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 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. Jesse Barlow’s research was supported by the National Science Foundation under grant no. CCF-0429481. A. Smoktunowicz Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, Warsaw, 00-661 Poland e-mail: [email protected] J. L. Barlow (B) Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA 16802-6822, USA e-mail: [email protected] J. Langou Department of Mathematics, University of Colorado at Denver and Health Sciences Center, Denver, USA e-mail: [email protected] 300 A. Smoktunowicz et al. The classical Gram–Schmidt (CGS) orthogonal factorization is analyzed in a recent work of Giraud et al. [7] and in a number of other sources [3,10,14,1, 4,9,13, Sect. 6.9], [2, Sect. 2.4.5]. For a matrix A ∈ Rm×n (m ≥ n) with rank(A) = n, in exact arithmetic, the algorithm produces a factorization