# Fastest way of multiplying two $n×n$ matrices for fixed $n=1,2,3,\ldots$?

While the question of what is the asymptotically fastest matrix multiplication algorithm is still open, and tremendous improvements were made between 1968 and 1990 (Strassen, Coppersmith-Winograd), I was wondering in the opposite direction, whether it is known what the fastest matrix multiplication algorithm is for matrices of fixed size $$n=1,2,3,\ldots$$

More specifically, given fixed costs for basic scalar operations, is there some table out there on the web containing this data? How do these algorithms look like for, say, $$n=1\ldots 20$$? What is smallest $$n$$ for which the answer is unknown? (I suspect it will be rather small due to combinatorial explosion issues.)

• So by “fastest algorithm” you mean the formula with the least number of arithmetic operations? Did you try searching the OEIS? Aug 18, 2021 at 21:06
• @JossevanDobbendeBruyn yes I do mean number of arithmetic operations. I suppose some authors will only count multiplications, while others will count both multiplications and additions. Aug 18, 2021 at 23:50
• For $n=3$, a classic result is by Laderman (1976) that 23 multiplications suffice. And Bläser (2003) showed that at least 19 are needed. See Heule, Kauers & Seidl (2021) for a recent listing of known results: doi.org/10.1016/j.jsc.2020.10.003 Aug 19, 2021 at 17:10
• For $n=1$ (i.e. multiplication of numbers) there is an algorithm by Harvey and van der Hoeven (2019) in $O(d\log d)$ where $d$ is the number of digits. It's conjectured that this is best possible. Aug 24, 2021 at 12:49
• I think including $n=1$ is nice. At least you can then start the table with something that is certainly known! Aug 25, 2021 at 6:54