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Is there an algorithm that will allow me to numerically compute the limit of a function f(x) in a principled way?

The most naive algorithm would be to continue to compute the function for larger values of x. The first problem is how to figure out the 'large' values for x to compute the function for. How do I know when to stop?

Can we construct some error bars for this calculation perhaps based on some kind of statistical rationale?

Any books for further reading will be much appreciated.

J. M. ain't a mathematician
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Henry B.
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3 Answers3

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To expand on T's answer, convergence acceleration methods assume that the error terms of the sequence you are interested in are of a certain form.

One of the general-purpose methods for accelerating the convergence of a sequence is the Shanks transformation. For the case of a sequence of partial sums of a series, the Shanks transformation amounts to constructing a sequence of Padé approximants, which hopefully converge faster to the limit of the sequence of partial sums. (The justly famous Wynn ε algorithm is an efficient realization of the Shanks transformation)

Another popular technique is Richardson extrapolation, which assumes that the error can be expressed as a power series of some form. This is in effect the application of the usual algorithms for polynomial interpolation to estimate the limit of a sequence. (Richardson extrapolation is the machinery behind the Romberg algorithm for accelerating the convergence of the trapezoidal rule, since the error of the trapezoidal rule is expressible as a series in powers of the panel size.)

I have been deliberately vague here since you have given absolutely no information on the nature of your sequence. There are many convergence acceleration methods to choose from (a good reference is Brezinski and Redivo-Zaglia's Extrapolation Methods: Theory and Practice), and the best sequence transformation to use depends a lot on the provenance of your sequence.

J. M. ain't a mathematician
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  • Thanks for taking the time to answer. This is an abstract question. I don't have a sequence in mind. I have had this problem before and I just wanted some feedback on how to handle it for future use. – Henry B. Oct 23 '10 at 01:21
  • I think Stalker claimed some optimality properties of his method as well as some practical speedup over Romberg's algorithm for integration and Richardson's method for extrapolation. – T.. Oct 23 '10 at 02:05
  • Yes @T.., that's exactly what I was getting at with "(the) methods assume that the error terms of the sequence you are interested in are of a certain form". Certainly, if the error expansion matches the assumptions of Stalker's algorithm while being a poor fit for Richardson, then Stalker's method would have vastly better performance. It's really dependent on the nature of the sequence. – J. M. ain't a mathematician Oct 23 '10 at 02:13
  • In fact, most experts on sequence transformation recommend that you run not just one, but several algorithms, on your sequence of interest since for a sequence encountered in the wild, it might be inconvenient or impossible to determine the expansion of the error. – J. M. ain't a mathematician Oct 23 '10 at 02:20
  • http://www.nr.com/ , my reference for all numeric stuff, has a quote something like (Extrapolation is much more hazardous than interpolation, as many former stock market analysts can attest.) Parentheses to show it is by memory, not lookup. This is very dependent on how good your model is. The naive thing would be to model the dependence on 1/x as x goes to 0. This obscures the fact that you have data over one range and are extrapolating over another, where things can go wonky. The multiple runs J.M. recommends are one approach to evaluating modeling error. It will catch some and miss some. – Ross Millikan Oct 23 '10 at 04:43
  • Speaking of NR... the commentary by Ernst Joachim Weniger should be read after reading the section on extrapolation methods. In this respect, I would say sequence acceleration methods are a bit like Newton's method for rootfinding: it can succeed exceedingly well or fail spectacularly, depending on the input. – J. M. ain't a mathematician Oct 23 '10 at 04:49
  • As an aside, there is a FORTRAN routine included with Brezinski/Redivo Zaglia for exactly the task of comparing several sequence transformations (as well as the various transformations themselves). – J. M. ain't a mathematician Oct 23 '10 at 04:50
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You have to assume something about f(x). For the case where the approach to the limit is expected to have an asymptotic expansion:

John Stalker, A convergence speeding algorithm with applications to numerical integration, Advances in Applied Mathematics Volume 22, Issue 1, January 1999, Pages 119-153

T..
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In many cases you have a good idea of the limiting behavior of the function. For instance, you may have a conjecture and you want to test it. Then you can do some transformation on your function to a convenient form. For example, I have seen people plot $f(1/x)$ with $x\to0$ and guess the shape of $g(x)=f(1/x)$ from there. You can repeat this process to get more and more accurate pictures.

timur
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