Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES

@article{Paige2006ModifiedG,
  title={Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES},
  author={Christopher C. Paige and Miroslav Rozlozn{\'i}k and Zdenek Strakos},
  journal={SIAM J. Matrix Analysis Applications},
  year={2006},
  volume={28},
  pages={264-284}
}
The generalized minimum residual method (GMRES) [Y. Saad and M. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869] for solving linear systems Ax = b is implemented as a sequence of least squares problems involving Krylov subspaces of increasing dimensions. The most usual implementation is Modified Gram-Schmidt GMRES (MGS-GMRES). Here we show that MGS-GMRES is backward stable. The result depends on a more general result on the backward stability of a variant of the MGS algorithm… CONTINUE READING

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