Please, give me some tips for a HIGH PERFORMANCE C library for linear algebra (matrix algebra, eigenvalues, eigenvectors etc.). Can be both open-source or closed-source.
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ATLAS, maybe?
Edit: if you're open to C++, you should definitely check Eigen, it's a very neat library, and pretty fast too, according to the benchmarks.
Again if you are actually looking/open for modern C++ code, Armadillo is getting really hyped/popular. Also see their own benchmarking against IT++ and Newmat.
EDIT Thanks to comments from osgx:
I know both LAPACK and GSL and I can recommend both of them. LAPACK is very low-level library and GSL may be more comfortable to use, but speaking of good performance -- BLAS-based library it is.