How to determine BLAS/LAPACK implementation used internally for numerical matrix operations?

Question

Is there a command which reveals which implementation of BLAS and LAPACK are used in Mathematica's matrix operations such as Eigensystem? I asked a related question on StackOverflow and one user mentioned that in Julia, the BLAS/LAPACK implementation can be found by executing versioninfo(). Several users who tried my code there had varying results, with some observing Mathematica to execute faster, and others observing Julia executing faster.

In my case, my Julia installation appears to make use of the OpenBLAS implementation, and it runs between 3 to 6 times slower than Mathematica's Eigensystem for randomly-generated arrays of size $1000\times1000$ to $2000\times2000$.

In the Mathematica documentation's tutorial/SomeNotesOnInternalImplementation, it mentions "For dense arrays, LAPACK algorithms extended for arbitrary precision are used when appropriate" and "BLAS technology is used to optimize for particular machine architectures", but nothing more.

EDIT: So in response to Kuba's comment, apparently one of the Julia devs noted that there is anomalous behavior in Julia with regards to eigenvector computation speed as a function of BLAS thread number. In short, using more threads in Julia's use of OpenBLAS appears to slow things down considerably. For reference, in Mathematica:

SetSystemOptions["MKLThreads" -> 1];
First@Timing@Eigensystem[RandomReal[{-500, 500}, {1000, 1000}]]
SetSystemOptions["MKLThreads" -> 2];
First@Timing@Eigensystem[RandomReal[{-500, 500}, {1000, 1000}]]
SetSystemOptions["MKLThreads" -> 3];
First@Timing@Eigensystem[RandomReal[{-500, 500}, {1000, 1000}]]
SetSystemOptions["MKLThreads" -> 4];
First@Timing@Eigensystem[RandomReal[{-500, 500}, {1000, 1000}]]
(*Out:*)
1.747211
1.466409
1.341609
1.357209

So I guess there's nothing wrong with Mathematica's implementation.

Answer 1

As indicated in the comments, machine-precision linear algebra operations in Mathematica use the Intel MKL library optimized implementation of BLAS/LAPACK.

That is the case for all platforms where MKL is available: Windows, Linux and Mac OS X (there will be no obvious MKL library files present in the layout on OS X in versions 10.1 through 11.2 due to static linking).

As of the version 11.3, the Raspberry Pi port is using OpenBLAS 0.2.20.

I am not aware of a built-in way to query the exact version of MKL being used. Generally, it tends to be the latest stable MKL available as of the Mathematica release date, and is the same version on all platforms (exceptions are of course possible).

The following LibraryLink snippet, which works only on Linux, will return the MKL version information.

src = "#include <stdio.h>
  #include <stdlib.h>
  #include <WolframLibrary.h>

  void mkl_get_version_string(char * buff, int len);

  DLLEXPORT mint WolframLibrary_getVersion( ) {
      return WolframLibraryVersion;
  }

  DLLEXPORT int WolframLibrary_initialize(WolframLibraryData libData) {
      return 0;
  }

  DLLEXPORT void WolframLibrary_uninitialize(WolframLibraryData libData) {
      return;
  }

  DLLEXPORT int mkl_version(WolframLibraryData libData, mint argc, MArgument * args, MArgument res) {
    char * buff = (char *)malloc(198 * sizeof(char));
    mkl_get_version_string(buff, 198);
    MArgument_setUTF8String(res, buff);
    return LIBRARY_NO_ERROR; 
  }";

Needs["CCompilerDriver`"];

lib = CreateLibrary[src, "mkl_version"];

LibraryFunctionLoad[lib, "mkl_version", {}, "UTF8String"][]

and, according to the results, the last few Mathematica versions have used

| 11.x.y | Intel(R) Math Kernel Library Version 2017.0.1 Product Build 20161005| 
| 11.0.x | Intel(R) Math Kernel Library Version 11.3.2 Product Build 20160120  |
| 10.4.x | Intel(R) Math Kernel Library Version 11.3.1 Product Build 20151021  |
| 10.3.x | Intel(R) Math Kernel Library Version 11.2.2 Product Build 20150120  |
| 10.2.0 | Intel(R) Math Kernel Library Version 11.2.2 Product Build 20150120  |
| 10.1.0 | Intel(R) Math Kernel Library Version 11.2.1 Product Build 20141023  |
| 10.0.x | Intel(R) Math Kernel Library Version 11.1.2 Product Build 20140122  |
|  9.0.x | Intel(R) Math Kernel Library Version 10.3.5 Product Build 20110720  |