Description of fast matrix multiplication algorithm: ⟨3×3×3:23⟩

Algorithm type

4 X^{2} Y^{2} Z^{2} + 2 X^{3} Y Z + 2 X Y^{3} Z + 2 X Y Z^{3} + 13 X Y Z

Algorithm definition

The algorithm ⟨3×3×3:23⟩ is taken from:

Julian David Laderman. A noncommutative algorithm for multiplying 3×3 matrices using 23 multiplications. Bulletin of the American Mathematical Society, 82(1):126--128, January 1976. [DOI]

Algorithm description

Algorithm symmetries

The following group of 24 isotropies acts as a permutation group on algorithm tensor representation:

maple tensor representation (compressed by bzip2);

These encodings are given in compressed text format using the maple computer algebra system. In each cases, the last line could be understood as a description of the encoding with respect to classical matrix multiplication algorithm. As these outputs are structured, one can construct easily a parser to its favorite format using the maple documentation without this software.

Commentaries

Laderman tensor decomposition requires at least 62 scalars operations and no variant is known in its orbits with a better constant.

Another decomposition with the same tensor rank is presented by Stapleton (2025) that requires 60 scalars operations.

Back to main table