# How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). However, the constant is so large that this algorithm is in fact slower in practice than naive matrix multiplication for small $n$. Similarly, the Coppersmith-Winograd algorithm, which has the lowest asymptotic complexity of all known matrix multiplication algorithms, has an exponent of $2.376$ and was discussed here previously.

Question: Recently, I made a claim in a submitted paper that the Smith normal form algorithm has super-cubical complexity and a reviewer countered by saying that actually, the complexity has been reduced to matrix multiplication time = $n^{2.37\ldots}$. I am not an expert on matrix algorithms and would happily change the offending line, but the experience has forced me to wonder, what are the practical implications of saying "X can be done in matrix multiplication time"? More precisely,

Does there exist an actual software implementation of Coppersmith Winograd? If not, is there a theoretical obstacle to its existence?

By a theoretical obstacle I don't mean something like "Well, it would only be better than existing techniques for $n$ larger than the number of atoms in the universe so what's the point?", but rather something like "the algorithm uses the axiom of choice, or the classification of finite simple groups" etc.

PS: Okay, so there is also this paper which apparently reduces the complexity of the Coppersmith-Winograd approach to $2.3737$ from $2.376$, so I stand corrected about CW being the fastest. The question still stands if we replace CW by the method of V. V. Williams.

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There are software implementations of the Strassen algorithm but am not sure if there is one for Coppersmith Winograd. The problem from an implementation perspective is that the data structure to handle these algorithms gets complex. As far as I know, there is no theoretical obstacle to the existence of such implementations, though I have not completely read the Coppersmith Winograd algorithm. – user11000 Jul 6 '12 at 21:03
Two points. The exponent for the Stassen algorithm is not exactly $2.8$ but $\log 7/\log2=2.807...$. The constant is not that big, a corse bound being $73.5$. – Denis Serre Jul 6 '12 at 21:34
I am a bit surprised that in your introduction you make it sound as if Strassen was impractical. Yes, for small n it is not used, but the cross over point are rather moderate (think hundred or so). – user9072 Jul 6 '12 at 21:35
I am quite surprised that you can compute Smith Normal Form in matrix-multiplication time. Would you happen to have the reference? – Igor Rivin Jul 7 '12 at 2:27
Igor: I was pointed to this paper by the reviewer: mrzv.org/publications/zzph-mmt/socg11 but I should warn you that I haven't yet had time to go through it and verify the claim – Vidit Nanda Jul 7 '12 at 2:55