### 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) |
---|---|

Pages (from-to) | 299-313 |

Number of pages | 15 |

Journal | Numerische Mathematik |

Volume | 105 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Cite this

*Numerische Mathematik*,

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

}

*Numerische Mathematik*, vol. 105, no. 2, pp. 299-313. https://doi.org/10.1007/s00211-006-0042-1

**A note on the error analysis of classical Gram-Schmidt.** / Smoktunowicz, Alicja; Barlow, Jesse Louis; Langou, Julien.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A note on the error analysis of classical Gram-Schmidt

AU - Smoktunowicz, Alicja

AU - Barlow, Jesse Louis

AU - Langou, Julien

PY - 2006/12/1

Y1 - 2006/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33751159603&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751159603&partnerID=8YFLogxK

U2 - 10.1007/s00211-006-0042-1

DO - 10.1007/s00211-006-0042-1

M3 - Article

AN - SCOPUS:33751159603

VL - 105

SP - 299

EP - 313

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 2

ER -