(* :Author: Christopher Moretti *) 

(* :Summary: 
NewtonsMethod[expr,{x,seed},n] applies Newton'sMethod to finding a root of 
expr(x)=0 with an initial value of seed and n 
iteration.NewtonsMethod[expr,{x,seed},n,Digits->k] will use k decimal places 
of accuracy in the intial calculation."
   *) 
(* :History: Version 1.0 by Christopher Moretti, 1998 *)

Clear[NewtonsMethod]
Options[NewtonsMethod]={Digits->10};
NewtonsMethod[f_,{x_,s_},n_,opts___]:=
        Block[ {k,seed,d},
                d=({Digits}/.{opts}/.Options[NewtonsMethod])[[1]];
                seed=N[s,d];
                Print[ N[{seed,f/.x->seed},d]];
                Do[
                        seed=N[seed-(( f/.x->seed)/(D[f,x]/.x->seed)),d];
                        Print[N[{seed,f/.x->seed},d]];,{k,1,n}];
                Return[seed];
                ];
NewtonsMethod::usage=
  "NewtonsMethod[expr,{x,seed},n] applies Newton'sMethod to finding a root of 
expr(x)=0 with an initial value of seed and n 
iteration.NewtonsMethod[expr,{x,seed},n,Digits->k] will use k decimal places 
of accuracy in the intial calculation."

(* Example: NewtonsMethod[Sin[x],{x,3},10,Digits->30] *)