The design and implementation of the MRRR algorithm
ACM Transactions on Mathematical Software (TOMS)
Algorithm 880: A testing infrastructure for symmetric tridiagonal eigensolvers
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Detecting Localization in an Invariant Subspace
SIAM Journal on Scientific Computing
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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.