by Beresford N. Parlett, Christof, V Ömel
http://digitalassets.lib.berkeley.edu/techreports/ucb/text/CSD-04-1367.pdf
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Abstract:
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. Key words. Multiple relatively robust representations, numerically orthogonal eigenvectors, symmetric tridiagonal matrix, tight clusters of eigenvalues, glued matrices, Wilkinson matrices. AMS subject classifications. 15A18, 15A23.
Citations
|
754
|
The Algebraic Eigenvalue Problem
– Wilkinson
- 1965
|
|
537
|
The Symmetric Eigenvalue Problem
– Parlett
- 1980
|
|
89
|
Accurate singular values of bidiagonal matrices
– Demmel, Kahan
- 1990
|
|
70
|
Computing accurate eigensystems of scaled diagonally dominant matrices
– Barlow, Demmel
- 1990
|
|
64
|
A divide and conquer method for the symmetric tridiagonal eigenproblem
– Cuppen
- 1981
|
|
55
|
Rank-one modification of the symmetric eigenproblem
– Bunch, Nielsen, et al.
- 1978
|
|
51
|
The rotation of eigenvectors by a perturbation
– Davis, Kahan
- 1970
|
|
47
|
Fernando's Solution to Wilkinson's Problem: An Application of Double Factorization
– Parlett, Dhillon
- 1997
|
|
40
|
Eisenstat, A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem
– Gu, C
- 1995
|
|
23
|
Relatively robust representations of symmetric tridiagonals
– Parlett, Dhillon
- 2000
|
|
23
|
Orthogonal eigenvectors and relative gaps
– Dhillon, Parlett
|
|
19
|
Computing an eigenvector with inverse iteration
– Ipsen
- 1997
|
|
17
|
Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices
– Dhillon, Parlett
- 2004
|
|
10
|
Current inverse iteration software can fail
– Dhillon
- 1998
|
|
10
|
Invariant subspaces for tightly clustered eigenvalues of tridiagonals
– Parlett
- 1996
|
|
7
|
LAPACK working note 162: The design and implementation of the MRRR algorithm
– Dhillon, Parlett, et al.
- 2004
|
|
7
|
Acta Numerica, chapter The new qd algorithms
– Parlett
- 1995
|
|
6
|
An implementation of the dqds algorithm (positive case
– Parlett, Marques
- 1999
|
|
4
|
A history of inverse iteration
– Ipsen
- 1996
|
|
1
|
Eigenvalues and eigenvectors of glued matrices. University of the MRRR algorithm can fail on tight eigenvalue clusters 15
– Parlett, Vömel
- 2004
|
