A note on the error analysis of classical Gram-Schmidt

Alicja Smoktunowicz, Jesse Louis Barlow, Julien Langou

Research output: Contribution to journalArticle

13 Scopus citations

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 languageEnglish (US)
Pages (from-to)299-313
Number of pages15
JournalNumerische Mathematik
Volume105
Issue number2
DOIs
StatePublished - Dec 1 2006

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

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