## 7 Conclusions  
In this paper the GMRES method was re-examined from the two aspects of preserving the orthogonality and maximizing the decreasing length of the residual vector, both of which can significantly enhance the stability and accelerate the convergence speed of GMRES. If \(\eta_{k}=\mathbf{r}_{k}^{\mathrm{T}}\mathbf{A}\mathbf{u}_{k}/\|\mathbf{A} \mathbf{u}_{k}\|^{2}\) in GMRES is kept to \(\eta_{k}=1\) during the iterations, it is stable; otherwise, the GMRES is unstable. For any iterative form of \(\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k}\), we can improve it to \(\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k}(\mathbf{r}_{k}^{\mathrm{T}} \mathbf{A}\mathbf{u}_{k})/\|\mathbf{A}\mathbf{u}_{k}\|^{2}\), such that the orthogonality is preserved automatically, and simultaneously the length of the residual vector is reduced maximally. It brings the GMRES method to a re-orthogonalized GMRES (ROGMRES) method, preserving the orthogonality automatically and having the property of absolute convergence. We derived the new iterative form \(\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}_{k}(\mathbf{r}_{k}^{\mathrm{T}} \mathbf{A}\mathbf{u}_{k})/\|\mathbf{A}\mathbf{u}_{k}\|^{2}\) also from the invariant-manifold theory.  
In order to find a better descent vector, we solved a maximal projection problem in a larger affine Krylov subspace to derive the descent vector. We proved that the maximal projection is equivalent to the minimal residual length in the \(m\)-dimensional affine Krylov subspace. After taking the automatically orthogonality preserving method into account, a new orthogonalized maximal projection algorithm (OMPA) method was developed. We derived formulas similar to GMRES in the affine Krylov subspace, namely the affine GMRES (A-GMRES) method, which had superior performance for the problems tested in the paper; it is even better than OMPA for some problems. The algorithm A-GMRES possessed two advantages: solving a simpler least squares problem in the affine Krylov subspace, and preserving the orthogonality automatically.  
Through the studies conducted in the paper, the novelty points and main contributions are summarized as follows.  
* The GMRES was examined from the viewpoints of maximizing the decreasing quantity of the residual and to maintain the orthogonality of the consecutive residual vector.
* A stabilization factor, \(\eta\), to maintain orthogonality was inserted into GMRES to preserve the orthogonality automatically.
* GMRES was improved in a larger space of the affine Krylov subspace.
* A new orthogonalized maximal projection algorithm, OMPA, was derived; a new affine A-GMRES was derived.
* The new algorithms ROGMRES, OMPA, and A-GMRES guarantee the absolute convergence.
* Through examples, we showed that the orthogonalization techniques are useful to stabilize the methods of GMRES, MPA, and A-GMRES.
* Numerical testings for five different problems confirmed that the methods of ROGMRES, ROGMRES(m), OMPA, OMPA(m), A-GMRES, and A-GMRES(m) can significantly accelerate the convergence speed compared with the original methods of GMRES and GMRES(m).  
Conceptualization, C.-S.L. and C.-W.C.; Methodology, C.-S.L. and C.-W.C.; Software, C.-S.L., C.-W.C. and C.-L.K.; Validation, C.-S.L., C.-W.C. and C.-L.K.; Formal analysis, C.-S.L. and C.-W.C.; Investigation, C.-S.L., C.-W.C. and C.-L.K.; Resources, C.-S.L. and C.-W.C.; Data curation, C.-S.L., C.-W.C. and C.-L.K.; Writing--original draft, C.-S.L. and C.-W.C.; Writing--review & editing, C.-W.C.; Visualization, C.-S.L., C.-W.C. and C.-L.K.; Supervision, C.-S.L. and C.-W.C.; Project administration, C.-W.C.; Funding acquisition, C.-W.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data presented in this study are available on request from the corresponding authors.  
The authors declare no conflicts of interest.