Numerical Linear Algebra with Applications · 1994 · 91 citations · 7 references
ComputingComputational EngineeringMathematicsComputational MechanicsFinite Element MethodsOptimizationIterative SolversMatrix ComputationNumerical AlgorithmsNumerical AnalysisNumerical Linear AlgebraContinuous ModelsGmresGmres RestartingIterative Methods For Linear SystemsLinear System SolversPreconditioning TechniquesSparse Matrix AlgorithmsSparse Matrix Techniques
Abstract The generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar 0 instead of r 0 , we obtain simpler Gram–Schmidt and Householder implementations of the GMRES method that do not require upper Hessenberg factorization. The Gram–Schmidt implementation also maintains the residual vector at each iteration, which allows cheaper restarts of GMRES( m ) and may otherwise be useful.
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