Download:
by Osni A. Marques, Beresford N. Parlett, Christof, V Ömel
http://digitalassets.lib.berkeley.edu/techreports/ucb/text/CSD-05-1392.pdf
Add To MetaCart
Abstract:
Abstract. The main advantage of inverse iteration over the QR algorithm and Divide & Conquer for the symmetric tridiagonal eigenproblem is that subsets of eigenpairs can be computed at reduced cost. The MRRR algorithm (MRRR = Multiple Relatively Robust Representations) is a clever variant of inverse iteration without the need for reorthogonalization. stegr, the current version of MRRR in LAPACK 3.0, does not allow for subset computations. The next release of stegr is designed to compute a (sub-)set of k eigenpairs with O(kn) operations. Because of the special way in which eigenvectors are computed, MRRR subset computations are more complicated than when using inverse iteration. Unlike the latter, MRRR sometimes cannot ignore the unwanted part of the spectrum. We describe the problems with what we call ’false singletons’. These are eigenvalues that appear to be isolated with respect to the wanted eigenvalues but in fact belong to a tight cluster of unwanted eigenvalues. This paper analyzes these complications and ways to deal with them. Key words. Multiple relatively robust representations, numerically orthogonal eigenvectors, symmetric tridiagonal matrix, subset computation, false singleton. AMS subject classifications. 15A18, 15A23.
Citations
|
557
|
The Symmetric Eigenvalue Problem
– Parlett
- 1980
|
|
380
|
LAPACK Users' Guide
– Anderson, Bai, et al.
- 1995
|
|
64
|
A divide and conquer method for the symmetric tridiagonal eigenproblem
– Cuppen
- 1981
|
|
56
|
Rank-one modification of the symmetric eigenproblem
– Bunch, Nielsen, et al.
- 1978
|
|
51
|
The rotation of eigenvectors by a perturbation
– Davis, Kahan
- 1970
|
|
48
|
The Rutherford-Boeing Sparse Matrix Collection
– S, Grimes, et al.
- 1997
|
|
47
|
Fernando’s solution to Wilkinson’s problem: an application of double factorization
– Parlett, Dhillon
- 1997
|
|
41
|
A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem
– Gu, Eisenstat
- 1995
|
|
24
|
Relatively robust representations of symmetric tridiagonals
– Parlett, Dhillon
- 2000
|
|
23
|
Orthogonal eigenvectors and relative gaps
– Parlett, Dhillon
|
|
19
|
Computing an eigenvector with inverse iteration
– Ipsen
- 1997
|
|
17
|
Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices
– Parlett, Dhillon
- 2004
|
|
12
|
de Geijn. A Parallel Eigensolver for Dense Symmetric Matrices based on Multiple Relatively Robust Representations
– Bientinesi, Dhillon, et al.
|
|
10
|
Current inverse iteration software can fail
– Dhillon
- 1998
|
|
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
|
|
3
|
LAPACK working note 168: pdsyevr. ScaLAPACK’s parallel MRRR algorithm for the symmetric eigenvalue problem
– Antonelli, Vömel
- 2005
|
