DeepMind likes coming up with new improved algorithms for these more specific cases, e.g. it was announced in 2025 that AlphaEvolve found a novel 4x4 complex valued algorithm that uses 48 multiplications.
Bibliography:
- fmm.univ-lille.fr/ attempts to keep an up-to-date list for various sizes
A "commutative matrix multiplication algorithm" is a matrix multiplication algorithm that requires the ring to be commutative. Such algorithms are inferior because you cannot use them to create more efficient algorithms for general matrix matrix multiplication by decomposing the bigger matrix into smaller ones.
For example, the Strassen algorithm is based on reduction to non-commutative 2x2 matrix multiplication optimized to be done in 7 multiplications rather than 8 as in the native algorithm.
For 3x3 matrix multiplication, the best algorithms as of 2025 are:and beating the Strassen algorithm using 3x3 matrices would require a non-commutative algorithm with 21 multiplications.
- commutative: 21 multiplications
- non-commutative: 23 multiplications
Bibliography:
Articles by others on the same topic
There are currently no matching articles.