A SIMPLICIAL ALGORITHM FOR COMPUTING AN INTEGER ZERO POINT OF A MAPPING WITH THE DIRECTION

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1、JournalofComputationalMathematics,Vol.24,No.6,2006,711–718.ASIMPLICIALALGORITHMFORCOMPUTINGANINTEGERZEROPOINTOFAMAPPINGWITHTHEDIRECTIONPRESERVINGPROPERTY*1)Chuang-yinDang(DepartmentofManufacturingEngineeringandEngineeringManagement,CityUniversityofHongKong,Kowlo

2、on,HongKong)AbstractnnAmappingf:Z→Rissaidtopossessthedirectionpreservingpropertyiffi(x)>0impliesfi(y)≥0foranyintegerpointsxandywithkx−yk∞≤1.Inthispaper,asimplicialalgorithmisdevelopedforcomputinganintegerzeropointofamapping0withthedirectionpreservingproperty.Wea

3、ssumethatthereisanintegerpointxwith00c≤x≤dsatisfyingthatmax1≤i≤n(xi−xi)fi(x)>0foranyintegerpointxwithnf(x)6=0ontheboundaryofH={x∈R

4、c−e≤x≤d+e},wherecanddaretwo⊤nfiniteintegerpointswithc≤dande=(1,1,···,1)∈R.Thisassumptionisimpliedbyoneoftwoconditionsfortheexistence

5、ofanintegerzeropointofamappingwiththe0preservingpropertyinvanderLaanetal.(2004).Underthisassumption,startingatx,thealgorithmfollowsafinitesimplicialpathandterminatesatanintegerzeropointofthemapping.Thisresulthasapplicationsingeneraleconomicequilibriummodelswithin

6、divisiblecommodities.Mathematicssubjectclassification:90C49.Keywords:IntegerZeroPoint,DirectionPreserving,SimplicialAlgorithm,Triangulation,Existence.1.IntroductionTheproblemweconsiderinthispaperistocomputeanintegerzeropointofamappingf:Zn→Rn.Theinterestsininteger

7、zeropointsorfixedpointsofamappinghavebeeninspiredbytheworkinIimura(2003)thoughthestatementoftheexistenceofadiscretefixedpointinIimura(2003)isincorrectandacorrectedstatementwasgiveninIimuraetal.(2004)afteranapplicationoftheintegrallyconvexsetdefinedinFavatiandTardel

8、la(1990).AbriefintroductiontotheapplicationsofdiscretefixedpointsofamappingineconomicscanbefoundinIimura(2003)andreferencestherein.FollowingthedefinitioninIimura(2003),wesaythatf(x)satisfiesthedirectionpreservingpropertyiffi(x)>0impliesfi(y)≥0foranyintegerpointsxan

9、dywithkx−yk∞≤1.Weassumethroughoutthispaperthatf(x)satisfiesthedirectionpreservingproperty.Recently,undertwodifferentconditions,basedonthe2n-rayalgorithminvanderLaanandT