### Abstract

We consider the QR decomposition of an m × n matrix X with full column rank, where m × n. Among the many algorithms available, the Cholesky QR algorithm is ideal from the viewpoint of high performance computing since it consists entirely of standard level 3 BLAS operations with large matrix sizes, and requires only one reduce and broadcast in parallel environments. Unfortunately, it is well-known that the algorithm is not numerically stable and the deviation from orthogonality of the computed Q factor is of order O((κ_{2}(X))^{2}u), where κ_{2}(X) is the 2-norm condition number of X and u is the unit roundoff. In this paper, we show that if the condition number of X is not too large, we can greatly improve the stability by iterating the Cholesky QR algorithm twice. More specifically, if κ_{2}(X) is at most O(u^{-1/2} ), both the residual and deviation from orthogonality are shown to be of order O(u). Numerical results support our theoretical analysis.

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

Pages (from-to) | 306-326 |

Number of pages | 21 |

Journal | Electronic Transactions on Numerical Analysis |

Volume | 44 |

Publication status | Published - 2015 |

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### Keywords

- Cholesky QR
- Communication-avoiding algorithms
- QR decomposition
- Roundoff error analysis

### ASJC Scopus subject areas

- Analysis

### Cite this

*Electronic Transactions on Numerical Analysis*,

*44*, 306-326.

**Roundoff error analysis of the Cholesky QR2 algorithm.** / Yamamoto, Yusaku; Nakatsukasa, Yuji; Yanagisawa, Yuka; Fukaya, Takeshi.

Research output: Contribution to journal › Article

*Electronic Transactions on Numerical Analysis*, vol. 44, pp. 306-326.

}

TY - JOUR

T1 - Roundoff error analysis of the Cholesky QR2 algorithm

AU - Yamamoto, Yusaku

AU - Nakatsukasa, Yuji

AU - Yanagisawa, Yuka

AU - Fukaya, Takeshi

PY - 2015

Y1 - 2015

N2 - We consider the QR decomposition of an m × n matrix X with full column rank, where m × n. Among the many algorithms available, the Cholesky QR algorithm is ideal from the viewpoint of high performance computing since it consists entirely of standard level 3 BLAS operations with large matrix sizes, and requires only one reduce and broadcast in parallel environments. Unfortunately, it is well-known that the algorithm is not numerically stable and the deviation from orthogonality of the computed Q factor is of order O((κ2(X))2u), where κ2(X) is the 2-norm condition number of X and u is the unit roundoff. In this paper, we show that if the condition number of X is not too large, we can greatly improve the stability by iterating the Cholesky QR algorithm twice. More specifically, if κ2(X) is at most O(u-1/2 ), both the residual and deviation from orthogonality are shown to be of order O(u). Numerical results support our theoretical analysis.

AB - We consider the QR decomposition of an m × n matrix X with full column rank, where m × n. Among the many algorithms available, the Cholesky QR algorithm is ideal from the viewpoint of high performance computing since it consists entirely of standard level 3 BLAS operations with large matrix sizes, and requires only one reduce and broadcast in parallel environments. Unfortunately, it is well-known that the algorithm is not numerically stable and the deviation from orthogonality of the computed Q factor is of order O((κ2(X))2u), where κ2(X) is the 2-norm condition number of X and u is the unit roundoff. In this paper, we show that if the condition number of X is not too large, we can greatly improve the stability by iterating the Cholesky QR algorithm twice. More specifically, if κ2(X) is at most O(u-1/2 ), both the residual and deviation from orthogonality are shown to be of order O(u). Numerical results support our theoretical analysis.

KW - Cholesky QR

KW - Communication-avoiding algorithms

KW - QR decomposition

KW - Roundoff error analysis

UR - http://www.scopus.com/inward/record.url?scp=84937410265&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937410265&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84937410265

VL - 44

SP - 306

EP - 326

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

SN - 1068-9613

ER -