The Newton-Raphson Method

Section 1.3 - An Example of When the Newton-Raphson Method Does the Unexpected

Example 1.3.1

F[x_] = Sin[x];

Plot[F[x],{x,0,2π}, PlotStyle->{Blue, Thickness[0.007]}]

(Subscript[x, 0]) = SetPrecision[1.97603146838000000,20];

DF[x] = D[F[x],x];

Newt[x_] = SetPrecision[x-F[x]/DF[x],20]

NewtMat = SetPrecision[NestList[Newt[#]&, (Subscript[x, 0]),30],20]

N[-3π]

(Subscript[TLx, 0]) = Expand[F[(Subscript[x, 0])]+(D[F[x],x]/.x-> (Subscript[x, 0]))(x-(Subscript[x, 0]))]

(Subscript[TLx, 1])= Expand[F[NewtMat[[2]]] + (D[F[x],x]/.x-> NewtMat[[2]]) (x-NewtMat[[2]])]

Plot[{F[x],(Subscript[TLx, 0]),(Subscript[TLx, 1])},{x,0,2π},PlotStyle->{{Blue,Thickness[0.008]},{Red,Thickness[0.006]},{Black,Thickness[0.006]}}]