Glued Matrices and the MRRR Algorithm
Inderjit S. Dhillon, Beresford Ν. Parlett, Christof Vömel · SIAM Journal on Scientific Computing · 2005
During the last ten years, Dhillon and Parlett devised a new algorithm (multiple relatively robust representations (MRRR)) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with $\mathcal{O}(n^2)$ cost. It has been incorporated into LAPACK version 3.0 as routine {\sc stegr}. We have discovered that the MRRR algorithm can fail in extreme cases. Sometimes eigenvalues agree to working accuracy and MRRR cannot compute orthogonal eigenvectors for them. In this paper, we describe and analyze these failures and various remedies.