In linear algebra, the Coppersmith–Winograd algorithm, named after Don Coppersmith and Shmuel Winograd, is the asymptotically fastest known algorithm for square matrix multiplication. It can multiply two n × n matrices in O(n2.3727) time (see Big O notation). This is an improvement over the trivial O(n3) time algorithm and the O(n2.807) time Strassen algorithm. It is possible to improve the exponent further; however, the exponent must be at least 2 (because an n × n matrix has n2 values, and all of them have to be read at least once to calculate the exact result).
It was known that the complexity of this algorithm is O(n2.3755).[1] In 2010, however, Stothers gave an improvement to the algorithm, O(n2.3737)[2] and independently Williams improved the bound to O(n2.3727).[3] These improvements were due to using more complicated tensor products.
The Coppersmith–Winograd algorithm is frequently used as a building block in other algorithms to prove theoretical time bounds. However, unlike the Strassen algorithm, it is not used in practice because it only provides an advantage for matrices so large that they cannot be processed by modern hardware.[4]
Henry Cohn, Robert Kleinberg, Balázs Szegedy and Christopher Umans have rederived the Coppersmith–Winograd algorithm using a group-theoretic construction. They also show that either of two different conjectures would imply that the optimal exponent of matrix multiplication is 2, as has long been suspected. [5]
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