In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity ( O ( n log 2 7 ) {\displaystyle O(n^{\log _{2}7})} versus O ( n 3 ) {\displaystyle O(n^{3})} ), although the naive algorithm is often better for smaller matrices. The Strassen algorithm is slower than the fastest known algorithms for extremely large matrices, but such galactic algorithms are not useful in practice, as they are much slower for matrices of practical size. For small matrices, highly optimized implementations of the conventional algorithm are generally faster because Strassen's additional additions and recursion overhead outweigh its reduction in multiplications.
Strassen's algorithm works for any ring such as plus/multiply, but not all semirings, particularly combinatorial matrix multiplication such as min-plus or boolean algebra, where the naive algorithm still works.
History Volker Strassen first published this algorithm in 1969 and thereby proved that the n 3 {\displaystyle n^{3}} general matrix multiplication algorithm was not optimal. The Strassen algorithm's publication resulted in more research about matrix multiplication that led to both asymptotically lower bounds and improved computational upper bounds.
Algorithm
Let A {\displaystyle A} , B {\displaystyle B} be two square matrices over a ring R {\displaystyle {\mathcal {R}}} , for example matrices whose entries are integers or the real numbers. The goal of matrix multiplication is to calculate the matrix product C = A B {\displaystyle C=AB} . The following exposition of the algorithm assumes that all of these matrices have sizes that are powers of two (i.e., A , B , C ∈ Matr 2 n × 2 n ( R ) {\displaystyle A,\,B,\,C\in \operatorname {Matr} _{2^{n}\times 2^{n}}({\mathcal {R}})} ), but this is only conceptually necessary — if the matrices A {\displaystyle A} , B {\displaystyle B} are not of type 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} , the "missing" rows and columns can be filled with zeros to obtain matrices with sizes of powers of two — though real implementations of the algorithm do not do this in practice. The Strassen algorithm partitions A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} into equally sized block matrices
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