### Abstract

Designing communication-avoiding algorithms is crucial for high performance computing on a large-scale parallel system. The TSQR algorithm is a communication-avoiding algorithm for computing a tall-skinny QR factorization, and TSQR is known to be much faster and as stable as the classical Householder QR algorithm. The Cholesky QR algorithm is another very simple and fast communication-avoiding algorithm, but rarely used in practice because of its numerical instability. Our recent work points out that an algorithm that simply repeats Cholesky QR twice, which we call CholeskyQR2, gives excellent accuracy for a wide range of matrices arising in practice. Although the communication cost of CholeskyQR2 is twice that of TSQR, it has an advantage that its reduction operation is addition whereas that of TSQR is a QR factorization, whose high-performance implementation is more difficult. Thus, CholeskyQR2 can potentially be significantly faster than TSQR. Indeed, in our experiments using 16384 nodes of the K computer, CholeskyQR2 ran about three times faster than TSQR for a 4194304 × 64 matrix.

Original language | English |
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Title of host publication | Proceedings of ScalA 2014 |

Subtitle of host publication | 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems - held in conjunction with SC 2014: The International Conference for High Performance Computing, Networking, Storage and Analysis |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 31-38 |

Number of pages | 8 |

ISBN (Electronic) | 9781479975624 |

DOIs | |

Publication status | Published - 2014 |

Event | 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, ScalA 2014 - New Orleans, United States Duration: 2014 Nov 17 → … |

### Other

Other | 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, ScalA 2014 |
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Country | United States |

City | New Orleans |

Period | 14/11/17 → … |

### Fingerprint

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Networks and Communications
- Computer Science Applications
- Software
- Electrical and Electronic Engineering

### Cite this

*Proceedings of ScalA 2014: 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems - held in conjunction with SC 2014: The International Conference for High Performance Computing, Networking, Storage and Analysis*(pp. 31-38). [7016731] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ScalA.2014.11

**CholeskyQR2 : A Simple and Communication-Avoiding Algorithm for Computing a Tall-Skinny QR Factorization on a Large-Scale Parallel System.** / Fukaya, Takeshi; Nakatsukasa, Yuji; Yanagisawa, Yuka; Yamamoto, Yusaku.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of ScalA 2014: 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems - held in conjunction with SC 2014: The International Conference for High Performance Computing, Networking, Storage and Analysis.*, 7016731, Institute of Electrical and Electronics Engineers Inc., pp. 31-38, 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, ScalA 2014, New Orleans, United States, 14/11/17. https://doi.org/10.1109/ScalA.2014.11

}

TY - GEN

T1 - CholeskyQR2

T2 - A Simple and Communication-Avoiding Algorithm for Computing a Tall-Skinny QR Factorization on a Large-Scale Parallel System

AU - Fukaya, Takeshi

AU - Nakatsukasa, Yuji

AU - Yanagisawa, Yuka

AU - Yamamoto, Yusaku

PY - 2014

Y1 - 2014

N2 - Designing communication-avoiding algorithms is crucial for high performance computing on a large-scale parallel system. The TSQR algorithm is a communication-avoiding algorithm for computing a tall-skinny QR factorization, and TSQR is known to be much faster and as stable as the classical Householder QR algorithm. The Cholesky QR algorithm is another very simple and fast communication-avoiding algorithm, but rarely used in practice because of its numerical instability. Our recent work points out that an algorithm that simply repeats Cholesky QR twice, which we call CholeskyQR2, gives excellent accuracy for a wide range of matrices arising in practice. Although the communication cost of CholeskyQR2 is twice that of TSQR, it has an advantage that its reduction operation is addition whereas that of TSQR is a QR factorization, whose high-performance implementation is more difficult. Thus, CholeskyQR2 can potentially be significantly faster than TSQR. Indeed, in our experiments using 16384 nodes of the K computer, CholeskyQR2 ran about three times faster than TSQR for a 4194304 × 64 matrix.

AB - Designing communication-avoiding algorithms is crucial for high performance computing on a large-scale parallel system. The TSQR algorithm is a communication-avoiding algorithm for computing a tall-skinny QR factorization, and TSQR is known to be much faster and as stable as the classical Householder QR algorithm. The Cholesky QR algorithm is another very simple and fast communication-avoiding algorithm, but rarely used in practice because of its numerical instability. Our recent work points out that an algorithm that simply repeats Cholesky QR twice, which we call CholeskyQR2, gives excellent accuracy for a wide range of matrices arising in practice. Although the communication cost of CholeskyQR2 is twice that of TSQR, it has an advantage that its reduction operation is addition whereas that of TSQR is a QR factorization, whose high-performance implementation is more difficult. Thus, CholeskyQR2 can potentially be significantly faster than TSQR. Indeed, in our experiments using 16384 nodes of the K computer, CholeskyQR2 ran about three times faster than TSQR for a 4194304 × 64 matrix.

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U2 - 10.1109/ScalA.2014.11

DO - 10.1109/ScalA.2014.11

M3 - Conference contribution

AN - SCOPUS:84988268780

SP - 31

EP - 38

BT - Proceedings of ScalA 2014

PB - Institute of Electrical and Electronics Engineers Inc.

ER -