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Because [[matrix multiplication]] is such a central operation in many [[numerical algorithm]]s, 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 (graph theory)|graph]].<ref name="skiena"/> Many different algorithms have been designed for multiplying matrices on different types of hardware, including [[parallel computing|parallel]] and [[distributed computing|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 [[Analysis of algorithms|takes time]] on the order of {{math|''n''<sup>3</sup>}} [[Field (mathematics)|field]] operations to multiply two {{math|''n'' × ''n''}} matrices over that field ({{math|Θ(''n''<sup>3</sup>)}} in [[big O notation]]). Better asymptotic bounds on the time required to multiply matrices have been known since the [[Strassen algorithm|Strassen's algorithm]] in the 1960s, but it is still unknown what the optimal time is (i.e., what the [[Computational complexity theory|complexity of the problem]] is). {{As of|2020|12}}, the matrix multiplication algorithm with best asymptotic complexity runs in {{math|O(''n''<sup>2.3728596</sup>)}} time, given by Josh Alman and [[Virginia Vassilevska Williams]],<ref name="aw20">{{Citation | last1=Alman | first1=Josh | last2=Williams | first2=Virginia Vassilevska | contribution=A Refined Laser Method and Faster Matrix Multiplication | year = 2020 | arxiv=2010.05846 | title = 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2021) |url=https://www.siam.org/conferences/cm/program/accepted-papers/soda21-accepted-papers}}</ref><ref>{{Cite web|last=Hartnett|first=Kevin|title=Matrix Multiplication Inches Closer to Mythic Goal|url=https://www.quantamagazine.org/mathematicians-inch-closer-to-matrix-multiplication-goal-20210323/|access-date=2021-04-01|website=Quanta Magazine|language=en}}</ref> however this algorithm is a [[galactic algorithm]] because of the large constants and cannot be realized practically.
==Iterative algorithm==
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