Numerical behaviour of the modified gram-schmidt GMRES implementation

摘要: In [6] the Generalized Minimal Residual Method (GMRES) which constructs Arnoldi basis and then solves transformed least squares problem was studied. It proved that GMRES with Householder orthogonalization-based implementation of process (HHA), see [9], is backward stable. practical computations, however, orthogonalization too expensive, it usually replaced by modified Gram-Schmidt (MGSA). Unlike HHA case, in MGSA orthogonality vectors not preserved near level machine precision. Despite this, MGSA-GMRES performs surprisingly well, its convergence behaviour ultimately attainable accuracy do differ significantly from those HHA-GMRES. As observed, but explained, [6], thelinear independence basis, precision, important. Until linear nearly lost, norms residuals match despite more significant loss orthogonality.