Abstract

In this paper, we show why the modified Gram-Schmidt algorithm generates a well-conditioned set of vectors. This result holds under the assumption that the initial matrix is not 'too ill-conditioned' in a way that is quantified. As a consequence we show that if two iterations of the algorithm are performed, the resulting algorithm produces a matrix whose columns are orthogonal up to machine precision. Finally, we illustrate through a numerical experiment the sharpness of our result.

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