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LAPACK Working Note 163: How the MRRR Algorithm Can Fail on Tight Eigenvalue Clusters

Authors:
Parlett, Beresford N.
Voemel, Christof
Technical Report Identifier: CSD-04-1367
March 11, 2005
CSD-04-1367.pdf

Abstract: 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 O(n^2) cost. It has been incorporated into LAPACK version 3.0 as routine 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. (Revised version).