Computational complexity of matrix multiplication: Difference between revisions

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Unlike algorithms with faster asymptotic complexity, Strassen's algorithm is used in practice. The [[numerical stability]] is reduced compared to the naïve 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.

== The matrixMatrix multiplication exponent ==

[[File:MatrixMultComplexity svg.svg|thumb|400px|right|Improvement of estimates of exponent {{math|ω}} over time for the computational complexity of matrix multiplication <math>O(n^\omega)</math>.]]