[Math] Newton’s Method – Slow Convergence

Question

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 🙂

Answer 1

Newton's method has quadratic convergence near simple zeros, but the derivative of $\frac{1}{2}x^2+x+1-e^x$ at $x=0$ is zero and so $x=0$ is not a simple zero.