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1、SIAMJ.NUMER.ANAL.1984SocietyforIndustrialandAppliedMathematicsVol.21,No.5,October1984001TENSORMETHODSFORNONLINEAREQUATIONS*ROBERTB.SCHNABEL?ANDPAULD.FRANK?Abstract.Anewclassofmethodsforsolvingsystemsofnonlinearequations,calledtensormethods,isintroduced.Tensormethodsaregeneralpurposemet
2、hodsintendedespeciallyforproblemswheretheJacobianmatrixatthesolutionissingularorill-conditioned.Theybaseeachiterationonaquadraticmodelofthenonlinearfunction,thestandardlinearmodelaugmentedbyasimplesecondorderterm.Thesecondordertermisselectedsothatthemodelinterpolatesfunctionvaluesfroms
3、everalpreviousiterations,aswellasthecurrentfunctionvalueandJacobian.Thetensormethodrequiresnomorefunctionandderivativeinformationperiterationandhardlymorestorageorarithmeticperiteration,thanastandardmethodbasedonNewton'smethod.Inextensivecomputationaltests,atensoralgorithmissignificant
4、lymoreefficientthanasimilaralgorithmbasedonthestandardlinearmodel,bothonstandardnonsingulartestproblemsandonproblemswheretheJacobianatthesolutionis.singular.1.Introduction.Thispaperintroducesanewclassofmethods,tensormethods,forsolvingthenonlinearequationsproblem(1.1)givenF:RRn,findx.Rs
5、uchthatF(x.)O,whereitisassumedthatF(x)isatleastoncecontinuouslydifferentiable.ThenovelfeatureofthesemethodsisthattheybaseeachiterationonaquadraticmodelofF(x)whosesecondordertermhasaspecial,restricted,form.Tensormethodsareespeciallyintendedtoimproveupontheperformanceofstandardmethodsonp
6、roblemswheretheJacobianmatrixofFatx.,F'(x.)R"',issingularorill-conditioned.Atthesametime,theyareintendedtobeatleastasefficientasstandardmethodsonproblemswhereF'(x.)isnonsingular,Theirstoragerequirementsandarithmeticoperationsperiterationarenotsignificantlyhigherthantherequirementsofsta
7、ndardmethods.Standardmethodsforsolving(1.1)baseeachiterationuponalinearmodelM(x)ofF(x)aroundthecurrentiteratexcRn,(1.2)M(xc+a)F(xc)+Jd,wheredR",JcR"'.Thesemethodscanbedividedintotwoclasses,thosewhereJcisthecurrentJacobianmatrixF'(xc)orafinitedifferenceapproximationtoit,andthosewhereJ
