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Matrix multiplication algorithm
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<p>Because <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a> is such a central operation in many <a href="/facts/Numerical_algorithm/QT9cYa9z">numerical algorithms</a>, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including <a href="/facts/Scientific_computing/PnVD19hw">scientific computing</a> and <a href="/facts/Pattern_recognition/o1s38opj">pattern recognition</a> and in seemingly unrelated problems such as counting the paths through a <a href="/facts/Graph_(graph_theory)/kw3eIBUe">graph</a>. Many different algorithms have been designed for multiplying matrices on different types of hardware, including <a href="/facts/Parallel_computing/nQzcpzQt">parallel</a> and <a href="/facts/Distributed_computing/o5nyhqTF">distributed</a> systems, where the computational work is spread over multiple processors (perhaps over a network). </p><p>Directly applying the mathematical definition of matrix multiplication gives an algorithm that <a href="/facts/Analysis_of_algorithms/pStoBgBn">takes time</a> on the order of <i>n</i>3 <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> operations to multiply two <i>n</i> × <i>n</i> matrices over that field (Θ(<i>n</i>3) in <a href="/facts/Big_O_notation/weFFjSWg">big O notation</a>). Better asymptotic bounds on the time required to multiply matrices have been known since the <a href="/facts/Strassen_algorithm/rrIGRSNp">Strassen's algorithm</a> in the 1960s, but the optimal time (that is, the <a href="/facts/Computational_complexity_of_matrix_multiplication/KCn5KUzd">computational complexity of matrix multiplication</a>) remains unknown. As of April 2024<a href="https://en.wikipedia.org/w/index.php?title=Matrix_multiplication_algorithm&action=edit">[update]</a>, the best announced bound on the <a href="/facts/Time_complexity/77T62gmf">asymptotic complexity</a> of a matrix multiplication algorithm is O(<i>n</i>2.371552) time, given by <a href="/facts/Virginia_Vassilevska_Williams/dd7jUV8W">Williams</a>, Xu, Xu, and Zhou. This improves on the bound of O(<i>n</i>2.3728596) time, given by Alman and Williams. However, this algorithm is a <a href="/facts/Galactic_algorithm/Q17Y5kP8">galactic algorithm</a> because of the large constants and cannot be realized practically. </p>