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.
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The work of the second author was supported in part by the US Department of Energy, Office of Basic Energy Science under LAB03-17 initiative, DOE contract No. DE-FG02-03ER25584, and in part by the TeraScale Optimal PDE Simulations (TOPS) SciDAC, DoE Contract No. DE-FC02-01ER25480
The work of the third author was supported by the project 1ET400300415 within the National Program of Research ‘‘Information Society’’ and by the GA AS CR under grant No. IAA1030405.
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Giraud, L., Langou, J., Rozložník, M. et al. Rounding error analysis of the classical Gram-Schmidt orthogonalization process. Numer. Math. 101, 87–100 (2005). https://doi.org/10.1007/s00211-005-0615-4
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DOI: https://doi.org/10.1007/s00211-005-0615-4