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Is it possible to do parallel linear algebra with arbitrary precision within Mathematica (in a simple manner, as is done for the machine precision)?

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Good question, but unfortunately I fear that the answer might be "no". Mathematica is mostly single-threaded; for machine precision it's only parallelized because the MKL is used. Things are improving, though, so you may be lucky in version 9 (but don't hold your breath). Other than that you'll have to roll your own or call out to a library, I guess. –  Oleksandr R. Nov 22 '12 at 2:08
    
I don't get it. Isn't it obvious to use parallelization in coputation intensive algoriths? –  mmal Nov 23 '12 at 21:48
    
Yes, I suppose so. Why don't you write your own arbitrary precision linear algebra library or parallelize Mathematica's implementation, then? I'm being facetious, of course. WRI has many very clever, talented, and hard-working people, but this is easier said than done. In fact I don't know of a single program or library that offers what you want, Mathematica-hosted or otherwise. That presumably reflects the reality that "obvious" does not imply "trivial". –  Oleksandr R. Nov 24 '12 at 1:47
    
So the question is what is nontrivial in doing this? –  mmal Nov 26 '12 at 10:53
    
I don't exactly know what the difficulties are and don't have time now to find out, but this is a topic of current research. I suspect the main issue is to find decompositions of the various problems that are both efficient and numerically stable. The people over at Computational Science will know more and may be able to suggest approaches for the particular problems of interest to you. –  Oleksandr R. Nov 26 '12 at 12:59
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