# Using graphs to analyse rate of convergence

I'm trying to analyse the convergence rate of numerical methods. Here is an example of what the graphs should look like in the case of linear, superlinear/quadratic and sublinear convergence: https://en.wikipedia.org/wiki/Rate_of_convergence#/media/File:ConvergencePlots.png

I did some numerical experiments and calculated $\frac{\|x_{k+1}-L\|}{\| x_k-L \|^2}$ and got values between $0$ and $1$ as expected for quadratic convergence. I then made graphs of $e_k$ against $k$. For some of my results the rate of convergence I got from my calculations was supported by the shape of my graph. However, in the cases where I had few iterations such as 6 iterations the graphs did not look like a graph you would associate with a quadratic rate of convergence. Does anyone know why this is? I was thinking that potentially because it's only 6 iterations there is not enough data to create a graph which truly reflects the rate of convergence? Alternatively if someone could point me in the direction of reading material that would cover such a topic.

• You may want to add the tag (reference-request). – mlc Apr 9 '17 at 16:54
• You generally need to be "close enough" for any convergence theorem to hold. I've found I need to use (say) 128 bit floats before I can say anything meaningful about the convergence rate, because lower precision is often contaminated by floating point error before a meaningful graph can be created. – user14717 May 22 '17 at 22:53