二元非线性方程组求根的牛顿迭代法[1]

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.:(1981-),,,,,:.:1004-4280(2009)04-0089-03(,250353):Taylor,Taylor;,MATLAB:;;;Tayor;Matlab:O242.1:ANewtonsmethodforthenonlinearfunctionoftwoindependentvariablesXURui2min(SchoolofMathematicsandPhysics,ShandongInstituteofLightIndustry,Jinan250353,China)Abstract:AccordingtotherelationbetweenTaylorslawforoneindependentvariableandNewtonsmethodaboutthenonlinearfunctionofoneindependentvariable,aNewtonsmethodforthenonlinearequationsetoftwoindependentvariableswasderivedbyTaylor’slawformoltivariatefunctionbasedonthis.itispossibletoverifythemethodthroughMATLAbtocalculatetherootoftheequation.Keywords:NewtonsIterativemethod;functionofoneindependentvariable;functionoftwoindependentvariables;Taylorformula;Matlab0f(x)=0[1],,f(x,y)=0g(x,y)=0,?Taylor,Taylor,matlab,11(Taylor)[2]f(x)x0(a,b)(n+1),x(a,b),f(x)=f(x0)+f(x0)(x-x0)+f(x0)2!(x-x0)2++f(n)(x0)n!(x-x0)n+Rn(x)Rn(x)=f(n+1)()(n+1)!(x-x0)n+1,x0x2(Taylor)[3]z=f(x,y)(x0,y0)©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(n+1),(x0+h,y0+k),f(x0+h,y0+k)=f(x0,y0)+h99x+k99yf(x0,y0)+12!h99x+k99y2f(x0,y0)++1n!h99x+k99ynf(x0,y0)+1(n+1)!h99x+k99yn+1f(x0+k,y0+k),(01).h99x+k99ymf(x0,y0)mp=0Cpmhpkm-p9mf9xp9ym-p|(x0,y0)3:[1]f(x)=0xk(f(xk)0,f(x)xk,f(x)f(xk)+f(xk)(x-xk),f(x)=0f(xk)+f(xk)(x-xk)=0,xk+1,xk+1xk+1=xk-f(xk)f(xk)(k=0,1,)2z=f(x,y)(x0,y0)2,(x0+h,y0+k),f(x0+h,y0+k)f(x0,y0)+h99xf(x,y)|x=x0+k99yf(x,y)|y=y0h=x-x0,k=y-y0f(x,y)=0f(xk,yk)+h99xf(x,y)|x=xk+k99yf(x,y)|y=yk=0f(xk,yk)+(x-xk)fx(xk,yk)+(y-yk)fy(xk,yk)=0z=g(x,y)(x0,y0)2,(x0+h,y0+k),g(x0+h,y0+k)g(x0,y0)+h99xg(x,y)|x=x0+k99xg(x,y)|y=y0h=x-x0,k=y-y0g(x,y)=0g(xk,yk)+h99xg(x,y)|x=xk+k99xg(x,y)|y=yk=0g(xk,yk)+(x-xk)gx(xk,yk)+(y-yk)gy(xk,yk)=0f(xk,yk)+(x-xk)fx(xk,yk)+(y-yk)fy(xk,yk)=0g(xk,yk)+(x-xk)gx(xk,yk)+(y-yk)gy(xk,yk)=0:gx(xk,yk)fy(xk,yk)-fx(xk,yk)gy(xk,yk)0x=xk+f(xk,yk)gy(xk,yk)-g(xk,yk)fy(xk,yk)gx(xk,yk)fy(xk,yk)-fx(xk,yk)gy(xk,yk)y=yk+g(xk,yk)fx(xk,yk)-f(xk,yk)gx(xk,yk)gx(xk,yk)fy(xk,yk)-fx(xk,yk)gy(xk,yk):x=xk+f(xk,yk)gy(xk,yk)-g(xk,yk)fy(xk,yk)gx(xk,yk)fy(xk,yk)-fx(xk,yk)gy(xk,yk)y=yk+g(xk,yk)fx(xk,yk)-f(xk,yk)gx(xk,yk)gx(xk,yk)fy(