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Wolfram's website declares that "a large fraction of Mathematica's numerical algorithms are based on original research at Wolfram Research" (cf. So Many Original Wolfram Algorithms). So, when I evaluate the discrete Fourier transform via System`Fourier (not Signal`SymbolicFourier or CUDALink`CUDAFourier), how do I identify if Mathematica makes use of certain original DFT methods pioneered by the Wolfram Research?
Well, I speculate that in the major computations (that is, excluding preprocessing and postprocessing), Mathematica essentially calls some external routine (rather than runs a Wolfram implementation). In the following directory:

System`Private`$DynamicLibraryPath\[LeftDoubleBracket]2\[RightDoubleBracket]
(*…\Wolfram Research\Mathematica\13.3\SystemFiles\Libraries\Windows-x86-64*)

I find something like "ipp*.dll" and "mkl*.dll". But since Intel IPP and Intel MKL both provide optimized Fourier Transform functions, how to confirm which one is called in fact? (The documentation does not mention this point.)
Besides, I find a third-party library "flint-17.dll" in that path as well. And I read that Mathematica offers faster polynomial factorization operations. Now that "many algorithms in Mathematica are now found with Wolfram Research's unique automated algorithm discovery methods", can I fairly state that this improved performance is chiefly credited to Mathematica? Or is this simply contributed to the FLINT library instead? (Actually, https://search.wolfram.com/?q=FLINT returns nothing about FLINT (at least) at present. How can the user know whether the pivot of a built-in functionality refers to (probably compiled) code developed by Wolfram Research or utilizes third-party subroutines (under valid license agreements)?)

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  • $\begingroup$ Wolframs claim is true, historically, because the system can develop numerical aproximations, plots and interpolations with a few line of code and visible results in the GUI. But what today is Mathematica is probably 100% import from research from the worldwide community implementing their task on the platform. For a commercial project its of course essential to shield its product from to much insight by users. The more so, because mathematical theorems and algorithms have not yet been shielded by a Texan village jugde as personal property underlying copyright rules. $\endgroup$
    – Roland F
    Commented Aug 12, 2023 at 6:54
  • $\begingroup$ @RolandF Thanks. One of my worries is whether this will be a marketing skill and then lead to a one-sided propaganda. Well, I am not in a position to query Wolfram Mathematica; one should not cut down the tree that gives one shade. I except that this question is not unreasonable inasmuch as Wolfram tech is mostly open. $\endgroup$
    – user688486
    Commented Aug 12, 2023 at 14:10

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