解线性方程组的预条件AOR迭代法

英文文献,解线性方程组的预条件AOR迭代法。

ImprovingAORmethodforconsistentlinearsystems

Yan-leiCHANG

Guo-fengZHANG Jing-yuZHAO§

SchoolofMathematicsandstatistics,LanzhouUniversity,Lanzhou730000,P.R.China

Abstract

ForsolvingthelinearsystemAX=b,di erentpreconditionedGauss-Seidelmethodshavebeenproposedbymanyauthors.Inthispaper,weconsiderpreconditionedAORiterativemethodswithpreconditioners(I+R),(I+S)and(I+S+R).Inaddition,theconvergenceandcomparisontheoremsoftwopreconditionedAORmethodswithpreconditioners(I+R)and(I+S+R),respectively,areestablished.Numericalexampleisalsogiventoillustrateourresults.

Keywords:Preconditioner;L-matrix;M-matrix;AORmethod;SORmethod;MathematicsSubjectClassi cation(2000):65F10,65F50

1Introduction

Considerthefollowinglinearsystem:

Ax=b,

(1.1)

whereA∈Rn×n,b∈Rnaregivenandx∈Rnisunknown.Foranysplitting,A=M Nwithdet(M)=0,thebasiciterativemethodforsolving(1.1)is

x(i+1)=M 1Nxi+M 1b,i=0,1,...

Forsimplicity,withoutlossofgenerality,weassumethroughoutthispaperthat

A=I L U,

(1.2)

whereIistheidentitymatrix,LandUarestrictlyloweranduppertriangularmatricesobtainedfromA,respectively.ThentheiterationmatricesoftheclassicalAORiterativemethodin[4]isde ned:

Lrw=(I rL) 1[(1 w)I+(w r)L+wU],(1.3)wherewandrarerealparameterswithw=0.

Thespectralradiusoftheiterativematrixisdecisivefortheconvergenceandstabilityofthe

method,andthesmalleritis,thefasterthemethodconvergeswhenthespectralradiusissmallerthan1.Inordertoacceleratetheconvergenceofiterativemethodsolvingthelinearsystem(1.1),preconditionedmethodsareoftenused.Thatis,

PAx=Pb,

The

project-sponsoredbySRFforROCS,SEMandChunhuiProgramme.address:changyl05@

Correspondingauthor.E-mail:gfzhang@§E-mailaddress:zhaojingyu1228@

E-mail

1

英文文献,解线性方程组的预条件AOR迭代法。

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whereP,calledthepreconditioner,isannon-singularmatrix.Denote

PA=D L U .

ApplyingtheAORmethod,wegetthecorrespondingpreconditionedAORiterativemethodswhoseiterativematricesare

1

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

wherewandrarerealparameterswithw=0,D ,L andU arethediagonal,strictlylower

triangularandstrictlyuppertriangularpartsofPA,respectively.

In[8],Morimotopresentedamodi edJacobiandamodi edGauss–SeideliterativemethodbyusingthepreconditionerP=I+R,where

000···00 000···00 . .........R= . .....0 000···00 an1 an20··· ann 10In[3],Gunawardenapresentedamodi edJacobiandamodi edGauss–SeideliterativemethodwithpreconditionerP=I+S,where

0 a120···0 00 a23···0 . .........S= . ..... 000··· an 1n 000···0In[9],Nikietal.consideredaGauss–SeideliterativemethodwithpreconditionerP=I+S+

¯,whereR=I+R

0 a120···00 00 a23···00 . ....¯.....R= . .....0 000···0 an 1n an1 an20··· ann 10In[5],Y.Lietal.consideredapreconditionedAORiterativemethodwithpreconditioner

P=I+S,

= Axb,(1.4) =(I+S)Aand whereAb=(I+S)bwith

0 a12 00 ...S= ... 0000

a23...00

······...······

.

an 1n 0

00...

Now,letusconsiderthe rstpreconditionedlinearsystem,

英文文献,解线性方程组的预条件AOR迭代法。

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= Axb,

=(I+R)Aand whereAb=(I+R)bwith

00 00 ...R= ... 00 an1 an2andthesecondpreconditionedlinearsystem

¯=¯Axb,

¯=(I+S+R)A=(I+R¯)Aand¯whereAb=(I

0 a120 00 a23 ...¯= ...R... 000 an1 an20Weexpressthecoe cientmatrixof(1.5)as

=D L U, A

¯)bwith+S+R)b=(I+R

···00···00

.......0

···0 an 1n ··· ann 10

00 .0 0 0

(1.5)

0···0···......

0···0···

00...0 ann 1

(1.6)

(1.7)

=diag(A ),L andU arestrictlyloweranduppertriangularmatricesobtainedfromA ,whereD

respectively.Bycalculation,weobtainthat

=I E,L =L R+RL+F,U =U,D

whereEandFarediagonalandstrictlylowertriangularmatricesobtainedfromRU.

Thecoe cientmatrixof(1.4)canbeexpressedas

=D L U, A

(1.8)

=diag(A ),L andU arestrictlyloweranduppertriangularmatricesobtainedfromA ,whereD

respectively.Bycalculation,wealsoobtainthat

=I E1,L =L+F1,U =U S+SU,D

whereE1andF1arediagonalandstrictlylowertriangularmatricesobtainedfromSL.

Thecoe cientmatrixof(1.6)canbeexpressedas

¯=D¯ L¯ U,¯A

(1.9)

¯=diag(A¯),L¯andU¯arestrictlyloweranduppertriangularmatricesobtainedfromA¯,whereD

respectively.Bycalculation,wealsoobtainthat

¯=I E1 E2,L¯=L R+RL+F1+F2,U¯=U S+SU.D

WhereE1andF1arediagonalandstrictlylowertriangularmatricesobtainedfromSL,E2and

F2arediagonalandstrictlylowertriangularmatricesobtainedfromRU.

英文文献,解线性方程组的预条件AOR迭代法。

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ApplyingtheAORmethodtothepreconditionedlinearsystems(1.4)(1.5)(1.6),respectively,wehavethecorrespondingpreconditionedAORiterativemethodwhoseiterativematricesare

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

andand

¯rw=(D¯ rL¯) 1[(1 w)D¯+(w r)L¯+wU¯],L

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

(1.10)(1.11)(1.12)

Thispaperisorganizedasfollows.In§2,somepreliminariesaregiven.In§3,weprovethe

convergencetheorems.In§4,wediscusscomparedtheorembetween(I+S)and(I+S+R)correspondingAORmethod.Numericaltestsarein§5.

2Preliminaries