Strassens

Strassens refers to Strassen's algorithm, named after Volker Strassen, who introduced it in 1969. It is a fast method for multiplying matrices that reduces the number of required multiplications in a divide-and-conquer approach from eight to seven when multiplying 2x2 blocks, yielding a theoretical speedup for large matrices.

The method works by partitioning each input matrix into four equal-sized blocks and computing seven auxiliary products, M1 through M7, using only additions and subtractions on these blocks. The results are then combined through specific linear combinations to form the four blocks of the product matrix. The process can be applied recursively to the submatrices, continuing until the base case of small matrices is reached. In practice, if the matrices are not sized as a power of two, padding is used to fit the division.

Complexity and performance considerations: Strassen’s algorithm achieves a time complexity of O(n^log2(7)) ≈ O(n^2.807), improving on the standard O(n^3) matrix multiplication in theory. In practice, the algorithm introduces more additions and subtractions, has higher memory usage, and can face numerical stability issues, so it is not universally faster for all sizes or hardware configurations. Variants such as the Strassen–Winograd algorithm reduce the number of additions even further and are used in some high-performance implementations.

Impact and context: Strassen’s result marked a breakthrough in computational linear algebra, spurring extensive research into fast matrix multiplication methods. While modern practical libraries often rely on highly optimized classical approaches for many sizes, Strassen-like algorithms remain relevant for very large matrices and specialized applications in scientific computing.