We present a non-commutative algorithm for the product of 3x5 by 5x5 matrices using 58 multiplications. This algorithm allows to construct a non-commutative algorithm for multiplying 5x5 (resp. 10x10, 15x15) matrices using 98 (resp. 686, 2088) multiplications. Furthermore, we describe an approximate algorithm that requires 89 multiplications and computes this product with an arbitrary small error.

翻译：我们用58个乘法为3x5 乘以 5x5 矩阵的3x5 乘以 5x5 乘以 5x5 乘以 3x5 乘以 5x5 乘以 5x5 乘以 3x5 乘以 5x5 乘以 5x5 乘以 5x5 乘以 5x5 乘以 3x5 乘以 5x5 乘以 5x5 乘以 5x5 乘以 3x5 乘以 5x 乘以 5x10 乘以 15x15 乘以 5x5 乘以 3x5 乘以 5x5 乘以 5x5 乘以 3x5 乘以 乘以 乘以 5x5x5x5x5 乘以 5x5 乘以 5x5 乘以 5x5x5x5x5 乘以 3x5 乘以 3x5 3x5 5 乘以 5 5 乘以 乘以 5 5x5x5 3xxx5 乘以 3x5 3x5x5 乘以 乘以 乘以 乘以 3x5x5x5x5x5x5 乘以 5x5 乘以 乘以 乘以 乘以 3x5x5x5x5x5x5x5x5 乘以 3x5x5x5 乘以 乘以 乘以 乘以 乘以 乘以 5x5x5x5x5 乘以 5xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx5 乘以 乘以 乘以58 乘以58 乘以 3xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx