On the inequivalence of bilinear algorithms for 3×33×3 matrix multiplication

• A heuristic for finding algorithms for 3 by 3 matrix multiplication is presented.
• Our heuristic is very simpler and faster than the previous published methods.
• Using the heuristic, many algorithms are obtained.
• Many algorithms are proved to be inequivalent to the published ones, theoretically.

Since Laderman showed an algorithm for 3×33×3 matrix multiplication using 23 scalar multiplications, Johnson and McLoughlin used a numerical optimization and human controlled method to give two parameterized algorithms in which the coefficients are rational numbers. The algorithms are inequivalent to Ladermanʼs one with respect to the transformation introduced by de Groote. We present a simple and fast numerical heuristic for finding valid algorithms. Then we show that many of the obtained algorithms are inequivalent to the published ones.