### Abstract

An error analysis result is given for classical Gram-Schmidt factorization of a full rank matrix A into A = QR where Q is left orthogonal (has orthonormal columns) and R is upper triangular. The work presented here shows that the computed R satisfies R ^{T} R = A ^{T} A + E where E is an appropriately small backward error, but only if the diagonals of R are computed in a manner similar to Cholesky factorization of the normal equations matrix. At the end of the article, implications for classical Gram-Schmidt with reorthogonalization are noted. A similar result is stated in Giraud et al. (Numer Math 101(1):87-100, 2005). However, for that result to hold, the diagonals of R must be computed in the manner recommended in this work.

Original language | English (US) |
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Pages (from-to) | 299-313 |

Number of pages | 15 |

Journal | Numerische Mathematik |

Volume | 105 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2006 |

### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Numerische Mathematik*,

*105*(2), 299-313. https://doi.org/10.1007/s00211-006-0042-1