An iterative method has been used to solve a non-linear equation f$(x)=0$. The table below show the iterations $x_k$ at $k$.

$$\begin{array}{c|c|} & \text{} & \text{} \\ \hline \text{k} & 0 &1&2&3&4 \\ \hline \text{x_k} & 1.2000 & 1.3000 & 1.3250 & 1.3313 &1.3329 \\ \hline \end{array}$$

What convergence rate does the method have?

Correct answer: $p=1$

How can you determine the convergence rate for a method like this? I know that linear convergence gives $p=1$ and quadratic convergence gives $p=2$. However, I am having a difficult time understanding how I can see whether data given follows a linear, quadratic or cubic convergence.

I would appreciate it if anyone could give me some guidelines on how to determine different convergence rates. Thank you!


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