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Yes, given your situation those changes makes sense. Glad you got things solved.
The exercise, as it turned out, was point out that the documentation should reflect the default for reals is MKL-BLAS/LAPACK and not one of the three methods for exact/symbolic . Don't know what is accomplished by the eigenvalues and complex...
Am update. The following also works correctly. fun= Compile[{{k, _Integer}, {u, _Real}}, Block[{f = 0.0`}, Which[k == 1 , f = u + 1, k == 5 , f = u + 2 ]; f ]];
For list type: Compile`CompilerFunctions[] // Sort You will find it discussed at https://mathematica.stackexchange.com/questions/1096/list-of-compilable-functions to make sure your not calling main evaluate after compilation type ...