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2 update on williams' algorithm

# SohowHow fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. 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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# So how fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. 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.