When are analytical solutions preferred over numerical solutions in practical problems?

Question

In most engineering or applied math papers that I read, the authors seem to obtain solutions to say, a system of differential equations, using numerical methods, rather than analytical techniques. This is understandable, since I'm aware that obtaining analytical solutions of non-linear systems and highly complex mathematical models is notoriously difficult, and would be unnecessarily time-consuming, especially when one can obtain approximate numerical solutions of the same, with relative ease and to any required degree of accuracy.

However, I was wondering if there are any practical problems where analytical solutions are actually preferred over numerical solutions? The only reason I could think of, for this would be for the purpose of setting standards, to evaluate the correctness of numerical solutions, in mathematical software, for instance. Are there other examples where obtaining analytical solutions serve a significant advantage over numerical solutions?

Answer 1

I am a physicist involved in computer simulations for more than 53 years. What I should say is that when an analytical solution exists (whatever its level of complexity could be), I shall always favor it.

Just suppose that the function you work is the solution of an ordinary differential equation. For sure, there are a lot of numerical methods which can do the job (but you can face serious numerical instabilities as pointed out by Callus). But now, admit that you have to adjust some parameters in the equation in order to match experimental data. Using numerical methods will require a very high effort; moreover the derivative of the objective function to be minimized should require the gradient and the hessian.

Complexity of an analytical solution often means that it contains complex functions. Fortunately, we have very good libraries for their computations.

Answer 2

I'd say that analytical solutions are preferable, if they are available (which is rarely, in practical problems). They are typically faster and more reliable.

Take root-finding as an example. If you can find a reasonably simple analytic formula for roots, you'd generally use it. Quadratic polynomials are an obvious example. Of course, for most functions, you need to use numerical methods to find roots. Good software is available, but, still, there are several things that can go wrong during the iterations.

For a polynomial of degree four, analytical methods of finding roots are available in theory, but getting them to produce accurate answers (that are not ruined by floating point errors) is surprisingly tricky, and many people would use numerical methods instead.

Cubic polynomials (as another example) are on the other side of the borderline, arguably -- I think most people would choose to use analytic solutions, for these.