The year 2005 have been marked by two new papers on the Classical Gram-Schmidt algorithm (see [1,2]). These results offer a better understanding of the Classical Gram-Schmidt algorithm. It is finally proved that the Classical Gram-Schmidt algorithm generates a loss of orthogonality bounded by the square of the condition number of the initial matrix. In the first part of the talk, I will quickly review the proof, explain its key points and its implication in the context of iterative methods. In the second part, I will focus on the new results that we have found related to the Classical Gram-Schmidt algorithm. In particular an a-posteriori reorthogonalization scheme extremely efficient is given in the context of iterative methods. (We borrow ideas developed in [3] in the context of GMRES-MGS.)

[1] A. Smoktunowicz and J. Barlow, A note on the error analysis of Classical Gram Schmidt, submitted to Numerische Mathematik (2005).

[2] Luc Giraud, Julien Langou, Miroslav Rozlozník, Jasper van den Eshof, Rounding error analysis of the classical Gram-Schmidt orthogonalization process, Numerische Mathematik 101(1) (July 2005) 87-100.

[3] Luc Giraud, Serge Gratton, Julien Langou, A rank- update procedure for reorthogonalizing the orthogonal factor from modified Gram-Schmidt, SIAM J. Matrix An. Appl. 25(4) (August 2004) 1163-1177.