Am I allowed to do non-rigorous numerical analysis? - MathOverflow

Am I allowed to do non-rigorous numerical analysis?

2011-10-19T13:05:54Z

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

Answer by Emil Jeřábek for Am I allowed to do non-rigorous numerical analysis?

2011-10-19T13:42:51Z

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).

Answer by Nilima Nigam for Am I allowed to do non-rigorous numerical analysis?

2011-10-20T00:15:26Z

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

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