### Abstract

This paper aims to propose the LU-Cholesky QR algorithms for thin QR decomposition (also called economy size or reduced QR decomposition). CholeskyQR is known as a fast algorithm employed for thin QR decomposition, and CholeskyQR2 aims to improve the orthogonality of a Q-factor computed by CholeskyQR. Although such Cholesky QR algorithms can efficiently be implemented in high-performance computing environments, they are not applicable for ill-conditioned matrices, as compared to the Householder QR and the Gram–Schmidt algorithms. To address this problem, we apply the concept of LU decomposition to the Cholesky QR algorithms, i.e., the idea is to use LU-factors of a given matrix as preconditioning before applying Cholesky decomposition. Moreover, we present rounding error analysis of the proposed algorithms on the orthogonality and residual of computed QR-factors. Numerical examples provided in this paper illustrate the efficiency of the proposed algorithms in parallel computing on both shared and distributed memory computers.

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

Article number | 102571 |

Journal | Parallel Computing |

DOIs | |

Publication status | Accepted/In press - 2019 Jan 1 |

### Fingerprint

### Keywords

- High-performance computing
- Numerical linear algebra
- Rounding error analysis
- Thin QR decomposition

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Hardware and Architecture
- Computer Networks and Communications
- Computer Graphics and Computer-Aided Design
- Artificial Intelligence

### Cite this

*Parallel Computing*, [102571]. https://doi.org/10.1016/j.parco.2019.102571

**LU-Cholesky QR algorithms for thin QR decomposition.** / Terao, Takeshi; Ozaki, Katsuhisa; Ogita, Takeshi.

Research output: Contribution to journal › Article

*Parallel Computing*. https://doi.org/10.1016/j.parco.2019.102571

}

TY - JOUR

T1 - LU-Cholesky QR algorithms for thin QR decomposition

AU - Terao, Takeshi

AU - Ozaki, Katsuhisa

AU - Ogita, Takeshi

PY - 2019/1/1

Y1 - 2019/1/1

N2 - This paper aims to propose the LU-Cholesky QR algorithms for thin QR decomposition (also called economy size or reduced QR decomposition). CholeskyQR is known as a fast algorithm employed for thin QR decomposition, and CholeskyQR2 aims to improve the orthogonality of a Q-factor computed by CholeskyQR. Although such Cholesky QR algorithms can efficiently be implemented in high-performance computing environments, they are not applicable for ill-conditioned matrices, as compared to the Householder QR and the Gram–Schmidt algorithms. To address this problem, we apply the concept of LU decomposition to the Cholesky QR algorithms, i.e., the idea is to use LU-factors of a given matrix as preconditioning before applying Cholesky decomposition. Moreover, we present rounding error analysis of the proposed algorithms on the orthogonality and residual of computed QR-factors. Numerical examples provided in this paper illustrate the efficiency of the proposed algorithms in parallel computing on both shared and distributed memory computers.

AB - This paper aims to propose the LU-Cholesky QR algorithms for thin QR decomposition (also called economy size or reduced QR decomposition). CholeskyQR is known as a fast algorithm employed for thin QR decomposition, and CholeskyQR2 aims to improve the orthogonality of a Q-factor computed by CholeskyQR. Although such Cholesky QR algorithms can efficiently be implemented in high-performance computing environments, they are not applicable for ill-conditioned matrices, as compared to the Householder QR and the Gram–Schmidt algorithms. To address this problem, we apply the concept of LU decomposition to the Cholesky QR algorithms, i.e., the idea is to use LU-factors of a given matrix as preconditioning before applying Cholesky decomposition. Moreover, we present rounding error analysis of the proposed algorithms on the orthogonality and residual of computed QR-factors. Numerical examples provided in this paper illustrate the efficiency of the proposed algorithms in parallel computing on both shared and distributed memory computers.

KW - High-performance computing

KW - Numerical linear algebra

KW - Rounding error analysis

KW - Thin QR decomposition

UR - http://www.scopus.com/inward/record.url?scp=85076601603&partnerID=8YFLogxK

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U2 - 10.1016/j.parco.2019.102571

DO - 10.1016/j.parco.2019.102571

M3 - Article

AN - SCOPUS:85076601603

JO - Parallel Computing

JF - Parallel Computing

SN - 0167-8191

M1 - 102571

ER -