GMRES
GMRES (Generalized Minimal Residual) is an iterative numerical method for solving large, sparse, non-symmetric systems of linear equations. It works by constructing an orthogonal basis for the Krylov subspace and minimizing the residual norm at each iteration, making it effective for problems where direct methods are computationally prohibitive. It is widely used in computational fluid dynamics, structural analysis, and other scientific computing applications.
Developers should learn GMRES when working on simulations or engineering problems that involve solving large linear systems from discretized partial differential equations, as it efficiently handles sparse matrices common in finite element or finite difference methods. It is particularly useful in high-performance computing contexts where memory and time constraints favor iterative solvers over direct factorization methods like Gaussian elimination.