Rounding error analysis of the classical Gram-Schmidt orthogonalization process

Authors:
Luc Giraud;Julien Langou;Miroslav Rozložník;Jasper van den Eshof
Affiliations:
CERFACS, 42 Avenue Gaspard Coriolis, 31057, Toulouse Cedex 1, France;Department of Computer Science, The University of Tennessee, 1122 Volunteer Blvd., 37996-3450, Knoxville, TN, USA;Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 2, CZ-182 07, Prague 8, TN, Czech Republic;Heinrich-Heine-Universität, Mathematisches Institut, Universitätsstrasse 1, D-40225, Düsseldorf, TN, Germany
Venue:
Numerische Mathematik
Year:
2005

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Abstract

This paper provides two results on the numerical behavior of the classical Gram-Schmidt algorithm. The first result states that, provided the normal equations associated with the initial vectors are numerically nonsingular, the loss of orthogonality of the vectors computed by the classical Gram-Schmidt algorithm depends quadratically on the condition number of the initial vectors. The second result states that, provided the initial set of vectors has numerical full rank, two iterations of the classical Gram-Schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level.