数值分析-7-ludecompositionnew

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1、7.LUDecompositionandMatrixInverse•Gausseliminationisdesignedtosolveasinglesystem[A]{x}={b}.•Itisinefficientwhenwehavesystemswiththesamecoefficients[A]butdifferentright-hand-sideconstants{b}.oWehavetorepeatforwardeliminationandbackwardsubstitutionforeachsystem.oTheresultsoftheforwardeliminati

2、onofthematrix[A]fordifferentsystemsarethesame,whileitisthemostexpensivepartofGausselimination.•Findbetterways7.1LUDecomposition•Separatethetime-consumingeliminationofthematrix[A]fromthemanipulationof{b}.•Multipleright-hand-sidevectors{b}canbeevaluatedinanefficientmanneronce[A]hasbeen“decompo

3、sed”.DecomposeA=LU,whereLisalowertriangularmatrixandUisanuppertriangularmatrix.Then,thesystembecomes:Ax=b
L{Ux}=b.Defineanintermediatevectord=Ux.
Ld=b.Useforwardsubstitutiontocalculated.
Ux=d.Usebackwardsubstitutiontofindx.100L0U1,1U2,1U2,1LU2,1L10L00UULU1,22,12,12,1Ax=LUx=L1,3L

4、2,31L000U2,1LU2,1x=bMMMO0MMMOLLLL1000LUn1,n2,n3,2,1100L0L10L01,2⇒Ld=L1,3L2,31L0d=bforwardsubstitutiontofinddMMMO0LLLL1n1,n2,n3,U1,1U2,1U3,1LU,1n0UULU2,23,2,2n⇒Ux=00U3,3LU,3nx=dbackwardsubstitutiontofindxMMMO000LUn,n-1-7.1.2GaussEliminationas

5、LUDecompositionI)ForwardeliminationinGausselimination)0()0()0()0(100L00a11a12a13La1nm10L000a)1(a)1(La)1(1,222232nmm1L0000a)2(La)2(1,32,3333nA=MMOOM0MMMOMmmmL1M000a(n−)2a(n−)2n−1,1n−2,1n−3,1n−,1n−1n−,1n(n−)1mn1,mn2,mn3,mn4,L10000an,nThesuperscript(n)ina

6、denoteshowmanytimeswechangethecoefficientduringforward(j−)1(j−)1elimination.m=a/afori>jisthepivotelementwheneliminatingxifromthejthi,ji,jj,j)1()1(equation.(m=a/a,m=a/a,m=a/a)1,21,21,11,31,31,12,32,32,2Compactrepresentation)0()0()0()0(aaaLa1112131n)1()1()1(maaLa1,222232nmma)2(La)2(1,3

7、2,3333nA=MMMOMmmma(n−)2a(n−)2n−1,1n−2,1n−3,1n−,1n−1n−,1n(n−)1mn1,mn2,mn3,mn,n−1an,nExamples:214[A]=321133Gausselimination:[A]
(pivotforrow2is3/2=1.5,andpivotforrow3is½=0.5)21421402−5.1*1=5.01−4*5.1=−5
pivotis2.5/0.5=5
05.0−5