Fast matrix multiplication via recursive $\langle$ 4x4x4:48 $\rangle$ algorithms into practice

Abstract

We present a fast algorithm for multiplying two 4x4 matrices using 48 multiplications and 216 other operations (addition, subtraction or scaling by a constant) over any ring containing an inverse of 2. Applied recursively, this algorithm reaches a cost bound with leading term $7.75 n^{\log_4(48)}$. This is, up to our knowledge, the best-known leading constant in the cost for matrix multiplication algorithms with exponent $log_4(48)$. We further present an alternative basis version of this algorithm, reducing the leading term to $6.5 n^{\log_4(48)}$ by minimizing the extra cost after the change of basis.