Strassen algorithm

In the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication. It is faster than the standard matrix multiplication algorithm.

Contents

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

History

Volker Strassen published the Strassen algorithm in 1969 and although his algorithm is only slightly faster than the standard algorithm for matrix multiplication, he was the first to point out that Gaussian elimination was not optimal. His paper started the search for even faster algorithms (e.g. Coppersmith-Winograd algorithm).

Algorithm

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

<math>\mathbf{C} = \mathbf{A} \mathbf{B} \qquad \mathbf{A},\mathbf{B},\mathbf{C} \in \mathbb{K}^{2^n \times 2^n}<math>

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

<math>

\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} <math>

with

<math>\mathbf{A}_{i,j}, \mathbf{B}_{i,j}, \mathbf{C}_{i,j} \in \mathbb{K}^{2^{n-1} \times 2^{n-1}}<math>

then

<math>\mathbf{C}_{1,1} = \mathbf{A}_{1,1} \mathbf{B}_{1,1} + \mathbf{A}_{1,2} \mathbf{B}_{2,1} <math>

<math>\mathbf{C}_{1,2} = \mathbf{A}_{1,1} \mathbf{B}_{1,2} + \mathbf{A}_{1,2} \mathbf{B}_{2,2} <math>

<math>\mathbf{C}_{2,1} = \mathbf{A}_{2,1} \mathbf{B}_{1,1} + \mathbf{A}_{2,2} \mathbf{B}_{2,1} <math>

<math>\mathbf{C}_{2,2} = \mathbf{A}_{2,1} \mathbf{B}_{1,2} + \mathbf{A}_{2,2} \mathbf{B}_{2,2} <math>

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

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

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

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

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

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

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

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

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 multiplications for each M_k) and express the C_i,j as

<math>\mathbf{C}_{1,1} = \mathbf{M}_{1} + \mathbf{M}_{4} - \mathbf{M}_{5} + \mathbf{M}_{7}<math>

<math>\mathbf{C}_{1,2} = \mathbf{M}_{3} + \mathbf{M}_{5}<math>

<math>\mathbf{C}_{2,1} = \mathbf{M}_{2} + \mathbf{M}_{4}<math>

<math>\mathbf{C}_{2,2} = \mathbf{M}_{1} - \mathbf{M}_{2} + \mathbf{M}_{3} + \mathbf{M}_{6}<math>

We iterate this division process n-times until the submatrices degenerate into numbers.

Numerical analysis

The standard matrix multiplications takes

<math>n^3 = n^{\log_{2}8}<math>

multiplications of the elements in the field K. We ignore the additions needed because, depending on K, 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

<math>n^{\log_{2}7}\approx n^{2.807}<math>.

External links

• MathWorld: Strassen's Formula (http://mathworld.wolfram.com/StrassenFormulas.html)

References

• Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354-356, 1969it:Algoritmo di Strassen

ja:Strassenのアルゴリズム