On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix

Let $\tilde\lambda$ be an approximate eigenvalue of multiplicity $m_c=n-r$ of an $n\times n$ real symmetric tridiagonal matrix $T$ having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting $m_c$ rows of $T-\tilde\lambda I$ so that the condition number of the $r\times n$ matrix $B$ formed of the remaining r rows is as small as possible. A special basis of $m_c$ vectors with local supports is constructed for the subspace kerB. These vectors are approximate eigenvectors of $T$ corresponding to $\tilde\lambda$. Another method for deleting $m_c$ rows of $T-\tilde\lambda I$ is also proposed. This method uses a rank-revealing $\mathrm{QR}$ decomposition; however, it requires a considerably larger number of arithmetic operations. For the latter algorithm, the condition number of $B$ is estimated, and orthogonality estimates for vectors of the special basis of $\operatorname{ker}B$ are derived.