A note on the error analysis of classical Gram –

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 R = A 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. The classical Gram–Schmidt (CGS) orthogonal factorization is analyzed in a recent work of Giraud et al. [5] and in a number of other sources [3,8, 11,1,4,7], [10, §6.9], [2, §2.4.5]. For a matrix A ∈ R (m ≥ n) with rank(A) = n, in exact arithmetic, the algorithm produces a factorization