The usual formulation of Newton's method goes like -
$f(x) = f(a) + (x-a)f'(a)$
As $x$ is a root of $f$, $f(x) = 0$
And so -
$x = a - \frac{f(a)}{f'(a)}$
I was wondering if there was any way to include the second derivative too.
$f(x) = f(a) + (x-a)f'(a) + \frac{(x-a)^2f''(a)}{2}$
There is no obvious way to isolate $x$ here. Can we formulate some kind of iterative technique to make use of the above formula for find the root?
