解线性方程组的预条件AOR迭代法(2)

arethecoe cientmatricesofthelinearsystems(1.1)and(1.5),Lemma3.3LetAandA

respectively.If0≤r≤w≤1(w=0)(r=1),if1 µn>0andAisanirreducibleL-matrix

rwassociatedtotheAORmethodappliedtothelinear.ThentheiterativematricesLrwandL

systems(1.1)and(1.5),respectively,arenon-negativeandLrwisirreducible.

Proof.FromthatAisanL-matrix,wehaveL≥0isastrictlylowertriangularmatrixandU≥0isastrictlyuppertriangularmatrix.So(I rL) 1=I+rL+r2L2+···+rn 1Ln 1≥0.By(1.3),wehave

Lrw=(I rL) 1[(1 w)I+(w r)L+wU]

=[I+rL+r2L2+···+rn 1Ln 1][(1 w)I+(w r)L+wU]=(1 w)I+(w r)L+wU+rL(1 w)I+rL[(w r)L+wU]+(r2L2+···+rn 1Ln 1)×[(1 w)I+(w r)L+wU]=(1 w)I+w(1 r)L+wU+T.

where

T=rL[(w r)L+wU]+(r2L2+···+rn 1Ln 1)×[(1 w)I+(w r)L+wU]≥0.SoLrwisnon-negative.Wecanalsogetthat(1 w)I+w(1 r)L+wUisirreducibleforAisirreducible,henceLrwisirreducible.By(1.10),wehave

rw=(D rL ) 1[(1 w)D +(w r)L +wU ]L

1L ) 1[(1 w)I+(w r)D 1L +wD 1U ]=(I rD

1L +wD 1U +T. =(1 w)I+w(1 r)D

where

=rD 1L [(w r)D 1L +wD 1U ]+[r2(D 1L )2+···+rn 1(D 1L )n 1]T

1L +wD 1U ]≥0.×[(1 w)I+(w r)D

≥0andL rw≥0fromD ≥0,L ≥0andU ≥0.AsL rw,wehaveL rwisSowehaveT

non-negative.

英文文献,解线性方程组的预条件AOR迭代法。

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Similarly,wehavethefollowinglemma.

¯arethecoe cientmatricesofthelinearsystems(1.1)and(1.6).Lemma3.4LetAandA

respectively.If0≤r≤w≤1(w=0)(r=1),AisanirreducibleL-matrixand1 µn>

¯rwassociatedto0,1 aii+1ai+1i>0,(i=1,2,...,n 1).ThentheiterativematricesLrwandL

theAORmethodappliedtothelinearsystems(1.1)and(3.3),respectively,arenon-negativeandLrwisirreducible.Theorem3.5LetLrw(1.10),respectively.If1 µn>0.Then

rw)≤ρ(Lrw)if(1)ρ(L

rw)=ρ(Lrw)if(2)ρ(L

rw)≥ρ(Lrw)if(3)ρ(L

rwbetheiterativematricesoftheAORmethodgiven(1.3)andandL

0≤r≤w≤1(w=0)(r=1),AisanirreducibleL-matrixandifρ(Lrw)<1;ρ(Lrw)=1;ρ(Lrw)>1.

Proof.FromLemma3.3,itisclearthatLrwisnon-negativeandirreduciblematrices.Thus,fromLemma2.1thereexistsapositivevectorx,suchthat

Lrwx=λx,

whereλ≡ρ(Lrw)or,equivalently,

[(1 w)I+(w r)L+wU]x=λ(I rL)x.

Therefore,forthisx>0,

rwx λx=(D rL ) 1[(1 w)D +(w r)L +wU ]x λxL

rL ) 1[(1 w)D +(w r)L +wU λ(D rL )]x=(D

rL ) 1[(1 w)I (1 w)E+(w r)L+(w r)=(D

×( R+RL+F)+wU λ(I E)+λr(L R+RL+F)]x

rL ) 1[(w 1+λ)E+(w r)( RA E)+λr( RA E)]x=(D

rL ) 1[(λ 1)(1 r)E+(λ 1)w r+λrR(I rL)]x=(D 1w r+λr rL )[(1 r)Ex+=(λ 1)(DR(I rL)x]

(3.1)

rL )=(λ 1)(D

1

C.

