Iterated Gauss–Seidel GMRES

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869] is an iterative method for approximately solving linear systems , with initial guess residual . employs the Arnoldi process to generate Krylov basis vectors (the columns ). It well known that this can be viewed as a factorization matrix at each iteration. Despite loss orthogonality, unit roundoff condition number modified Gram–Schmidt formulation was shown backward stable in seminal paper by Paige et al. Matrix Anal. Appl., 28 (2006), 264–284]. We present iterated Gauss–Seidel (IGS-GMRES) based on ideas Ruhe [Linear Algebra 52 (1983), 591–601] Świrydowicz [Numer. Linear (2020), 1–20]. IGS-GMRES maintains orthogonality level or depending choice one two iterations; iterations, computed remain orthogonal working accuracy smallest singular value remains close one. resulting thus stable. show implemented only single synchronization point per iteration, making it relevant large-scale parallel computing environments. also demonstrate that, unlike MGS-GMRES, relative corresponding approximate solution no longer stagnates above machine precision even highly nonnormal systems.