I have a paper where I am trying to show that the growth of a certain function is exponential of the order $a^n$. I would like to compute $a$, at least approximately. The base $a$ satisfies a very complicated formula. I do considerable amount of completely rigorous analysis to prove that this formula works, and this is really the interesting part of the paper.

This formula cannot be evaluated exactly, but there is a complicated method using numerical analysis techniques (numerical integration, root-finding, numerical optimization, and so forth) to compute it approximately.

I can use an off-the-shelf numerical package to compute $a$, seemingly to high precision. However, to get a rigorous bound, I would have to go through the entire recipe for computing $a$, showing that all the relevant functions are sufficiently smooth, all of the relevant local minima are in fact the global minima, the functions have the right concavity, the number of digits of accuracy is sufficiently large in each step, and so on. This would be extremely difficult, tedious, and frankly unenlightening --- if you look at a graph of the function it is clear that it has the right smoothness, and I don't want to waste a huge amount of space proving it has these properties.

Is it OK if I just numerically solve for $a$ without proving that my numerical solution is sufficiently accurate? Does it matter that this paper is in the subject of computer science, not pure math? Is there any better alternative?

Thanks for any advice

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    $\begingroup$ It definitely matters what the subject of the paper is, and more especially, who the audience is and where you intend to publish it. The standards of such things are very different in different fields, and if your intended audience does not consist of pure mathematicians then this site is probably not a useful place to get an answer to your question. $\endgroup$ Oct 19 '11 at 13:29
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    $\begingroup$ Meta thread started. tea.mathoverflow.net/discussion/1178/… Personally, I like this question. $\endgroup$ Oct 19 '11 at 13:32
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    $\begingroup$ The short answer is, yes it is OK not to prove a sufficiently accurate numerical estimation. After you proved that a exists and presented a formula, rigorously computing or giving very good estimations of the actual value of a will not be important, especially if this is difficult, tedious and uninteresting. It can be useful to remark about the outcomes of the numeric programs and mention the fact that they dont compute a rigorously. If you can give some estimation rigorously and painlessly this may good to add. $\endgroup$
    – Gil Kalai
    Oct 19 '11 at 14:00
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    $\begingroup$ @David Harris: It might help if you explain why you're bothering to compute $a$ numerically at all. What are you using $a$ for, or what will your readers use $a$ for? If the value is not going to be used for anything except to satisfy idle curiosity, then it should be fine to report a non-rigorously computed value and label it as such. On the other hand, if you think someone else may want to use your reported value of a for subsequent rigorous computations, or might record it in some standards table, then that's a different story. $\endgroup$ Oct 19 '11 at 14:28
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    $\begingroup$ I think there's an important meta-question implicit here, about the role of rigor in pure mathematics (it's implicit in Emil's answer and in the comments). The defining characteristic of pure mathematics isn't that you prove everything rigorously, but rather that you maintain a clear distinction between what you have or haven't proved. Of course it's valuable to prove whatever you reasonably can, but you are always free to include heuristic or approximate remarks as well, as long as you make their status clear. $\endgroup$
    – Henry Cohn
    Oct 19 '11 at 16:54

When I review papers with such assertions, here is what I look for:

  1. A clear description of the problem, and any known features of the quantity one is interested in (unique root, local minimizer, etc);

  2. A clear description of the method used;

  3. Information on the stopping/ error criteria used. This latter is rather important - one may stop an algorithm when the successive approximations are 'close' in some norm, or when some residual measure is smaller than some threshold (presuming one's not exceeded a specified total number of iterations.)

With this information, and a sufficiently modest claim "the computed quantity 'a' appears to provide a good approximation to the desired result'', this reviewer would be happy.

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    $\begingroup$ It is worth mentioning on this page that Dr. Nigam teaches and researches items involving numerical analysis, so "this reviewer's happiness" carries significant mathematical currency. Gerhard "Has Change For A Five" Paseman, 2011.10.19 $\endgroup$ Oct 20 '11 at 1:03

I’d say do whatever you find most convenient and/or useful for the reader, but describe explicitly in the paper what you did. If you approximate the constant using a numerical computation without rigorously verifying all the smoothness conditions, write that you approximated the constant using a numerical computation without rigorously verifying all the smoothness conditions. Then it’s up to the reader whether they trust the value or not (if they actually care).

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    $\begingroup$ Right. You can do anything you want; the question is what your colleagues (and the referee) will think of you afterwards. You know them better than I do, but as long as you're clear about what you've done and what you haven't, they can't really take too much umbrage. $\endgroup$
    – Ben Webster
    Oct 19 '11 at 13:47

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