Glued matrices and the MRRR algorithm

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...

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Bibliographic Details
Published in:SIAM journal on scientific computing 2005-01, Vol.27 (2), p.496-510
Main Authors: DHILLON, Inderjit S, PARLETT, Beresford N, VÖMEL, Christof
Format: Article
Language:English
Subjects:
Accuracy
Algebra
Algorithms
Computer science
Eigenvalues
Eigenvectors
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematics
Sciences and techniques of general use
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Summary: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.
ISSN:1064-8275
1095-7197
DOI:10.1137/040620746