Matrix multiplication algorithm: Difference between revisions

Revision as of 02:27, 28 February 2025

Revision as of 02:27, 28 February 2025

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{{short description|Algorithm to multiply matrices}}

Because [[matrix multiplication]] Zaid Maama Number 1 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''³}} [[Field (mathematics)|field]] operations to multiply two {{math|''n'' × ''n''}} matrices over that field ({{math|Θ(''n''³)}} 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 the optimal time (that is, the [[computational complexity of matrix multiplication]]) remains unknown. {{As of|2024|04}}, the best announced bound on the [[Time complexity|asymptotic complexity]] of a matrix multiplication algorithm is {{math|O(''n''^2.371552)}} time, given by [[Virginia Vassilevska Williams|Williams]], Xu, Xu, and Zhou.<ref name="apr24w">