解线性方程组的预条件AOR迭代法(3)

≥[(I E1 E2)+E1+E2 rL rE1 rF1 rRL+rR](1 ρ(Lrw))xw1

≥[(E1+E2)+(I E1 E2) rL rE1 rF1 rRL+rR rF2](1 ρ(Lrw))xw1

=[(E1 rE1+E2)+(I E1 E2) r(L R+RL+F1+F2)](1 ρ(Lrw))x.wDenoteE1 rE1+E2=D,then

¯=1[D+(I E1 E2) r(L R+RL+F1+F2)](1 ρ(Lrw))xAx

w1¯ rL¯](1 ρ(Lrw))x.=[D+Dw

denote

¯ rL¯) 1[(1 w)D¯+(w r)L¯+wU¯]=ER FR,¯=(I+S+R)A=1(DA

ww

ER=

Then

1¯¯),FR=1[(1 w)D¯+(w r)L¯+wU¯].(D rLww

¯=ER(I L¯rw)xAx

1

≥(D+wER)(1 ρ(Lrw))x.w

¯rw)x≥1E 1D(1 ρ(Lrw))x+(1 ρ(Lrw))x.(I L

wR

¯rw)x>(1 ρ(Lrw))x,L¯rwx<ρ(Lrw)x.(I L

So

¯rw)<ρ(Lrw)<1.Formtheorem2.1.11([1]),weobtainρ(L

¯wbetheiterativematricesoftheSORmethodgiven(1.3)and(1.11),Corollary4.LetLwandL

respectively.If0<w<1,Aisannon-singularirreducibleM-matrixandaii+1ai+1i>0,i=

¯w)<ρ(Lw)ifρ(Lw)<1.2,3,...,n 1,µn>0Thenρ(L

4Comparedtheoremof(I+S)and(I+S+R)

Inthissection,weestablishcomparedtheorembetween(I+S)and(I+S+R)corresponding

AORmethod.

英文文献,解线性方程组的预条件AOR迭代法。

10

rwandL¯Theorem4.1LetAbeannon-singularM-matrix,L n 1rwbetheiterativematricesofthe

AORmethodsgiven(1.12)and(1.11),respectively.Ifk=1 ankakj≥0,j=1,2,...,n 1,0≤ n 1

¯

k=1ankakn<1,and0≤r≤w≤1(w=0)(r=1).Thenρ(Lrw)≤ρ(Lrw)ifρ(Lrw)<1.Proof.From(1.4)(1.6),wehave

¯ A =(I+S+R)A (I+S)A=RA≥0.A

1≥A¯ 1≥0.ThenA

Denote

=(I+S)A=1(D rL ) 1[(1 w)D +(w r)L +wU ]=ES FS,A

ww

ES=

Bycalculation,weobtain

1 ),FS=1[(1 w)D +(w r)L +wU ].(D rLww

1 1

ER≥0,ES≥0,FR≥0,FS≥0.

=ES FS,andA¯=ER FRareregularsplitting.Moreover,So,A

FS FR=

1 +(w r)L +wU ] 1[(1 w)D¯+(w r)L¯+wU¯][(1 w)D

ww1

=[(1 w)E2+(w r)(R RL F2)]w≥0.

1FS≥A¯ 1FR≥0.A

x,

ThenFS≥FR≥0,and

bythemonotoneoffunctionf(x)=

wehave

¯rw)≤ρ(L rw).ρ(L

wandL¯Corollary5.LetAbeannon-singularM-matrix,L n 1wbetheiterativematricesofthe

SORmethodgiven(1.12)and(1.11),respectively.Ifk=1 ankakj≥0,j=1,2,...,n 1,0≤ n 1

¯

k=1ankakn<1,and0≤r≤w≤1(w=0)(r=1).Thenρ(Lw)≤ρ(Lw)ifρ(Lw)<1.

5Numericalexample

1

1 1

1

. ..

1

Nowletusconsiderthefollowingexamplestoillustratetheresultsobtained:

Example1.Thecoe cientmatrixAisgivenby

111

···1 11 1

1 ··· 111

1 ··· 111A= 1··· ..... ..........

1111

···

英文文献,解线性方程组的预条件AOR迭代法。

11

Table1:spectralradiiforfourmethods(AOR)

n50100150200

w0.90.950.90.8

r0.80.80.70.65

ρ(Lrw)0.23590.21500.29030.3858

rw)ρ(L0.23550.21480.29020.3857

rw)ρ(L0.22630.20700.28430.3811

¯rw)ρ(L0.22590.20680.28420.3810

Thefollowingtabledisplaysthespectralradiusofthecorrespondingiterativematrixwithdi erent

parameterswandr.ThedigitalsareformedbyMatlab6.51program:FortheSORmethod,wehavethefollowingresults:

Table2:spectralradiiforfourmethods(SOR)

n50100150200

w0.950.90.80.9

r0.950.90.80.9

ρ(Lw)0.15290.22990.34850.2458

w)ρ(L0.15250.22980.34840.2457

w)ρ(L0.14110.22150.34270.2390

¯w)ρ(L0.14070.22140.34260.2389

6References

References

[1]A.BermanandR.J.Plemmoms,Nonnegativematricesinthemathematicalsciences,SIAM,

Philadelphia,PA,1994.[2]D.J.Evans,M.M.Martins,M.E.Trigo,TheAORiterativemethodfornewpreconditioned

linearsystems,Comput.Appl.Math.132(2001)461–466.[3]A.D.Gunawardena,S.K.Jain,L.Snyder,Modi editerativemethodsforconsistentlinearsys-tem,LinearAlgebraAppl.154/156(1991)123–143.[4]A.Hadjimos,Acceleratedoverrelaxationmethod,put.32(1978)149–157.

[5]Y.Li,C.Li,S.Wu,ImprovingAORmethodforconsistentlinearsystems,Comput.Appl.

Math.186(2007)379–388.[6]W.Li,Theconvergenceofthemodi edGauss–Seidelmethodsforconsistentlinearsyatems,J.

Comput.Appl.Math,154(2003)97–105.[7]W.Li,AnoteonthepreconditionedGauss–Seidelmethodforlinearsystems,put.

Appl.Math,182(2005)81–90.[8]M.Morimoto,H.Kotakemori,T.KohnoandH.Niki,TheGauss–seidelmethodwithprecondi-tioner(I+R),TransactionsoftheJapanSocietyforIndustrialandAppliedMathematics,13(2003)439–445.

英文文献,解线性方程组的预条件AOR迭代法。

12

[9]H.Niki,K.Harada,M.Morinoto,andM.Sakakihara,Thesurveyofpreconditionersusedfor

acceleratingtherateofcinvergenceintheGauss–seidelmethod,put.Appl.Math,164–165(2004)587–600.[10]R.S.Varga.MatrixIterativeAnalysis,Springer,Berlin,2000.

[11]D.M.Young,IterativeSolutionofLargeSystems,AcademicPress,NewYork,1971.