xk,yk)-fx(xk,yk)gy(xk,yk)(1)gfx-fgx|(xk,yk)=g(xk,yk)fx(xk,yk)-f(xk,yk)gx(xk,yk)fgy-gfy|(xk,yk)=f(xk,yk)gy(xk,yk)-g(xk,yk)fy(xk,yk)gxfy-fxgy|(xk,yk)=gx(xk,yk)fy(xk,yk)-fx(xk,yk)gy(xk,yk)(1)x=xk+fgy-gfy|(xk,yk)gxfy-fxgy|(xk,yk)y=yk+gfx-fgx|(xk,yk)gxfy-fxgy|(xk,yk)(2):xk+1=xk+fgy-gfy|(xk,yk)gxfy-fxgy|(xk,yk)yk+1=yk+gfx-fgx|(xk,yk)gxfy-fxgy|(xk,yk)(3)(3)k=1,2,,(xk,yk),|(xk+1,yk+1)|(0),(xk,yk)(Newton)09©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=0xey-sin(xy)=0x=1,y=1,:f=xy-ex+ey-4g=xey-sin(xy),fx=y-ex,gx=ey-ycos(xy)fy=x+ey,gy=xey-xcos(xy)xk+1=xk+fgy-gfy|(xk,yk)gxfy-fxgy|(xk,yk)yk+1=yk+gfx-fgx|(xk,yk)gxfx-fxgy|(xk,yk):fgy-gfy|(xk,yk)=(xy-ex+ey-4)[xey-xcos(xy)]-[xey-sin(xy)](x+ey)gxfy-fxgy|(xk,yk)=(ey-ycos(xy))(x+ey)-(y-ex)[xey-xcos(xy)]gfx-fgx|(xk,yk)=(y-ex)[xey-sin(xy)]-[ey-ycos(xy)](xy-ex+ey-4)matlab[4,5]:x=1.1572e-005y=1.6094f=8.1770e-006g=3.9235e-005i=5:x=1,y=1,!5,:[1],.[M].:,2001.[2],,,.()[M].:,1983.[3],,,.()[M].:,1983.[4],.MATLAB[M].:,2004.[5].MATLAB[M].:,2005.(85)33,,FMSSOHCEM,,,KCl/SiO2,,SO21248ppm,KCl/SiO2:[1]MeijR.Asamplingmethodbasedonactivatedcarbonforgaseousmercuryinambientairandfluegases[J].Water,Air,andSoilPollution,1991,56:117-129.[2]PrestboE.M.,BloomN.S..Mercuryspeciationadsorption(MESA)methodforcombustionfluegas:methodology,artifacts,intercomparison,andatmosphericimplications[J].Water,Air,andSoilPollution,1995,80:145-158.[3]MarkS.Germani,WilliamH.Zoller.Vapor-phaseconcentrationsofarsenic,selenium,bromine,iodine,andmercuryinthestackofacoal-firedpowerplant[J].Environ.Sci.Technol.,1988,22(9):1079-1085.[4]AnthonyCarpi.Mercuryfromcombustionsource:areviewofthechemicalspeciesemittedandtheirtransportintheatmosphere[J].Water,AirandSoilPollution,1997,98:241-254.[5]PavlishJH,SondrealEA,MannMD,etal.Statusreviewofmercurycontroloptionsforcoal-firedpowerplants[J].FuelProcessTechnology,2003,82(2-3):89-165.[6],,,.300MW[J].,2008,37(4):22-27.[7]Babur.Nott.Intercomparisonofstackgasmercurymeasurementmethods[J].Water,Air,andSoilPollution,1995,80:1311-1314.[8]TakahisaYokoyama,KazuoAsakura.Mercuryemissionfromacoal-firedpowerplantinJapan[J].TheScienceoftheTotalEnvironment,2000,259:97-103.19