> The asymptotically fast algorithms (such as Strassen's[2]) outperform the naive algorithm (i.e. by definition, such is in the article) when matrices get large

> The asymptotically fast algorithms (such as Strassen's[2]) outperform the naive algorithm (i.e. by definition, such is in the article) when matrices get large. This is true only if approaching the matrix multiplication problem purely from a theoretical perspective. But the reality is that computers have a specific architecture designed to be as fast as possible, while still respecting the laws of physics. The libraries such as BLAS deal with those architectures, and it happens that there, an O(n³) triple loop with the optimizations described in the Goto paper is what gives you the fastest performance. Maybe if we were dealing with 1b*1b matrices, Strassen's like algorithms would start to shine, but with those sizes you already exceed the maximum space on a computer, in which case you then start looking at parallel algorithms on clusters/supercomputers where the algorithms become very different (see things such as SUMMA: Scalable Universal Matrix Multiplication Algorithm). In those cases, the computation happening on each processor would still be the fast O(n³) BLAS code.