Re-Orthogonalized/Affine GMRES and Orthogonalized Maximal Projection Algorithm for Solving Linear Systems

GMRES is one of the most powerful and popular methods to solve linear systems in the Krylov subspace; we examine it from two viewpoints: to maximize the decreasing length of the residual vector, and to maintain the orthogonality of the consecutive residual vector. A stabilization factor, $\eta$, to measure the deviation from the orthogonality of the residual vector is inserted into GMRES to preserve the orthogonality automatically. The re-orthogonalized GMRES (ROGMRES) method guarantees the absolute convergence; even the orthogonality is lost gradually in the GMRES iteration. When $\eta <1/2$, the residuals’ lengths of GMRES and GMRES(m) no longer decrease; hence, $\eta <1/2$ can be adopted as a stopping criterion to terminate the iterations. We prove $\eta =1$ for the ROGMRES method; it automatically keeps the orthogonality, and maintains the maximality for reducing the length of the residual vector. We improve GMRES by seeking the descent vector to minimize the residual in a larger space of the affine Krylov subspace. The resulting orthogonalized maximal projection algorithm (OMPA) is identified as having good performance. We further derive the iterative formulas by extending the GMRES method to the affine Krylov subspace; these equations are slightly different from the equations derived by Saad and Schultz (1986). The affine GMRES method is combined with the orthogonalization technique to generate a powerful affine GMRES (A-GMRES) method with high performance.