Computational complexity of matrix multiplication information
Algorithmic runtime requirements for matrix multiplication
Unsolved problem in computer science:
What is the fastest algorithm for matrix multiplication?
(more unsolved problems in computer science)
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical relevance.
Directly applying the mathematical definition of matrix multiplication gives an algorithm that requires n3 field operations to multiply two n × n matrices over that field (Θ(n3) in big O notation). Surprisingly, algorithms exist that provide better running times than this straightforward "schoolbook algorithm". The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication".[1] The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science.
As of January 2024[update], the best bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.371552).[2][3] However, this and similar improvements to Strassen are not used in practice, because they are galactic algorithms: the constant coefficient hidden by the big O notation is so large that they are only worthwhile for matrices that are too large to handle on present-day computers.[4][5]
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Volker Strassen (Aug 1969). "Gaussian elimination is not optimal". Numerische Mathematik. 13 (4): 354–356. doi:10.1007/BF02165411. S2CID 121656251.
^Vassilevska Williams, Virginia; Xu, Yinzhan; Xu, Zixuan; Zhou, Renfei. New Bounds for Matrix Multiplication: from Alpha to Omega. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). pp. 3792–3835. arXiv:2307.07970. doi:10.1137/1.9781611977912.134.
^Nadis, Steve (March 7, 2024). "New Breakthrough Brings Matrix Multiplication Closer to Ideal". Retrieved 2024-03-09.
^Iliopoulos, Costas S. (1989). "Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix" (PDF). SIAM Journal on Computing. 18 (4): 658–669. CiteSeerX 10.1.1.531.9309. doi:10.1137/0218045. MR 1004789. Archived from the original (PDF) on 2014-03-05. Retrieved 2015-01-16. The Coppersmith–Winograd algorithm is not practical, due to the very large hidden constant in the upper bound on the number of multiplications required.
^Robinson, Sara (November 2005). "Toward an Optimal Algorithm for Matrix Multiplication" (PDF). SIAM News. 38 (9). Even if someone manages to prove one of the conjectures—thereby demonstrating that ω = 2—the wreath product approach is unlikely to be applicable to the large matrix problems that arise in practice. [...] the input matrices must be astronomically large for the difference in time to be apparent.
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