Analysis of Augmented Krylov Subspace Methods

Residual norm estimates are derived for a general class of methods based on projection techniques on subspaces of the form $ K_m + {\cal W}$, where $K_m$ is the standard Krylov subspace associated with the original linear system and ${\cal W}$ is some other subspace. These "augmented Krylov subspace methods" include eigenvalue deflation techniques as well as block-Krylov methods. Residual bounds are established which suggest a convergence rate similar to one obtained by removing the components of the initial residual vector associated with the eigenvalues closest to zero. Both the symmetric and nonsymmetric cases are analyzed.