A note on the error analysis of classical Gram-Schmidt

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.