Abstract
It has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved in an implicit way, via unitary similarity transformations, from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which aside from poles at infinity and zero, also contain finite non-zero poles. Furthermore, the algorithms are generalized to deal with block rational Krylov subspaces and techniques to exploit the symmetry when working with Hermitian matrices are also presented. For each variant of the algorithm numerical experiments illustrate the power of the new approach. The experiments involve matrix functions, Ritz-value computations, and the solutions of matrix equations.
| Original language | English |
|---|---|
| Pages (from-to) | 100-124 |
| Number of pages | 25 |
| Journal | Electronic Transactions on Numerical Analysis |
| Volume | 43 |
| Publication status | Published - 2014 |
| Externally published | Yes |
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Keywords
- Extended Krylov
- Iterative methods
- Krylov
- Rational Krylov
- Rotations
- Similarity transformations
ASJC Scopus subject areas
- Analysis
Cite this
Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices. / Mach, Thomas; Pranić, Miroslav S.; Vandebril, Raf.
In: Electronic Transactions on Numerical Analysis, Vol. 43, 2014, p. 100-124.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices
AU - Mach, Thomas
AU - Pranić, Miroslav S.
AU - Vandebril, Raf
PY - 2014
Y1 - 2014
N2 - It has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved in an implicit way, via unitary similarity transformations, from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which aside from poles at infinity and zero, also contain finite non-zero poles. Furthermore, the algorithms are generalized to deal with block rational Krylov subspaces and techniques to exploit the symmetry when working with Hermitian matrices are also presented. For each variant of the algorithm numerical experiments illustrate the power of the new approach. The experiments involve matrix functions, Ritz-value computations, and the solutions of matrix equations.
AB - It has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved in an implicit way, via unitary similarity transformations, from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which aside from poles at infinity and zero, also contain finite non-zero poles. Furthermore, the algorithms are generalized to deal with block rational Krylov subspaces and techniques to exploit the symmetry when working with Hermitian matrices are also presented. For each variant of the algorithm numerical experiments illustrate the power of the new approach. The experiments involve matrix functions, Ritz-value computations, and the solutions of matrix equations.
KW - Extended Krylov
KW - Iterative methods
KW - Krylov
KW - Rational Krylov
KW - Rotations
KW - Similarity transformations
UR - http://www.scopus.com/inward/record.url?scp=84928891265&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84928891265&partnerID=8YFLogxK
M3 - Article
VL - 43
SP - 100
EP - 124
JO - Electronic Transactions on Numerical Analysis
JF - Electronic Transactions on Numerical Analysis
SN - 1068-9613
ER -