## 1 Introduction
For seeking a fast convergent iterative algorithm, we must keep the orthogonality, and simultaneously maximize the projection quantity \(\left\|\mathbf{W}\mathbf{y}\right\|=\left\|\mathbf{A}\mathbf{u}\right\|\). Later on, we will modify the GMRES method from these two aspects. [PERSON] [12] derived an effective iterative algorithm based on the maximal projection method to solve Equation (1) in the column space of \(\mathbf{A}\). Recently, [PERSON] et al. [13] applied the maximal projection method and the minimal residual norm method to solve the least-squares problems. There are two reasons to cause the slowdown and even the stagnation of GMRES: one is the loss of orthogonality, and other is that its descent vector, \(\mathbf{u}=\mathbf{V}\mathbf{y}\), is not the best one. There are different methods to accelerate and improve GMRES [14; 15; 16; 17]. The restart technique is often applied to the Krylov subspace methods, which, however, generally slows their convergence. Another way to accelerate the convergence is updating the initial value [18; 19]. The preconditioning technology is often used to speed up GMRES and other Krylov subspace methods [20; 21; 22; 23; 24]. The multipreconditioned GMRES [25] is also proposed by using two or more preconditioners simultaneously. The polynomial preconditioned GMRES and GMRES-DR were also developed in [26]. As an augment of the Krylov subspace, some authors sought better descent vectors in a larger space to accelerate the convergence of GMRES [27; 28; 29]. Recently, [PERSON] et al. [30] presented mathematically important and practically relevant phenomena that uncover the behavior of GMRES through a discussion of computed examples; they considered the conjugate gradient (CG) and GMRES methods crucially important for further development and practical applications, as well as other Krylov subspace methods. **Contribution and Novelty:**  
The major contribution and novelty of the present paper are that we propose a quite simple strategy to overcome the slowdown and stagnation behavior of GMRES by inserting a stabilization factor, \(\eta=\mathbf{r}^{\mathrm{T}}\mathbf{A}\mathbf{u}/\|\mathbf{A}\mathbf{u}\|^{2}\), into the iterative equation. In doing so, the orthogonality property of the residual vector and the stepwise decreasing property of the residual length are automatically preserved. We extend the space of GMRES to an affine Krylov subspace for seeking a better descent vector. When the maximal projection method and the affine technique are combined into the orthogonalization technique, two very powerful iterative algorithms for solving linear systems with high performance are developed. **Highlight:**  
* Examine the iterative algorithm for linear systems from the viewpoint of maximizing the decreasing quantity of the residual and maintaining the orthogonality of the consecutive residual vector. * A stabilization factor to measure the deviation from the orthogonality is inserted into GMRES to preserve the orthogonality automatically. * The re-orthogonalized GMRES guarantees the preservation of orthogonality, even if the orthogonality is gradually lost in the iterations by means of GMRES. * Improve GMRES by seeking the descent vector to minimize the length of the residual vector in a larger space of the affine Krylov subspace. * The new algorithms are all absolute convergence. **Outline:**  
The Arnoldi process and the conventional GMRES and GMRES(m) are described in Section 2. In Section 3, we propose a new algorithm modified from GMRES by inserting a stabilization factor, \(\eta\), which is named a re-orthogonalized GMRES (ROGMRES). We discuss ROGMRES from two aspects: preserving the orthogonality and maximally decreasing the length of the residual vector; we also prove that ROGMRES can automatically maintain \(\eta=1\), and preserve the good property of orthogonality and the maximality of reducing the residual.