A note on the error analysis of classical Gram–Schmidt

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 RT R = AT 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.