Post-Modern GMRES
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The GMRES algorithm of Saad and Schultz (1986) for nonsymmetric linear systems relies on the Arnoldi expansion of the Krylov basis. The algorithm computes the QR factorization of the matrix B = [ r_0, AV_k ] at each iteration. Despite an 𝒪(ϵ)κ(B) loss of orthogonality, the modified Gram-Schmidt (MGS) formulation was shown to be backward stable in the seminal papers by Paige, et al. (2006) and Paige and Strakoš (2002). Classical Gram-Schmidt (CGS) exhibits an 𝒪(ϵ)κ^2(B) loss of orthogonality, whereas DCGS-2 (CGS with delayed reorthogonalization) reduces this to 𝒪(ϵ) in practice (without a formal proof). We present a post-modern (viz. not classical) GMRES algorithm based on Ruhe (1983) and the low-synch algorithms of Świrydowicz et al (2020) that achieves 𝒪(ϵ) A v_k_2 /h_k+1,k loss of orthogonality. By projecting the vector A v_k, with Gauss-Seidel relaxation, onto the orthogonal complement of the space spanned by the computed Krylov vectors V_k where V_k^TV_k = I + L_k + L_k^T, we can further demonstrate that the loss of orthogonality is at most 𝒪(ϵ)κ(B). For a broad class of matrices, unlike MGS-GMRES, significant loss of orthogonality does not occur and the relative residual no longer stagnates for highly non-normal systems. The Krylov vectors remain linearly independent and the smallest singular value of V_k is not far from one. We also demonstrate that Henrici's departure from normality of the lower triangular matrix T_k ≈ ( V_k^TV_k )^-1 in the modified Gram-Schmidt projector P = I - V_kT_kV_k^T is an appropriate quantity for detecting the loss of orthogonality. Our new algorithm results in an almost symmetric correction matrix T_k.
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