Flexible GMRES with Deflated Restarting
SIAM Journal on Scientific Computing
An Augmented Stability Result for the Lanczos Hermitian Matrix Tridiagonalization Process
SIAM Journal on Matrix Analysis and Applications
GMRES Methods for Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
On Efficient Numerical Approximation of the Bilinear Form $c^*A^{-1}b$
SIAM Journal on Scientific Computing
Estimates of the Norm of the Error in Solving Linear Systems with FOM and GMRES
SIAM Journal on Scientific Computing
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems
SIAM Journal on Scientific Computing
On the generation of Krylov subspace bases
Applied Numerical Mathematics
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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 applied to solving a linear least squares problem, and uses other new results on MGS and its loss of orthogonality, together with an important but neglected condition number, and a relation between residual norms and certain singular values.