The paper presents the Generalized Minimal Residual (GMRES) algorithm for solving nonsymmetric linear systems. GMRES is derived from the Arnoldi process for constructing an $ l_2 $-orthogonal basis of Krylov subspaces. It is a generalization of the MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and ORTHODIR. GMRES minimizes the residual norm over a Krylov subspace at each step and is more stable and efficient than GCR and ORTHODIR. Key features of GMRES include its ability to handle indefinite systems, its mathematical equivalence to GCR and ORTHODIR, and its efficiency in terms of storage and arithmetic operations. GMRES is particularly useful for systems where the coefficient matrix is not positive real, as it does not break down unless it has already converged. The algorithm requires only half the storage of GCR and fewer arithmetic operations. The paper also discusses the theoretical aspects of GMRES, including its convergence properties and the conditions under which it breaks down. It compares GMRES with other methods like GCR and ORTHODIR, showing that GMRES is more efficient and reliable. Numerical experiments demonstrate that GMRES performs well, especially with preconditioning, and that it can converge for sufficiently large $ m $ in the restarted version GMRES(m). The results show that GMRES is effective for solving large sparse nonsymmetric systems of linear equations.The paper presents the Generalized Minimal Residual (GMRES) algorithm for solving nonsymmetric linear systems. GMRES is derived from the Arnoldi process for constructing an $ l_2 $-orthogonal basis of Krylov subspaces. It is a generalization of the MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and ORTHODIR. GMRES minimizes the residual norm over a Krylov subspace at each step and is more stable and efficient than GCR and ORTHODIR. Key features of GMRES include its ability to handle indefinite systems, its mathematical equivalence to GCR and ORTHODIR, and its efficiency in terms of storage and arithmetic operations. GMRES is particularly useful for systems where the coefficient matrix is not positive real, as it does not break down unless it has already converged. The algorithm requires only half the storage of GCR and fewer arithmetic operations. The paper also discusses the theoretical aspects of GMRES, including its convergence properties and the conditions under which it breaks down. It compares GMRES with other methods like GCR and ORTHODIR, showing that GMRES is more efficient and reliable. Numerical experiments demonstrate that GMRES performs well, especially with preconditioning, and that it can converge for sufficiently large $ m $ in the restarted version GMRES(m). The results show that GMRES is effective for solving large sparse nonsymmetric systems of linear equations.