Roundoff error analysis of the Cholesky QR2 algorithm

Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa, Takeshi Fukaya

研究成果: Article

11 引用 (Scopus)

抜粋

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.

元の言語English
ページ(範囲)306-326
ページ数21
ジャーナルElectronic Transactions on Numerical Analysis
44
出版物ステータスPublished - 2015

ASJC Scopus subject areas

  • Analysis

フィンガープリント Roundoff error analysis of the Cholesky QR2 algorithm' の研究トピックを掘り下げます。これらはともに一意のフィンガープリントを構成します。

  • これを引用

    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.