Outer products are faster with b*b' in matlab?

I have a column vector with about 6800 elements, many of which are 0. How can I quickly calculate the multiplication of the column vector and its transpose?

a = rand(77,1);
b = [a; zeros(6800,1)]
tic
v = b*b';
% 0.053078s
toc
tic
v = bsxfun(@times, b, b.');
%0.066428s
toc

[Bruno Luong]

How so you intend to use v later? The best way would be do not compute v explicitly (1 rank matrix) but store it as simply as b. Then when you need to multiply v by an x a column vector, you will do

a = b'*x; % a is a scalar
vx = b * a;

You could store b as sparse matrix as suggested by @Sandeep Mishra

[z cy]

Thanks!

I need to sum up the vector v that is iterated N times.

n1 = 0;

n2= 0;

for i = 1:100

a = rand(77,1);

b = [a; zeros(6800,1)]

tic

n1 = n1 + b*b';

toc

tic

n2 = n2 + bsxfun(@times, b, b.');

toc

end

[Bruno Luong]

If the 0s is always at the same place why bother to multiply them in to loop

A = rand(77,100);
tic
n1 = 0;
for i = 1:100
    a = A(:,i);
    b = [a; zeros(6800,1)];
    n1 = n1 + b*b';
end
toc

Elapsed time is 7.638378 seconds.

tic
n3 = 0;
for i = 1:100
    a = A(:,i);
    n3 = n3 + a*a';
end
n3(end+6800,end+6800) = 0;
toc

Elapsed time is 0.168009 seconds.

norm(n1-n3,'fro') % Check is they are equal

ans = 0

[z cy]

Thanks！

I have found the faster way as you shown!

Just use A*A', while A is a 6800*100 matrix, then I can get the final result.

[Bruno Luong]

If you know A befre hand yes. You never state that in your question.

Answer 1

[Sandeep Mishra]

Hi Z,

From the code snippet, it's clear that you are aiming to calculate the product of a matrix with its transpose, where a significant portion of the matrix elements are zeros.

To improve efficiency, MATLAB’s ‘sparse’ matrix can be employed.

By converting the matrix into a sparse format, you can effectively reduce memory usage since sparse matrices only store non-zero elements. This can result in a faster computation, especially when dealing with large matrices where most elements are zero.

Refer to the following code snippet to implement the functionality:

a = rand(77,1); 
b = [a; zeros(6800,1)];
% Converting matrix to sparse matrix 
b_sparse = sparse(b);  

tic 
% Calculating Sparse product 
v_sparse = b_sparse * b_sparse';  
% Converting Sparse product result to Full matrix form
v_full = full(v_sparse); 
toc 

Elapsed time is 0.004505 seconds.

For more information, refer to the following MathWorks Documentation:

1. ‘sparse’ function: https://www.mathworks.com/help/releases/R2022b/matlab/ref/sparse.html
2. 'full' function: https://www.mathworks.com/help/releases/R2022b/matlab/ref/full.html

I hope this approach helps in addressing the issue effectively.