数值分析session-3

1Summaryoflastsession-Chapter4.SolvingSystemsofLinearEquations4.0.Introduction4.1.MatrixAlgebra:BasicConcepts,ElementaryOperation,EquivalentSystem,InverseMatrix.4.2.LUFactorization:EasytoSolveSystem,LUFactorization,LDUFactorization,DoolittleFactorization,Crout’sFactorization,CholeskyFactorization.24.3PivotingandConstructingAlgorithm-BasicGaussianEliminationHereisasimplesystemoffourequationsinfourunknownsthatwillbeusedtoillustratetheGaussianalgorithm:62241286103139364BasicGaussianElimination:12341234(1)2711838Inthefirststepoftheprocess,wesubtract2timesthefirstequationfromsecond,andthenwesubtract1/2timesxxxxthefirstequationfromthird.Finally,wesubtract1timesthefirstequationfromthefourth.Thenumbers2,1/2and1arecalledthemultipliersforthefirststepintheeliminationprocess.3Thenumber6,usedasthedivisorintheformingeachofthesemultipliers,iscalledtheforthisstep.Afterthefirststephasbeencompleted,thespivotelement1234ystembecome622412042210(2)01281210231426Althoughthefirstrowwasusedintheprocess,itxxxxwasnotchanged.Inthisfirststep,werefertorow1asthe.Inthenextstepoftheprocess,row2isusedasthepivotrowand4isthepivotelemenpivotrowt.Wesubtract3timesthesecondrowfromthethird,andthen1/2timesthesecondrowissubtractedfromthefourth.4.3PivotingandConstructingAlgorithm-BasicGaussianElimination41234Themultipliersforthisstepare3and1/2.Theresultis622412042210(3)002590041321Thefinalstepconsistsofsuxxxx1btracting2timesthethirdrowfromthefourthsothat2isboththemultiplierandthepivotelement.Theresultingsystemis6224042200250003xx2341210(4)93xx4.3PivotingandConstructingAlgorithm-BasicGaussianElimination5Thissystemisuppertriangularandtotheorigionalsysteminthesencethatthesolutionsofthetwosystemsarethesame.Thefinalsystemiseasiequivalentlysolvedbythefourthrowandworkingbackwarduptherows.Thesolutionis1321x4.3PivotingandConstructingAlgorithm-BasicGaussianElimination6Themultipliersusedintransformingthesystemcanbeexhibitedinaunittriangularmatrix()10002100(5)1/231011/221NoticethateachmultijLlLiplieriswritteninthelocationcorrespondingtothe0entryinthematrixitwasreasonableforcreating.Thecoefficientmatrixofthefinalsystemisanuppertriangularmatrix62240422(6)00250003U4.3PivotingandConstructingAlgorithm-BasicGaussianElimination7Thesetwomatricesgivethe-factorizationof,whereisthecoefficientmatrixoftheorigionalsystem.Thus,10006224210012861013103139326411811212LUAA62240422(7)002500034.3PivotingandConstructingAlgorithm-BasicGaussianElimination81.2,.i.e000000.ofrowsecondeoccupyth1and2tscoefficienThe.22Hence,.2example,forus,givesprocessneliminatioethenth,,,,,ofrowsand,,,,ofrowsthedenoteweIf.fromgetcanweprocessthereversingbythen,fromobtainedwashowknowweIftrue.bemustswhythiseetohardnotisIt222143214443424133323122211143212121212243214321llllllllllllUUUUAAAALUUUAAAAUUUUUUAAAAAUAAU4.3PivotingandConstructingAlgorithm-BasicGaussianElimination9331231231233132Similarly,(1/2)3,andfinallywehave1/231/23Thecoefficients1/2,3,and1mustthereforeoccupythethirdrowofi.e.1/2,3,llUAAUAAUUUUUL3341424344and1,andcorrespondlythefourthrowofis1,-1/2,2and1.TodescribeformallytheprogressoftheGaussianalgorithm,weinterpretitasasulllllL(1)(2)(n)ccessionof1majorstepsresultinginasequenceofmatricesasfollows:n-AAAAL4.3PivotingandConstructingAlgorithm-BasicGaussianElimination10(k)Attheconclusionofstep-1,thematrixwillhavebeenconstructed;itsappearanceisshowninthefollowingdisplayinwhichlinesareplacedaroundthethrokkA()()()()()111,1111()()()()1,11,1,(k)wandjustbeforethethcolumntoillustratethestructureproducedbytheeliminationprocess:0kkkkkkkjnkkkkkkkkkjkaaaaaaaaaALLLMOMMMKLK1,()()()()()()1,1,1,()()()()()()00000000knkkkkkkjknkkkkkkjknkkkikijinkkknknjnnaaaaaaaaaaaaKKKKKLMMMMMKKKMMMMMKKK4.3PivotingandConstructingAlgorithm-BasicGaussianElimination11(1)Nextwedescribehowisobtainedfrom.Toproduce0'sincolumnbelowthepivotelement,wesubtractmultiplesofrowfromtherowsbeneathit.Rowsk(k)(k)kkkakAA(k)(k1)(k)(k)(k)(k)12arenotalerted.Theformulaisthereforeifif1and10ijijijikkkkj,,,kaikaaaaaikjkL(8)if1andikjk4.3PivotingandConstructingAlgorithm120.iselementspivottheofanyifdownbreakwillprocessneliminatioentireisthatth(9)and(8)fromclearbeshouldItangular.uppertriandangularlowertriunitwithmatrixofionfactorizatGaussianstandardtheisHere(9)1if0if11ifbydefineandsetThenwe(k)(k)ULALUALAUk-ikikiaalkkkiik(n)4.3PivotingandConstructingAlgorithm13ijijiijiij1iij3ij2ij2ijijiij1kij1i1kkijiijkkj1i1kkkkkikiijii1i1kkkjiki1ki1kkkjikkkjikn1kkjikijkjikkkjnkjkjkij1kij(k)kkaaaaaaaaaaaaaaaaalalalululjijk0uik0laau1kjkiaaa)1()()()()()()()1()()()()()()()()()()()()()()()()()(havewefacts,theseUsing.letNow.if,