how to solve systems of linear equations

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1、CHAPTER17HowtoSolveSystemsofLinearEquationsTogetnumericalanswersoutofanylinearmodel,onemustintheendobtainthesolutionofasystemoflinearequations.Tocarryoutthistaskefficientlyhasthereforeahighpriority;itisnotsurprisingthatithasengagedtheattentionofsomeoftheleadingmathematicians.Twomethodsstillincur

2、rentuse,GaussianeliminationandtheGauss-Seideliteration,weredevisedbythePrinceofMathematicians.ThegreatJacobiinventedaniterativemethodthatbearshisname.Theavailabilityofprogrammable,high-performancecomputerswithlargememories-andremember,yesterday'shigh-performancecomputeristoday'spocketcomputer-ha

3、sopenedthefloodgates;thesizeandscopeoflinearequationsthatcouldbesolvedefficientlyhasbeenenlargedenormouslyandtheroleoflinearmodelscorrespondinglyenhanced.Thesuccessofthisefforthasbeenduenotonlytothehugeincreaseincomputationalspeedandinthesizeofrapidaccessmemory,butinequalmeasuretonew,sophisticat

4、ed,mathematicalmethodsforsolvinglinearequations.AtthetimevonNeumannwasengagedininventingandbuildingaprogrammableelectroniccomputer,hedevotedmuchtimetoanalyzingtheaccumulationandamplificationofround-offerrorsinGaussianelimination.OthernotableearlyeffortsweretheverystablemethodsthatGivensandHouseh

5、olderfoundforreducingmatricestoJacobiform(seeChapter18).Itisinstructivetorecallthatinthe1940slinearalgebrawasdeadasasubjectforresearch;itwasreadytobeentombedintextbooks.Yetonlyafewyearslater,inresponsetotheopportunitiescreatedbytheavailabilityofhigh-speedcomputers,veryfastalgorithmswerefoundfort

6、hestandardmatrixoperationsthatastoundedthosewhothoughttherewerenosurprisesleftinthissubject.Inthischapterwedescribeafewrepresentativemodernalgorithmsforsolvinglinearequations.Includedamongthem,inSection4,istheconjugategradientmethoddevelopedbyLanczos,Stiefel,andHestenes.LinearAlgebraandItsApplic

7、ations.SecondEdition,byPeterD.LaxCopyrightr;2007JohnWiley&Sons,Inc.246HOWTOSOLVESYSTEMSOFLINEAREQUATIONS247Thesystemsoflinearequationsconsideredinthischapterareoftheclassthathaveexactlyonesolution.Suchasystemcanbewritteninth