Coppersmith–Winograd algorithm

In the mathematical discipline of linear algebra, the Coppersmith–Winograd algorithm is the fastest currently known algorithm for square matrix multiplication. It can multiply two ${\displaystyle n\times n}$ matrices in ${\displaystyle O(n^{2.376})\!\ }$ time (see Big O notation). This is an improvement over the trivial ${\displaystyle O(n^{3})\!\ }$ time algorithm and the ${\displaystyle O(n^{2.807})\!\ }$ time Strassen algorithm. It might be possible to improve the exponent further; however, the exponent must be at least 2 (because an ${\displaystyle n\times n}$ matrix has ${\displaystyle n^{2}}$ values, and all of them have to be read at least once to calculate the exact result).

The Coppersmith–Winograd algorithm is frequently used as building block in other algorithms to prove theoretical time bounds, but it appears to be not particularly practical for implementations.

A newer approach by Henry Cohn, Robert Kleinberg, Balázs Szegedy and Christopher Umans gets the exponent 2.41 via a group-theoretic approach.

References

• Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9:251–280, 1990.

Biography

• Shmuel Winograd

External links

• Article by Cohn/Kleinberg/Szegedy/Umans