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迭代求解非Hermitian正定线性方程组的衍生多分裂方法（英文）.docx

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迭代求解非Hermitian正定线性方程组的衍生多分裂方法（英文） IterativeMethodsforSolvingNon-HermitianPositiveDefiniteLinearSystems:ASurveyofDerivativeMulti-SplittingMethods Introduction Linearsystemsofequationsplayanintegralroleinvariousscientificandengineeringcomputations.Themethodofchoiceforsolvingsuchsystemsdependsontheparticularpropertiesofthesystemmatrix.HermitianpositivedefinitesystemsareoftensolvedusingiterativemethodssuchasConjugateGradient(CG)orPreconditionedCG(PCG)techniques.However,fornon-Hermitianpositivedefinitesystems,thesemethodsmaynotbeapplicable,andalternativemethodsarerequired. Thereareseveraliterativemethodsforsolvingnon-Hermitianpositivedefinitesystems.Thesemethodsarebasedontheideaofsplittingthecoefficientmatrixintoseveralsub-matricesandthenusingtheresultingfactorizationtoconstructiterativeschemes.Inthispaper,weshallsurveysomeofthederivativemulti-splittingmethodsforsolvingnon-Hermitianpositivedefinitelinearsystems. DerivativeMulti-SplittingMethods Thederivativemulti-splittingmethodsarebasedontheconceptofsplittingthecoefficientmatrixintoseveralsub-matricestofacilitatetheconstructionofiterativeschemes.Thesemethodsarederivedusingvariousderivativesofthesystemmatrix,whicharethenusedtoconstructthesub-matrices.Thebasicideaistousethederivativestosplitthecoefficientmatrixintoaproductofseveralsub-matrices,whereeachsub-matrixcanbeeasilyinvertedorapproximated.Theresultingfactorizationisthenusedtoconstructefficientiterativeschemesforsolvingthesystemofequations. Oneofthewell-knownderivativemulti-splittingmethodsistheRestrictedAdditiveSchwarzMethod(RASM).Thismethodisderivedbyusingthefirstderivativeofthesystemmatrix,whichisthenusedtoconstructasub-matrixthatcanbeeasilyinverted.TheRASMalgorithmisaniterativemethodthatinvolvesthesolutionofrestrictedsub-problemsateachiteration.TheRASMmethodisknowntoexhibitgoodconvergenceproperties,especiallywhenusedwithappropriatepreconditioners. Anotherderivativemulti-splittingmethodistheSymmetricBlockJacobi(SBJ)method.Thismethodisderivedbyusingthesecondderiva