Stability of Linear GMRES Convergence with Respect to Compact Perturbations

GMRES algorithms over 35 years

“…Simoncini and Szyld [242] addressed the same problem using spectral projectors. Moret [189] discussed superlinear convergence of GMRES in a complex separable Hilbert space; see also [33] for a generalization of Moret's result. Since any nonincreasing convergence curve is possible for GMRES [126], on the other hand, stagnation may occur even for the original GMRES method without restarting.…”

“…where A ∈ R N ×N , D ∈ R S×S , and B ∈ R N ×S . A block GMRES algorithm for solving (33) was described by Robbé and Sadkane [212] in 2002; see also [238,292]. It is known that a family of shifted systems can be interpreted as a Sylvester equation of the form (33) where D is a diagonal matrix; see [247,84].…”

“…A block GMRES algorithm for solving (33) was described by Robbé and Sadkane [212] in 2002; see also [238,292]. It is known that a family of shifted systems can be interpreted as a Sylvester equation of the form (33) where D is a diagonal matrix; see [247,84]. GMRES has also been extended to solve least square problems [137], tensor equations [57], and quaternion linear systems [148].…”