### Abstract

We consider Givens QR factorization of banded Hessenberg-Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial.

Original language | English |
---|---|

Pages (from-to) | 659-682 |

Number of pages | 24 |

Journal | Numerical Linear Algebra with Applications |

Volume | 12 |

Issue number | 7 |

DOIs | |

Publication status | Published - Sep 2005 |

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### Keywords

- Banded Hessenberg-Toeplitz matrices
- Convergence
- Discrete and continuous QR factorization

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics

### Cite this

*Numerical Linear Algebra with Applications*,

*12*(7), 659-682. https://doi.org/10.1002/nla.428

**Asymptotic properties of the QR factorization of banded Hessenberg-Toeplitz matrices.** / Chang, Xiao W.; Gander, Martin J.; Karaa, Samir.

Research output: Contribution to journal › Article

*Numerical Linear Algebra with Applications*, vol. 12, no. 7, pp. 659-682. https://doi.org/10.1002/nla.428

}

TY - JOUR

T1 - Asymptotic properties of the QR factorization of banded Hessenberg-Toeplitz matrices

AU - Chang, Xiao W.

AU - Gander, Martin J.

AU - Karaa, Samir

PY - 2005/9

Y1 - 2005/9

N2 - We consider Givens QR factorization of banded Hessenberg-Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial.

AB - We consider Givens QR factorization of banded Hessenberg-Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial.

KW - Banded Hessenberg-Toeplitz matrices

KW - Convergence

KW - Discrete and continuous QR factorization

UR - http://www.scopus.com/inward/record.url?scp=25144447659&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=25144447659&partnerID=8YFLogxK

U2 - 10.1002/nla.428

DO - 10.1002/nla.428

M3 - Article

VL - 12

SP - 659

EP - 682

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 7

ER -