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Is it possible that built in functions in Mathematica (like Minimize[expr,{x1,x2,...}]) will start to work via CUDA after installation of CUDA module for Mathematica?
I don’t believe so, no. Mathematica’s CUDALink module currently provides only a handful of GPU accelerated functions – some basic image processing operations, BLAS style linear algebra calls, Fourier Transforms and simple parallel reductions (argmin, argmax, and summation). There is also tools for integrating user written CUDA code, and for generating CUDA code symbolically. Outside of that, the rest of Mathematica’s core functionality remains CPU only.
You can see full details of current CUDA and OpenCL support here.