Newton's Method - Slow Convergence I'm using Newton's method to find the root of the equation $\frac{1}{2}x^2+x+1-e^x=0$ with $x_0=1$. Clearly the root is $x=0$, but it takes many iterations to reach this root. What is the reason for the slow convergence? Thanks for any help :)

If
$f(x)
=\frac{1}{2}x^2+x+1-e^x
$,
$f'(x)
=x+1-e^x
$,
so
$f'(0) = 0$.
Since the iteration is
$x^*
= x-\dfrac{f(x)}{f'(x)}
$,
the next value will be
far away
if you start at $x=0$.
You were probably saved by rounding error.
It would have helped if
your question showed what your iterations
actually were.
