I have an application that is somewhat similar to the situation of computing Gaussian quadrature nodes and weights: simply put, I need to compute the eigenvalues and the last two (normalized) components of the eigenvectors of a symmetric tridiagonal matrix.
I am aware of the Golub-Welsch scheme that uses the implicit QL/QR algorithm for Gaussian quadrature rule generation; the difference between it and my situation is that one uses the first component of the eigenvectors for Golub-Welsch.
I can modify the relevant LAPACK routines for my specific situation, but I was wondering if there were any new developments in this area. From my limited searching, it does not seem that MRRR or divide-and-conquer allow for computing only selected components.
Literature pointers would be extremely appreciated.