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I am studying some numerical optimization methods, but I am not an expert in numerical maths. Question: If the convergence rate is linear, then the logarithm log(x_n) of the distance x_n to the minimum plotted against n gives a straight line. Now for more general convergence rates, i.e. super linear, is there any reference where similar graphical estimate methods are explained? Ok, I can do it myself, but writing a paper for engineers, I don't want to reinvent the wheel and I am lazy. Thanks Karl

Karl
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Convergence results are usually plotted in Log-Log plots. See for example here.

In a Log-Log plot, the order of convergence is simply the slope of your data as suggested by this picture (from the same reference). Log-Log plots are pretty much the standard to visualize the order of convergence.

enter image description here

Thomas
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  • Thanks sonystarmap,but this is not exactly what I'm looking for. Having a sequence $x_n \to 0$, let $y_n=\log(|x_n|)$ and $|x_{n+1}|\approx\lambda |x_|^{\alpha}$, then $\alpha$ is the convergence speed. It can be estimated by linear regression from: y_{n+1} \approx \log(\almbda) +\alpha y_n. – Karl Sep 14 '14 at 06:48
  • I'm not sure, I fully get the problem. We have $x_n \rightarrow 0$ with order $\alpha$ and $y_n=\log(x_n)$. Do you now want to visualize $y_n$ and see the meaning of $\alpha$ visualized in a plot? – Thomas Sep 14 '14 at 06:57
  • Yes, I want to estimate $\alpha$ by linear regression from data $y_{n+1} \approx \log(\lambda) +\alpha y_n$. I know how to do it, but this is definitely not new and there must be some textbooks explaining this in detail. I would like to make reference to some text to avoid explanations. – Karl Sep 14 '14 at 07:31
  • Hmm, then I'm not sure if I can help you. I'll look around if I'm able to find any references, but I don't have that much hope. – Thomas Sep 14 '14 at 08:06
  • Thanks for the help anyway, sonystarmap. It was just a question to avoid some work ;-). – Karl Sep 14 '14 at 08:52
  • @Karl for more general orders, you can use different types of fits in logspace. For example, $q$-linear vs. $q$-quadratic convergence can be determined by a power law fit in logspace – whpowell96 Jan 06 '20 at 19:03