options - How to find out which method Mathematica selected?

There are commands like NonlinearModelFit[] or NDSolve[] that have the option Method it typically defaults to Automatic. How can you check after the evaluation of the command which method Mathematica picked?

Answer

I think you can actually see (most of) what Mathematica is doing by using Trace[..., TraceInternal -> True].

For example,

Select[Flatten[
  Trace[NDSolve[y'[x] == x && y[0] == 0, y, {x, 0, 6}], 
   TraceInternal -> True]], ! FreeQ[#, Method | NDSolve`MethodData] &]

shows the DE was evaluated using NDSolve`LSODA and Newton's method. (I think)

And

Select[Flatten[
  Trace[NDSolve[{Derivative[1][x][t]^2 + x[t]^2 == 1, x[0] == 1/2}, 
    x, {t, 0, 10 Pi}, SolveDelayed -> True], 
   TraceInternal -> True]], ! FreeQ[#, Method | NDSolve`MethodData] &]

used NDSolve`IDA.

As an aside, here's something I just learnt from Trott's Mathematica guidebook for numerics, to see all of the methods and suboptions for NDSolve

{#, First /@ #2} & @@@ 
 Select[{#, Options[#]} & /@ (ToExpression /@ 
   DeleteCases[Names["NDSolve`*"],(* PDE method only *) "NDSolve`MethodOfLines"]), 
   (Last[#] =!= {}) &]

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