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Would an ODE solver written in C perhaps using the GSL library have significant speed advantages compared with Mathematica 8.0 NDSolve? How would it fair in terms of accuracy?
My understanding is that compiled code could in principle be faster, but that these days NDSolve uses a lot of compiled code itself already somehow?
Also are there any options for using things like MathLink or Mathematica’s compile function to speed solving an ODE up?
Mathematica’s focus is on usability. They do use numerical libraries. So the speed would be the same as the best available library or worse (in almost all cases). for example, i heard they use eigen for matrix stuff.
the other thing that you should consider is that although they optimize functions that they provide, your own functions are not optimized. so the derivative that you calculate at each step would be faster in c.
to my friends that decide between mathematica and c++, i tell to go with mathematica since they should focus on getting results fast rather than building the fastest code.