Strassen’s matrix multiplication algorithm is an algorithm for multiplying two matrices. It was proposed by Volker Strassen in 1969 and is notable for its lower time complexity than the standard matrix multiplication algorithm.
The standard algorithm for multiplying two n x n matrices has a time complexity of O(n3). Strassen’s algorithm improves on this by dividing each matrix into four submatrices of size n/2 x n/2, recursively computing the products of these submatrices using seven matrix multiplications, and then combining these products to obtain the final result.
The time complexity of Strassen’s algorithm is given by the recurrence relation T(n) = 7T(n/2) + O(n2), which can be solved to give a time complexity of O(nlog2(7))O(n2.81). Thus, Strassen’s algorithm is more efficient than the standard algorithm for large values of n.
Here is an example of how Strassen’s algorithm works for multiplying two 2 x 2 matrices:
Consider the matrices A and B:
A = | a11 a12 |
| a21 a22 |
B = | b11 b12 |
| b21 b22 |We can divide each matrix into four submatrices of size 1 x 1:
A = | A11 A12 | B = | B11 B12 |
| A21 A22 | | B21 B22 |We can then compute seven products of these submatrices:
P1 = A11 * (B12 - B22)
P2 = (A11 + A12) * B22
P3 = (A21 + A22) * B11
P4 = A22 * (B21 - B11)
P5 = (A11 + A22) * (B11 + B22)
P6 = (A12 - A22) * (B21 + B22)
P7 = (A11 - A21) * (B11 + B12)We can then combine these products to obtain the final result:
C11 = P5 + P4 - P2 + P6
C12 = P1 + P2
C21 = P3 + P4
C22 = P5 + P1 - P3 - P7This example illustrates how Strassen’s algorithm works by recursively dividing the matrices into submatrices and computing the products of these submatrices using a small number of matrix multiplications. While the algorithm has a lower time complexity than the standard algorithm for large matrices, it requires more memory and is often not used in practice for matrices of small to moderate size.