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I have a fair background in pure mathematics and right now my project is a numerical implementation of a certain algorithm. I have some numerical background, but not all that much, so the question is simple:

In my algorithm, I have to calculate the value of certain parameter $$h_r = \lim_{t \to \infty} I(t,r),$$ which we have proven exists. Calculating the expression $I$ is computationally fairly expensive, as it involves solving a non-linear PDE. The parameter $r$ indexes a suitably dense set of directions (unit vectors) and all the calculations have to be done for all of them.

How do I calculate the limit? I can guess at a few options, but I don't know what are the best practices, or if I am missing some elegant solution that is common knowledge.

  1. Select a large enough $t$, heuristically or by experiment. This seems to be a bad idea except for a test to see if the algorithm as a whole works.
  2. Use an increasing sequence of $t$ with some stopping condition, such as $|I(t_j,r) - I(t_{j+1},r)|$ (or, better yet, relative change) being small.
  3. Calculate how large $t$ needs to be, in order for the error of $I$ to be sufficiently small. This might turn out to be very difficult, given the complicated nature of $I$.
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  • $\begingroup$ Please edits the tags and the title to improve the question. $\endgroup$ – Tommi Jun 12 '17 at 12:34
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This is a bit primitive, but has a good chance to work.

Plot $I$ versus $1/t$. Fit a smooth curve and find the intercept. This works a lot better if you have some theory predicting the behavior of $I$. For example, an error analysis might give $$I(t)=I_\infty+\frac{C}{t^2}+\cdots$$ If you have two different values of $t$, you can write two equations and solve them to find the unknowns $I_\infty$ and $C$. With more values, plot $I(t)$ against $t^{-2}$. Does it look like a straight line? Or only if $t$ is big enough. Then fit a straight line to the values that appear to lie on one.

The human eye is very good at detecting patterns, such as spotting a linear trend. You can make various hypotheses about plots that might be linear and pick the ones that are.

The more formal versions of this trick define Richardson's deferred approach to the limit, which you might Google. Good luck!

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