A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications

Abstract

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm achieves an error bound exponent of only log4(${\hat{I}}^3$$\infty$,2) $\approx$2.335. In practice, it also reaches a better accuracy w.r.t. max-norm, when compared to previously known such fast algorithms. Furthermore, we propose a straight line program of this algorithm, giving a leading constant in its complexity bound of $316/32 n^{2+log\_4(3)} + o(n^{2+log\_4(3))$ operations over any ring containing an inverse of 2.