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：我们提出了一种使用7个系数乘积的非交换算法，用于计算2x2矩阵乘法。相较于先前已知的同类快速算法，该算法在实践中同时实现了更高的精度，并通过交替基稀疏化获得了当前已知最优首项的时间复杂度上界。为构建此算法，我们研究了控制数值矩阵乘法稳定性与精度的矩阵范数与张量范数上界。首先，我们通过最小化斯特拉森2x2矩阵乘法张量分解唯一轨道上的增长因子来降低这些上界。其次，我们开发了启发式方法以最小化实现给定双线性公式所需的操作数，同时进一步提升其精度。第三，我们执行了交替基稀疏化操作，该操作改善了时间复杂度常数并基本保持了整体精度。