A note on augmented unprojected Krylov subspace methods

Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov in which is augmented with fixed spanned by vectors deemed be helpful accelerating convergence or conveying knowledge of the solution. Recently, survey was published, framework describing vast majority such proposed [Soodhalter et al., GAMM-Mitt., 43 (2020), Art. e202000016]. In many these methods, one generated system matrix composed projector that depends on space. However, it not requirement projected used. There built using subspaces original matrix, also fit into general framework. this note, we observe gains implementation benefits considering unprojected We demonstrate applying idea R3GMRES method [Dong Electron., Trans., Numer., Anal., 42 (2014), pp. 136–146] obtain simplified connect algorithm early based flexible preconditioning [Saad, SIAM J. Matrix Anal. Appl., 18 (1997)].