be new.  Our results are illustrated via several numerical examples.

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