+λr

WhereC=(1 r)Ex+w rR(I rL)x≥0. rwx λx≤0.Therefore,L rwx≤λx.ByLemma2.2,weget(1)If0<λ<1,thenL

rw)≤λ=ρ(Lrw).ρ(L

rwx λx=0.ByLemma2.2,wegetρ(L rw)=λ=ρ(Lrw).(2)Ifλ=1,thenL

rwx λx≥0.Therefore,L rwx≥λx.ByLemma2.2,weget(3)Ifλ>1,thenL

rw)≥λ=ρ(Lrw).ρ(L

¯rwbetheiterativematricesoftheAORmethodgiven(1.3)andTheorem3.6LetLrwandL

(1.11),respectively.If0≤r≤w≤1(w=0)(r=1),AisanirreducibleL-matrixand1 µn>0,1 aii+1ai+1i>0,(i=1,2,...,n 1).Then

¯rw)≤ρ(Lrw)ifρ(Lrw)<1;(1)ρ(L

¯rw)=ρ(Lrw)ifρ(Lrw)=1;(2)ρ(L

¯rw)≥ρ(Lrw)ifρ(Lrw)>1.(3)ρ(L

英文文献,解线性方程组的预条件AOR迭代法。

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Proof.FromLemma3.4,itisclearthatLrwisnon-negativeandirreduciblematrices.Thus,fromLemma2.1thereexistsapositivevectorx,suchthat

Lrwx=λx,

whereλ≡ρ(Lrw)or,equivalently,

[(1 w)I+(w r)L+wU]x=λ(I rL)x.

¯ rL¯) 1≡B,E3=E1+E2,L¯=L+D.WeknowB≥0,E3≥0andL¯≥0.Denote(D

Therefore,forthisx>0,

¯rwx λx=(D¯ rL¯) 1[(1 w)D¯+(w r)L¯+wU¯]x λxL

¯+(w r)L¯+wU¯ λ(D¯ rL¯)]x=B[(1 w)D

=B[(1 w)(I E3)+(w r)(L+D)+w(U S+SU) λ(I E3)+λr(L+D)]x=B[(w 1)E3+(w r)D+w( S+SU)+λE3+λrD]x.DenoteD=H E3,H= R+RL+RU+SL= RA+SL.Byusingof(3.2),weget

¯rwx λx=B[(λ 1)(1 r)E3+(w r)H+w( S+SU)+λrH]xL

=B[(λ 1)(1 r)E3+(λ 1)(S+R)(I rL)+r(1 λ)RA+r(λ 1)SL]x

w r+λr

=(λ 1)B[(1 r)E3+S+R(I rL)]x.

w

+λr

WhereN=[(1 r)E3+S+w rR(I rL)]x≥0.Henceweobtainthat¯rwx λx≤0.Therefore,L¯rwx≤λx.ByLemma2.2,weget(1)If0<λ<1,thenL

¯rw)≤λ=ρ(Lrw).ρ(L

¯rwx λx=0.ByLemma2.2,wegetρ(L¯rw)=λ=ρ(Lrw).(2)Ifλ=1,thenL

¯rwx λx≥0.Therefore,L¯rwx≥λx.ByLemma2.2,weget(3)Ifλ>1,thenL

¯rw)≥λ=ρ(Lrw).ρ(L

(3.2)

RemarkItiswellknownthat,whenw=r,AORiterationisreducedtoSORiteration.Sowecaneasilygetthefollowingcorollaries.

wbetheiterativematricesoftheSuccessiveOverrelaxation(SOR)Corollary1.LetLwandL

iterativemethodassociatedto(1.1)and(1.10),respectively.UnderthehypothesisofLemma3.3.Weget

w)≤ρ(Lw)ifρ(Lw)<1;(1)ρ(L

w)=ρ(Lw)ifρ(Lw)=1;(2)ρ(L

w)≥ρ(Lw)ifρ(Lw)>1.(3)ρ(L

¯wbetheiterativematricesoftheSuccessiveOverrelaxation(SOR)Corollary2.LetLwandL

iterativemethodassociatedto(1.1)and(1.11),respectively.UnderthehypothesisofLemma3.4.Weget

¯w)≤ρ(Lw)ifρ(Lw)<1;(1)ρ(L

¯w)=ρ(Lw)ifρ(Lw)=1;(2)ρ(L

¯w)≥ρ(Lw)ifρ(Lw)>1.(3)ρ(L

Forobtainingstrictlyinequality,wesupposesomeconditionandobtainthefollowingtheoremandlemma.

英文文献,解线性方程组的预条件AOR迭代法。

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¯arethecoe cientmatricesofthelinearsystemsLemma3.7LetA∈Rn×n(n≥3)andA

(1.1)and(1.6),respectively.IfAisirreduciblewithaii+1ai+1i>0(i=1,2,...,n 1)andthere

¯isirreducible.SoL¯wrisirreducible.existsani,i≤n 2suchthatani=0.ThenA

¯isirreducible.Theassertionforn=3isobvious.NowweonlyProof.FirstweshowthatA

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