Lapack Working Note 194: a Refined Representation Tree for Mrrr

In order to compute orthogonal eigenvectors of a symmetric tridiagonal matrix without Gram-Schmidt orthogonalization, the MRRR algorithm finds a shifted LDL factorization (representation) for each eigenvalue such that the local eigenvalue is a singleton, that is defined to high relative accuracy and has a large relative gap. MRRR’s representation tree describes how, by successive shifting and refinement, each eigenvalue becomes relatively isolated. Its shape plays a crucial role for complexity: deeper trees are associated with more eigenvalue refinement to resolve clustering of eigenvalues. Motivated by recently observed deteriorating complexity of the LAPACK 3.1 MRRR kernels for certain matrices of large dimension, we here re-examine and refine the representation tree concept. We first describe the discovery of what we call a spectrum peeling problem: even though the matrix at hand might not have a spectrum with clusters within clusters, the representation tree might still contain a long chain of large nodes. We then formulate a refined proposal for the representation tree that aims at avoiding the unwarranted work while preserving tight accuracy bounds where possible. The trade-off between performance and accuracy in our solution is discussed by practical examples. AMS subject classifications. 65F15, 65Y15.