Canonical Matrix Multiplication

Data Structures and Algorithms with Object-Oriented Design Patterns in Python

Canonical Matrix Multiplication

Given an matrix A and an matrix B, the product C=AB is an matrix. The elements of the result matrix are given by

Accordingly, in order to compute the produce matrix, C, we need to compute mp summations each of which is the sum of n product terms. An algorithm to compute the matrix product is given in Program . The algorithm given is a direct implementation of Equation .

program3253
Program: DenseMatrix class __times__ method.

The algorithm begins by checking to see that the matrices to be multiplied have compatible dimensions. That is, the number of columns of the first matrix must be equal to the number of rows of the second one. This check takes O(1) time in the worst case.

Next a matrix in which the result will be formed is constructed (line 6). The running time for this is O(mp). For each value of i and j, the innermost loop (lines 10-11) does n iterations. Each iteration takes a constant amount of time.

The body of the middle loop (lines 9-12) takes time O(n) for each value of i and j. The middle loop is done for p iterations, giving the running time of O(np) for each value of i. Since, the outer loop (lines 7-12) does m iterations, its overall running time is O(mnp). Finally, the result matrix is returned on line 13. This takes a constant amount of time.

In summary, we have shown that lines 4-5 are O(1); line 6 is O(mp); lines 7-12 are O(mnp); and line 13 is O(1). Therefore, the running time of the canonical matrix multiplication algorithm is O(mnp).