The paper introduces the Generalized Minimal Residual (GMRES) algorithm, a method for solving nonsymmetric linear systems. GMRES is derived from the Arnoldi process, which constructs an $l_2$-orthogonal basis of Krylov subspaces. It is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and ORTHODIR, but offers several advantages. GMRES cannot break down unless it has already converged, and it requires less storage and fewer arithmetic operations compared to GCR. The paper also discusses the practical implementation of GMRES, including efficient methods for computing the residual norm and the approximate solution. Numerical experiments are provided to compare GMRES with other methods, demonstrating its effectiveness and reliability.The paper introduces the Generalized Minimal Residual (GMRES) algorithm, a method for solving nonsymmetric linear systems. GMRES is derived from the Arnoldi process, which constructs an $l_2$-orthogonal basis of Krylov subspaces. It is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and ORTHODIR, but offers several advantages. GMRES cannot break down unless it has already converged, and it requires less storage and fewer arithmetic operations compared to GCR. The paper also discusses the practical implementation of GMRES, including efficient methods for computing the residual norm and the approximate solution. Numerical experiments are provided to compare GMRES with other methods, demonstrating its effectiveness and reliability.