GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
Youcef Saad, Martin H. Schultz
Abstract
We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2-orthogonal basis of Krylov subspaces. It can be considered as a generalization of Paige and Saunders’ MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and to ORTHODIR. The new algorithm presents several advantages over GCR and ORTHODIR.
Topics & Concepts
Generalized minimal residual methodKrylov subspaceResidualMathematicsConjugate residual methodConjugate gradient methodLinear subspaceLinear systemAlgorithmApplied mathematicsOrthogonalizationIterative methodNorm (philosophy)GeneralizationMathematical optimizationComputer scienceMathematical analysisPure mathematicsMachine learningPolitical scienceLawArtificial neural networkGradient descentMatrix Theory and AlgorithmsStatistical and numerical algorithmsElectromagnetic Scattering and Analysis