An\'alise do desempenho do m\'etodo GMRES

In this work, we are interested in the resolution of two kind of problems that have many practical applications. The first one is the resolution of a large linear system of equations, in which the coefficient matrix is sparse. We propose the iterative method GMRES (Generalized Minimal RESidual), introduced by Saad and Schultz, in 1986. At each iteration of this algorithm, one minimizes the residual norm in the Krylov subspace, using the Arnoldi method (equivalent to the Gram-Schimdt process) to compute an orthonormal basis for this subspace.The second problem is the resolution of nonlinear systems of equations. The classical Newton method is very attractive to solve this kind of problem, because it has fast convergence. But it requires the resolution of a linear system at each iteration, which can be too expensive. So, we propose the Inexact Newton method, which uses an iterative method to solve the linear equations. In this work, we combine the Newton method with GMRES applied to the linear system for the Newton step. Finally we present some numerical experiments in order to analyse the performance of Newton--GMRES. We use MATLAB codes written by C. T. Kelley, in 1995.