For a homework problem, we are told to perform the first four iterations of the Newton-Raphson method for a function
$$f(x) = \exp(2x^2 - 3) - \frac18(x^2+1) = 0\textrm{.}$$
The solution we seek is $x^* \approx -0.857963$. Consider the starting value $x_0 = -0.8$.
This part I have no problem with. It then asks us to "check the error decrease and estimate the convergence order of the method". This is where I get confused, and unfortunately the TA was not helpful.
I have gathered that the equation I should be using seems to be
$$|\alpha-x_n+1| \leq c|\alpha-x_n|^p$$
where $\alpha$ is the root, the $x$ values are coming from the iterations, $c$ is a constant, and $p$ is the convergence order. The TA suggested I use some values from the iterations to make a system of equations and solve that. Is this on the right track?
Since it just asks you to estimate using the errors you obtained, you can take natural log of your equation:
$$\ln{e_{n+1}}=c+p\ln{e_n}$$
where $e_n$ is the absolute error at the $n$th step. Plot $\ln{e_{n+1}}$ versus $\ln{e_n}$. Estimate the slope and intercept of the line.
A simpler and rough way to do it is to just look at the error of your iterations. If the zeros in your error grows linearly, then the method converges linearly. If the zeros in your error doubles in every iteration, then the method converges quadratically.