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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 , although the naive algorithm is often better for smaller matrices.
The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". It is based on a way of multiplying two 2 × 2 -matrices which require only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations.
Strassen's algorithm improves on naive matrix multiplication through a divide-and-conquer approach. The key observation is that multiplying two 2 × 2 matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of 11 additional addition and subtraction operations).
The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication , and starts to outperform them in practice for numbers beyond about 10,000 to 100,000 decimal digits. [ 2 ]
Computing the k th power of a matrix needs k – 1 times the time of a single matrix multiplication, if it is done with the trivial algorithm (repeated multiplication). As this may be very time consuming, one generally prefers using exponentiation by squaring , which requires less than 2 log 2 k matrix multiplications, and is therefore much ...
Volker Strassen (born April 29, 1936) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz. [1]For important contributions to the analysis of algorithms he has received many awards, including the Cantor medal, [2] the Konrad Zuse Medal, [3] the Paris Kanellakis Award for work on randomized primality testing, [4] the Knuth ...
In the theory of matrix multiplication algorithms, Pan in 1978 published an algorithm with running time (). This was the first improvement over the Strassen algorithm after nearly a decade, and kicked off a long line of improvements in fast matrix multiplication that later included the Coppersmith–Winograd algorithm and subsequent developments.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.