Roundoff error analysis of the Cholesky QR2 algorithm

Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa, Takeshi Fukaya

Research output: Contribution to journalArticle

7 Citations (Scopus)

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((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.

Original languageEnglish
Pages (from-to)306-326
Number of pages21
JournalElectronic Transactions on Numerical Analysis
Volume44
Publication statusPublished - 2015

Fingerprint

QR Algorithm
Cholesky
Rounding error
Condition number
Orthogonality
Error Analysis
Deviation
QR Decomposition
Broadcast
Theoretical Analysis
High Performance
Norm
Numerical Results
Unit
Computing
Standards

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

Yamamoto, Y., Nakatsukasa, Y., Yanagisawa, Y., & Fukaya, T. (2015). Roundoff error analysis of the Cholesky QR2 algorithm. Electronic Transactions on Numerical Analysis, 44, 306-326.

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

In: Electronic Transactions on Numerical Analysis, Vol. 44, 2015, p. 306-326.

Research output: Contribution to journalArticle

Yamamoto, Y, Nakatsukasa, Y, Yanagisawa, Y & Fukaya, T 2015, 'Roundoff error analysis of the Cholesky QR2 algorithm', Electronic Transactions on Numerical Analysis, vol. 44, pp. 306-326.
Yamamoto Y, Nakatsukasa Y, Yanagisawa Y, Fukaya T. Roundoff error analysis of the Cholesky QR2 algorithm. Electronic Transactions on Numerical Analysis. 2015;44:306-326.
Yamamoto, Yusaku ; Nakatsukasa, Yuji ; Yanagisawa, Yuka ; Fukaya, Takeshi. / Roundoff error analysis of the Cholesky QR2 algorithm. In: Electronic Transactions on Numerical Analysis. 2015 ; Vol. 44. pp. 306-326.
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