When modified Gram–Schmidt generates a well‐conditioned set of vectors
AbstractIn 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.