On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix
Sergei Konstantinovich Godunov, Alexander N. Malyshev · Computational Mathematics and Mathematical Physics · 2008
Let $$ \tilde \lambda $$ be an approximate eigenvalue of multiplicity m c = 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 m c 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 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 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.