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Concept
Matrix multiplication algorithm
Summary
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. 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. , the best announced bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.37188) time, given by Duan, Wu and Zhou announced in a preprint. This improves on the bound of O(n2.3728596) time, given by Josh Alman and Virginia Vassilevska Williams. However, this algorithm is a galactic algorithm because of the large constants and cannot be realized practically.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries
From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Input: matrices A and B
Let C be a new matrix of the appropriate size
For i from 1 to n:
For j from 1 to p:
Let sum = 0
For k from 1 to m:
Set sum ← sum + Aik × Bkj
Set Cij ← sum
Return C
This algorithm takes time Θ(nmp) (in asymptotic notation).
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