title :
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When modified Gram-Schmidt generates a well-conditioned set of vectors.

speaker : Julien Langou.
Conference : Seventh Copper Mountain Conference on Iterative Methods. 
Location : Colorado, USA, 
Date : March 28, 2002, 
Note : winner in the student paper competition - NB: there were 3 winners.


abstract :
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   Orthogonalization methods play a key role
    in many iterative methods. In this talk,
    we establish new properties for the
    modified Gram-Schmidt algorithm. We show
    why the modified Gram-Schmidt algorithm
    applied to a matrix generates a
    well-conditioned set of vectors. This
    result holds under the assumption that the
    initial matrix on which the algorithm is
    applied 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.

    We conclude by giving some applications
    of our results on orthogonalization schemes
    for iterative methods.