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 $\normal{R}=\normal{A}+E$ 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.   A similar result is stated in [Giraud at 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. 
Alicja Smoktunowicz, Jesse L. Barlow, Julien Langou