Does an ODE written using GNU gsl outperform Mathematica's NDSolve?

Question

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?

Answer 1

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.

Answer 2

NDSolve and other numerical functions in Mathematica automatically compile your operand (e.g. the RHS of an ODE) to an intermediate "bytecode" language (the same one used by the Compile function). If you like you can specify CompilationTarget -> "C" and the function will be compiled all the way to C code and linked back in to Mathematica... You can see the generated C code yourself in this previous question on the Mathematica Stack Exchange:

https://mathematica.stackexchange.com/questions/821/how-well-does-mathematica-code-exported-to-c-compare-to-code-directly-written-fo/830#830

Of course, it's always possible in principle to hand-write a faster algorithm... But there are a lot of things to optimize that Mathematica will do automatically. You probably don't want to be responsible for manually optimizing the computation of a sparse matrix of partial derivatives in an optimization problem for example.