|
Data Structures and Algorithms
with Object-Oriented Design Patterns in Ruby |
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 .
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 (lines 5-6). The running time for this is O(mp). For each value of i and j, the innermost loop (lines 10-12) does n iterations. Each iteration takes a constant amount of time.
The body of the middle loop (lines 8-14) 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 7-15) does m iterations, its overall running time is O(mnp). Finally, the result matrix is returned on line 16. This takes a constant amount of time.
In summary, we have shown that line 4 is O(1); lines 5-6 is O(mp); lines 7-15 are O(mnp); and line 16 is O(1). Therefore, the running time of the canonical matrix multiplication algorithm is O(mnp).
Copyright © 2004 by Bruno R. Preiss, P.Eng. All rights reserved.
