Matrix multiplication algorithm
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in seemingly unrelated problems such as counting the paths through a graph.[1] Many different algorithms have been designed for multiplying matrices on different types of hardware, including parallel and distributed systems, where the computational work is spread over multiple processors (perhaps over a network).
Directly applying the mathematical definition of matrix multiplication gives an algorithm that takes time on the order of n3 field operations to multiply two n × n matrices over that field (Θ(n3) in big O notation). Better asymptotic bounds on the time required to multiply matrices have been known since the Strassen's algorithm in the 1960s, but the optimal time (that is, the computational complexity of matrix multiplication) remains unknown. As of April 2024, the best announced bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.371552) time, given by Williams, Xu, Xu, and Zhou.[2] [3] This improves on the bound of O(n2.3728596) time, given by Alman and Williams.[4][5] However, this algorithm is a galactic algorithm because of the large constants and cannot be realized practically.
Parallel and distributed algorithms[edit]
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The divide-and-conquer algorithm sketched earlier can be parallelized in two ways for shared-memory multiprocessors. These are based on the fact that the eight recursive matrix multiplications in