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1、AnIntroductiontotheConjugateGradientMethodWithouttheAgonizingPainEdition114JonathanRichardShewchukAugust4,1994SchoolofComputerScienceCarnegieMellonUniversityPittsburgh,PA15213AbstractTheConjugateGradientMethodisthemostprominentiterativemethodforsolvingsparsesystemsoflinearequations.Unfort
2、unately,manytextbooktreatmentsofthetopicarewrittenwithneitherillustrationsnorintuition,andtheirvictimscanbefoundtothisdaybabblingsenselesslyinthecornersofdustylibraries.Forthisreason,adeep,geometricunderstandingofthemethodhasbeenreservedfortheelitebrilliantfewwhohavepainstakinglydecodedth
3、emumblingsoftheirforebears.Nevertheless,theConjugateGradientMethodisacompositeofsimple,elegantideasthatalmostanyonecanunderstand.Ofcourse,areaderasintelligentasyourselfwilllearnthemalmosteffortlessly.TheideaofquadraticformsisintroducedandusedtoderivethemethodsofSteepestDescent,ConjugateDi
4、rections,andConjugateGradients.EigenvectorsareexplainedandusedtoexaminetheconvergenceoftheJacobiMethod,SteepestDescent,andConjugateGradients.OthertopicsincludepreconditioningandthenonlinearConjugateGradientMethod.Ihavetakenpainstomakethisarticleeasytoread.Sixty-sixillustrationsareprovided
5、.Denseproseisavoided.Conceptsareexplainedinseveraldifferentways.Mostequationsarecoupledwithanintuitiveinterpretation.SupportedinpartbytheNaturalSciencesandEngineeringResearchCouncilofCanadaundera1967ScienceandEngineeringScholarshipandbytheNationalScienceFoundationunderGrantASC-9318163.The
6、viewsandconclusionscontainedinthisdocumentarethoseoftheauthorandshouldnotbeinterpretedasrepresentingtheofficialpolicies,eitherexpressorimplied,ofNSERC,NSF,ortheU.S.Government.Keywords:conjugategradientmethod,preconditioning,convergenceanalysis,agonizingpainContents1.Introduction12.Notation
7、13.TheQuadraticForm24.TheMethodofSteepestDescent65.ThinkingwithEigenvectorsandEigenvalues95.1.EigendoitifItry::::::::::::::::::::::::::::::::::::::95.2.Jacobiiterations:::::::::::::::::::::::::::::::::::::::115.3.AConcreteExample:::::::::::::::::::::::::::::::::::::
