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.

  • $\begingroup$ You may want to add the tag (reference-request). $\endgroup$ – mlc Apr 9 '17 at 16:54
  • $\begingroup$ 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. $\endgroup$ – user14717 May 22 '17 at 22:53

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