Modify an algorithm to perform vector operations by eliminating the inner most for loop
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Let A and B be square matrices (both stored column-wise) in R^{nxn} with B an Upper Triangular matrix. Write the MATLAB algorithm that gives C = A x B.
Here's my algorithm
function C = scalarMultRegUpper(A,B)
[n,n] = size(A);
[n,n]=size(B);
C=zeros(n,n);
for j=1:n
for k=1:n
for i=1:n
C(i,j)=C(i,j) + B(k,j)*A(i,k);
end
end
end
Now, I'm asked to modify my algorithm to perform vector operations by eliminating the inner most for loop. How to do that? How will the algorithm change?
11 个评论
Ameer Hamza
2020-9-19
What is the relevance of B being an Upper Triangular matrix? You are using the naive algorithm of matrix multiplication. I think you are expected to use a more efficient version for the upper triangular matrix B. Also, vectorizing these loops can simply be replaced with
C = A*B;
Vladimir Sovkov
2020-9-19
The most fundamental and beatuful thing about Matlab (matrix laboratory) is that it supports matrix operations, so you can compute that product in just one line of code: C = A*B;
Pascale Bou Chahine
2020-9-19
Ameer Hamza
2020-9-19
I am not sure what is an optimal algorithm for this case. It seems like a home problem. Have you studied something similar?
Pascale Bou Chahine
2020-9-19
Vladimir Sovkov
2020-9-19
编辑：Vladimir Sovkov
2020-9-19
If B is upper triangular, then B(k,j)=0 for k>(n+1-j) if the diagonal elements are nonzero (if they are zero, it is k>(n-j)). Hence in your code, you can restrict the k-loop to
for k=1:(n+1-j)
However anyway, C=A*B will work faster.
When working with big matrices having many zero elementss, using the sparse matrix format can be efficient, but I do not think it is justified for the triangular matrices; you can try it.
Pascale Bou Chahine
2020-9-19
Pascale Bou Chahine
2020-9-19
Vladimir Sovkov
2020-9-19
I implied that the upper triangular is the matrix of the form 
, while in your undestanding (maybe, more common) it is 
. This makes the things even easier:
, while in your undestanding (maybe, more common) it is . This makes the things even easier:for k=1:j
Rather obvious, isn't it?
Pascale Bou Chahine
2020-9-19
Vladimir Sovkov
2020-9-20
Analogously. Analyze which elements of A are zero with every fixed k and exlude them from the loop over i. The product of upper triangular matrices is the upper triangular matrix.
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2020-9-19
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