GMRES Algorithm: A Classical Iterative Method for Solving Large Linear Systems

In numerical algorithms, besides the previously mentioned conjugate gradient (CG) method, another classical approach is the GMRES (Generalized Minimal Residual) algorithm. GMRES is specifically designed to solve large-scale matrix equation problems, especially when the matrix is nonsymmetric or indefinite - scenarios where the CG algorithm becomes inapplicable. The algorithm operates iteratively by constructing an orthogonal basis using Arnoldi iteration and minimizing the residual norm within a Krylov subspace. This approach enables GMRES to effectively handle ill-conditioned systems through successive approximations to the solution. In practical implementations, GMRES typically requires restarting strategies (GMRES(m)) to manage memory usage when dealing with extremely large systems. Key computational steps involve orthogonalization procedures and solving a least squares problem at each iteration. Therefore, in real-world applications, selecting the appropriate algorithm based on specific problem characteristics - such as matrix symmetry, definiteness, and conditioning - becomes crucial for obtaining accurate solutions to matrix equations efficiently.