数值分析 chapter4

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1、1oÙõª¼êZ%C.KF(Lagrange)þ(û)Úî1)Lagrangeõª9{L«©9å!:2)ûÚNewtonõªDA(Hermite)3)Hermiteõªpg":9©ã...ng^4)©ã$gZ%C5)ng^Z²%C6)Z%CSK¯Ì7)Z²%CIKJJIIJI11
67£¶w«'4DepartmentofMathematics,SoutheastUniversity,2011òÑ1)¼ê'Xy=f(x)´¼êL:y=f(x)(i=0,1,2···,n);.KF(Lagrange)ii2)¼ê)ÛLªy=f(

2、x),éE,.þ(û)Úî©9å!:^{ü¼ê(´õª)P(x)Cq¼êf(x).DA(Hermite)½½½ÂÂÂ1¼êy=f(x)3«m[a,b]þk½Â,®3:a≤pg":9©ã...x0

3、+anx,(2)JIK¡P(x)õª.12
673AÛþ,{Ò´¦y=P(x),¦ÙÏL½n+£1:(xi,yi),i=0,1,···,n.¶w«'4DepartmentofMathematics,SoutheastUniversity,2011òÑ1.KF(Lagrange).KF(Lagrange)þ(û)Úî1.1Äõª©9å!:¯¯¯KKK¦ngõªlk(x),¦÷vDA(Hermite)l(x)=0,l(x)=0,···,l(x)=0,l(x)=1,pg":9©ã...k0k1kk−1kkng^lk(xk+1)=0,

4、···,lk(xn)=0.Z%C=Z²%C1(j=k)SKlk(xj)=(3)0(j6=k).¯Ìd^(3)x,x,···,x,x,···,x´ngõIK01k−1k+1nªlk(x)":,¤±lk(x)knÏfµJJIIx−x0,x−x1,···,x−xk−1,x−xk+1,···,x−xn.JI¤±k13
67l(x)=A(x−x)(x−x)···(x−x)(x−x)···(x−x)£kk01k−1k+1nnY¶w«=Ak(x−xi)(4)'4i=0i6=kDepartmentofMathematics,SoutheastUniversity,20

5、11òÑÙ¥A½~ê.dl(x)=1,=.KF(Lagrange)kkkYnþ(û)ÚîA(x−x)=1©9å!:kkii=0DA(Hermite)i6=kpg":9©ã...ng^1Ak=⇒Z%CQn(xk−xi)Z²%Ci=0SKi6=k¯ÌQn(x−xi)IKi=0Ynx−xi6=kiJJIIlk(x)=Qn=.(5)xk−xiJI(xk−xi)i=0i6=ki=014
67i6=kl(x)¡ngÄõª.k=0,1,···,n,n+£k1Äõªl0(x),l1(x),···,ln(x).¶w«'4Departmen

6、tofMathematics,SoutheastUniversity,2011òÑ1.2Lagrangeõª.KF(Lagrange)þ(û)Úî

7、^Äõª,÷v^(1)ngõª©9å!:±L«DA(Hermite)XnP(x)=f(x)l(x).(6)pg":9©ã...kkng^k=0Z%C¯¢þ,duP(x)´ngõª,Z²%CXnSKP(xi)=f(xk)lk(xi)=f(xi)li(xi)=f(xi),(i=0,1,···,n.)¯Ìk=0IK(6)¡ngLagrangeõª,PLn(x),=JJIIXnXnYnx−

8、xiJILn(x)=f(xk)lk(x)=f(xk)(7)k=0k=0i=0xk−xi15
67i6=k£5551l0(x),l1(x)···,ln(x)5Ã',§´ngõªmPn¶w«

9、Ä,1,x,x2,···,xn´Ù

10、Ä.l(x),l(x)···,l(x)¡01n'4ngLagrangeļê.DepartmentofMathematics,SoutheastUniversity,2011òѽ½½nnn1x,x,···,x´pÉ!:,K3gêØ.KF(Lagrange)01nLngõªLn(x),¦þ(û)Úî©9å!:Ln(xi

11、)=f(xi),(i=0,1,···,