# 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.