### 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 |

### Fingerprint

### Keywords

- Extended Krylov
- Iterative methods
- Krylov
- Rational Krylov
- Rotations
- Similarity transformations

### ASJC Scopus subject areas

- Analysis

### Cite this

*Electronic Transactions on Numerical Analysis*,

*43*, 100-124.

**Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices.** / Mach, Thomas; Pranić, Miroslav S.; Vandebril, Raf.

Research output: Contribution to journal › Article

*Electronic Transactions on Numerical Analysis*, vol. 43, pp. 100-124.

}

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 -