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Newton's method.


Can someone provide a little help on this problem?:

Suppose I wanted to write a sequence to perform the Newton inversion of
a polynomial which inverts a series 1 + a(1)x + a(2)x^2+... through a
particular degree.  The sequence would look like the following:

(* Newton's inversion of a polynomial *)
guess = 1;

precision = 1;

While[precision < degree,
      precision *=2;
      If[precision > degree, precision = degree];
      temp = iterate[poly, precision - 1];
      guess = guess + guess*(1 - temp*guess);  (*The Newton iteration*)
      guess = iterate[guess, precision -1];
];

(* Now guess is 1 / poly through degree of 'degree' *)

----------------
iterate would be a function, iterate[f, n] that returns a polynomial f
taken through degree n.  This sequence effectively doubles the precision
each pass through it.  I'm a little stumped on how to implement the
iterate funtion.  Any help greatly appreciated.  Thanks.


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