We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2x2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy.

翻译：我们提出一种非交换算法，用于通过7个系数乘积实现2x2矩阵乘法。与先前已知的同类快速算法相比，该算法在实际计算中同时达到了更高的精度，并具有当前最优主导项的时间复杂度（通过交替基稀疏化获得）。为构建此算法，我们考虑控制数值矩阵乘法稳定性与精度的矩阵及张量范数界。首先，通过最小化沿Strassen 2x2矩阵乘法张量分解唯一轨道的增长因子来降低这些界。其次，我们开发启发式方法，在进一步改进特定双线性公式精度的同时最小化其运算量。最后，我们执行交替基稀疏化，在基本保持整体精度的前提下优化时间复杂度常数。