With zero initial guess, the generalized minimal residual method minimizes the Euclidean norm of the linear-system residual over each Krylov subspace. The Arnoldi decomposition reduces this to , followed by . Givens rotations solve the small problem in operations once the decomposition is available. For an invertible matrix , the same iterate minimizes the error in the weighted norm induced by , because .
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The Generalized Minimal Residual (GMRES) method is an iterative algorithm used to solve large, sparse systems of linear equations, particularly those that arise from discretizing partial differential equations. It is particularly effective for nonsymmetric and non-positive definite matrices. ### Key Features of GMRES: 1. **Iterative Method**: GMRES is an iterative method, meaning it generates a sequence of approximations to the solution rather than working towards an exact solution in a finite number of steps.