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Section8.4Newton-Raphson

We can find a solution to our equation C(t) =\Phi(C(t)) more efficiently by using Newton-Raphson. Note that \Phi is a power series (actually polynomial in u) and

\Phi(C_0+\de) \approx \Phi(C_0) +\Phi'(C_0)\de;
this works provided \de is divisible by a power of t. So if C_0 is an approximate solution to \Phi(C(t))=C(t), then our aim is to choose \de so that
C_0 + \de = \Phi(C_0) +\Phi'(C_0)\de,
which implies that
\de = \frac{\Phi(C_0)-C_0}{1-\Phi'(C_0)}.
Hence our proposed update formula is now
C_{n+1} = C_n + \frac{\Phi(C_n)-C_n}{1-\Phi'(C_n)}
Here
\Phi'(C_n(t)) = 2tC_n(t).

We can run a quick test. With

we find that the first 22 terms of nr(nr(nr(1+t))) are correct.

Your task is to write an efficient implementation of this method.