Thank you all for your answers. Not knowing the algorithm used by Mathematica is quite bothersome for me because without that I don't know how to consider the answers, notably regarding the questions of completeness and soudness in the following senses :

• Completeness : given a generic problem, does Mathematica explore all the possibilities and then return the best answer ? or does Mathematica only partially explore all the possibilites and then return an approximation of the best answer ?
• Soundness : given a generic problem, when Mathematica gives an answer, is the answer always correct ? or are there cases the answers may not be correct ?

I understand that Mathematica is a proprietary product and that SomeNotesOnInternalImplementation is probably the much they can disclose, but I hoped they would have revealed a little more.

"Given a choice of options, I can always try all of them and see which one gives the same result as Automatic. It wouldn't be giving anything away to reveal which option choice was used.":

But the choice will depend upon the particular differential equation, so still the documentation could not tell you in any general way which of the non-stiff Adams method or the stiff Gear backward differentiation formula is used for the explicit, or default, option Method -> Automatic.

Moreover, even if you find through experimentation that an Adams method was used, it won't tell you which order Adams method; for that you'd have to do additional experimentation (including, quite likely, implementing the particular order method).

In principle, Mathematica could be enhanced with an option that, at evaluation time, reports the choice. But that's a different level of functionality from what Mathematica currently has, and it could unduly complicate the underlying code, with unwanted effects on efficiency.

There are thousands of functions, so a simple list isn't going to work. For specific topics, you can search in documentation (for example search for "decomposition" or "matrix") usually there are tutorials as well that the documentation will point to. Another good source often is mathworld . Hope that helps!