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

Abstract

Let \( \tilde \lambda \) be an approximate eigenvalue of multiplicity mc = n − r of an n × n real symmetric tridiagonal matrix T having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting mc rows of T−\( \tilde \lambda \)I so that the condition number of the r × n matrix B formed of the remaining r rows is as small as possible. A special basis of mc 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 mc rows of T−\( \tilde \lambda \)I is also proposed. This method uses a rank-revealing 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 kerB are derived.