求解非线性方程的最优8阶史蒂芬森方法.pdfVIP

数 学 杂 志 Vo1．34(2014) NO．2 J．ofMath．(PRC) oPTIM AL EIGHTH．0RDER STEFFENSEN TYPE METHODSFOR NONLINEAR EQUATIONS WANG Xiao—feng，一．ZHANG Tie (．SchoolofMathematicsandPhysics，BohaiUniversity，JinzhD121013，Chin0) (2．CollegeofSciences，NortheasternUniversity，Shenyang110819，China) Abstract：Inthispaper，wepresentafamilyofthree—stepeighth—orderSteffensentypemeth— odsforsolvingnonlinearequationsbyusingweightfunctionmethods．Numericalexperimentsshow thatthesemethodsrequirelesstimetoconvergethantheotheroptimalmethods． Keywords： Steffensen’smethod；derivativefree；eighth-orderconvergence 2010MR SubjectClassification： 65H05；65B99 Documentcode： A ArticleID： 0255—7797(2O14)02-0205—09 1 Introduction Inthispaper，weconsideriterativemethodstofindasimplerootofanonlinearequation ／(x)=0，wheref：ICR RforanopenintervalIisascalarfunction．Theclassical Newton’Smethod[1]withsecond—orderconvergenceiswrittenas xn+1=x 一f(x)，x) (1．1) However，whenthefirstorderderivativeofthefunctionf(x)isunavailableorisexpensive tocompute，theNewton’Smethodisstillrestricted inpracticalapplications．Inorderto avoidcomputingthefirstorderderivative，Steffensen[2]proposedthefollowingsecond—order method X+1=X 一f(x)／Ix，Zn]一， (1．2) wheren= n+f(xn)，and，[-，】·isthefirstorderdivideddifference．Toimprovethelocal orderofconvergence，manyhigh—ordermethodswereproposedinopenliteratures，see[3—12] andreferencestherein．Reneta1．3【]proposedthefollowingfourth—ordermethods f = 一，()fix，]一， 【+1= 一f(Y)[， ，]+，【，]一．厂 ，]+3／(y． Xn)(一Zn)]一， (1．3) Received date：2012—11—26 Accepted date：2013—07—08 Foundationitem：SupportedbyNationalNaturalScienceFoundat