Strassen algorithm

Not to be confused with the Schönhage-Strassen algorithm for multiplication of polynomials.

In the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication. It is asymptotically faster than the standard matrix multiplication algorithm, but slower than the fastest known algorithm, and is useful in practice for large matrices.

1 History
2 Algorithm
3 Numerical analysis
4 See also
5 References
6 External links

[ History

Volker Strassen published the Strassen algorithm in 1969. Although his algorithm is only slightly faster than the standard algorithm for matrix multiplication, he was the first to point out that Gaussian elimination is not optimal. His paper started the search for even faster algorithms such as the Coppersmith–Winograd algorithm of Shmuel Winograd in 1980^{[dubious – discuss]} (which uses 7 binary multiplications, but 15 binary additions instead of 18 with the Strassen algorithm), and the more complex Coppersmith–Winograd algorithm published in 1987.

[ Algorithm

The left column represents 2x2 matrix multiplication. Naïve matrix multiplication requires one multiplication for each "1" of the left column. Each of the other columns represents a single one of the 7 multiplications in the algorithm, and the sum of the columns gives the full matrix multiplication on the left.

Let A, B be two square matrices over a ring R. We want to calculate the matrix product C as

$\mathbf{C} = \mathbf{A} \mathbf{B} \qquad \mathbf{A},\mathbf{B},\mathbf{C} \in R^{2^n \times 2^n}$

If the matrices A, B are not of type 2ⁿ x 2ⁿ we fill the missing rows and columns with zeros.

We partition A, B and C into equally sized block matrices

$\mathbf{A} = \begin{bmatrix} \mathbf{A}_{1,1} & \mathbf{A}_{1,2} \\ \mathbf{A}_{2,1} & \mathbf{A}_{2,2} \end{bmatrix} \mbox { , } \mathbf{B} = \begin{bmatrix} \mathbf{B}_{1,1} & \mathbf{B}_{1,2} \\ \mathbf{B}_{2,1} & \mathbf{B}_{2,2} \end{bmatrix} \mbox { , } \mathbf{C} = \begin{bmatrix} \mathbf{C}_{1,1} & \mathbf{C}_{1,2} \\ \mathbf{C}_{2,1} & \mathbf{C}_{2,2} \end{bmatrix}$

with

$\mathbf{A}_{i,j}, \mathbf{B}_{i,j}, \mathbf{C}_{i,j} \in R^{2^{n-1} \times 2^{n-1}}$

then

$\mathbf{C}_{1,1} = \mathbf{A}_{1,1} \mathbf{B}_{1,1} + \mathbf{A}_{1,2} \mathbf{B}_{2,1}$

$\mathbf{C}_{1,2} = \mathbf{A}_{1,1} \mathbf{B}_{1,2} + \mathbf{A}_{1,2} \mathbf{B}_{2,2}$

$\mathbf{C}_{2,1} = \mathbf{A}_{2,1} \mathbf{B}_{1,1} + \mathbf{A}_{2,2} \mathbf{B}_{2,1}$

$\mathbf{C}_{2,2} = \mathbf{A}_{2,1} \mathbf{B}_{1,2} + \mathbf{A}_{2,2} \mathbf{B}_{2,2}$

With this construction we have not reduced the number of multiplications. We still need 8 multiplications to calculate the C_i,j matrices, the same number of multiplications we need when using standard matrix multiplication.

Now comes the important part. We define new matrices

$\mathbf{M}_{1} := (\mathbf{A}_{1,1} + \mathbf{A}_{2,2}) (\mathbf{B}_{1,1} + \mathbf{B}_{2,2})$

$\mathbf{M}_{2} := (\mathbf{A}_{2,1} + \mathbf{A}_{2,2}) \mathbf{B}_{1,1}$

$\mathbf{M}_{3} := \mathbf{A}_{1,1} (\mathbf{B}_{1,2} - \mathbf{B}_{2,2})$

$\mathbf{M}_{4} := \mathbf{A}_{2,2} (\mathbf{B}_{2,1} - \mathbf{B}_{1,1})$

$\mathbf{M}_{5} := (\mathbf{A}_{1,1} + \mathbf{A}_{1,2}) \mathbf{B}_{2,2}$

$\mathbf{M}_{6} := (\mathbf{A}_{2,1} - \mathbf{A}_{1,1}) (\mathbf{B}_{1,1} + \mathbf{B}_{1,2})$

$\mathbf{M}_{7} := (\mathbf{A}_{1,2} - \mathbf{A}_{2,2}) (\mathbf{B}_{2,1} + \mathbf{B}_{2,2})$

which are then used to express the C_i,j in terms of M_k. Because of our definition of the M_k we can eliminate one matrix multiplication and reduce the number of multiplications to 7 (one multiplication for each M_k) and express the C_i,j as

$\mathbf{C}_{1,1} = \mathbf{M}_{1} + \mathbf{M}_{4} - \mathbf{M}_{5} + \mathbf{M}_{7}$

$\mathbf{C}_{1,2} = \mathbf{M}_{3} + \mathbf{M}_{5}$

$\mathbf{C}_{2,1} = \mathbf{M}_{2} + \mathbf{M}_{4}$

$\mathbf{C}_{2,2} = \mathbf{M}_{1} - \mathbf{M}_{2} + \mathbf{M}_{3} + \mathbf{M}_{6}$

We iterate this division process n-times until the submatrices degenerate into numbers (group elements).

Practical implementations of Strassen's algorithm switch to standard methods of matrix multiplication for small enough submatrices, for which they are more efficient. The particular crossover point for which Strassen's algorithm is more efficient depends on the specific implementation and hardware. It has been estimated that Strassen's algorithm is faster for matrices with widths from 32 to 128 for optimized implementations,^[1] and 60,000 or more for basic implementations.^[2]

[ Numerical analysis

The standard matrix multiplications takes

$n^3 = n^{\log_{2}8}$

multiplications of the elements in the ring R. We ignore the additions needed because, depending on R, they can be much faster than the multiplications in computer implementations, especially if the sizes of the matrix entries exceed the word size of the machine.

With the Strassen algorithm we can reduce the number of multiplications to

$n^{\log_{2}7}\approx n^{2.807}$ .

The reduction in the number of multiplications however comes at thee price of a somewhat reduced numerical stability.

[ See also

Z-order matrix representation

[ References

^ Skiena, Steven S. (1998), "§8.2.3 Matrix multiplication", The Algorithm Design Manual, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94860-7 .
^ Kakaradov, Boyko (2004), "Ultra-fast Matrix Multiplication: An Empirical Analysis of Highly Optimized Vector Algorithms", Stanford Undergraduate Research Journal 3: 33–36, http://surj.stanford.edu/2004/pdfs/kakaradov.pdf .

Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354-356, 1969
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 28: Section 28.2: Strassen's algorithm for matrix multiplication, pp.735–741.