Strassen's algorithm improves on naive matrix multiplication through a [[Divide-and-conquer algorithm|divide-and-conquer]] approach. The key observation is that multiplying two {{math|2 × 2}} matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of several additional addition and subtraction operations). This means that, treating the input {{math|''n''×''n''}} matrices as block {{math|2 × 2}} matrices, the task of multiplying {{math|''n''×''n''}} matrices can be reduced to 7 subproblems of multiplying {{math|''n/2''×''n/2''}} matrices. Applying this recursively gives an algorithm needing <math>O( n^{\log_{2}7}) \approx O(n^{2.807})</math> field operations.
Unlike algorithms with faster asymptotic complexity, Strassen's algorithm is used in practice. The [[numerical stability]] is reduced compared to the naïvenaive algorithm,<ref>{{Citation | last1=Miller | first1=Webb | title=Computational complexity and numerical stability | citeseerx = 10.1.1.148.9947 | year=1975 | journal=SIAM News | volume=4 | issue=2 | pages=97–107 | doi=10.1137/0204009}}</ref> but it is faster in cases where {{math|''n'' > 100}} or so<ref name="skiena">{{cite book |first=Steven |last=Skiena |author-link=Steven Skiena |title=The Algorithm Design Manual |url=https://archive.org/details/algorithmdesignm00skie_772 |url-access=limited |publisher=Springer |year=2008 |pages=[https://archive.org/details/algorithmdesignm00skie_772/page/n56 45]–46, 401–403 |doi=10.1007/978-1-84800-070-4_4|chapter=Sorting and Searching |isbn=978-1-84800-069-8 }}</ref> and appears in several libraries, such as [[Basic Linear Algebra Subprograms|BLAS]].<ref>{{cite book |last1=Press |first1=William H. |last2=Flannery |first2=Brian P. |last3=Teukolsky |first3=Saul A. |author3-link=Saul Teukolsky |last4=Vetterling |first4=William T. |title=Numerical Recipes: The Art of Scientific Computing |publisher=[[Cambridge University Press]] |edition=3rd |isbn=978-0-521-88068-8 |year=2007 |page=[https://archive.org/details/numericalrecipes00pres_033/page/n131 108]|title-link=Numerical Recipes }}</ref> It is very useful for large matrices over exact domains such as [[finite field]]s, where numerical stability is not an issue.
