Build a huge matrix with vector components in MATLAB - matlab

Question

How can I avoid using a double for loop in order to build a matrix pos like this code does:
pos=[0 0];
for i=1:m;
for j=1:n;
pos=[pos; i j];
end
end
m and n are numbers such as 500 and 900.
I have to find a better solution in order improve computation time.
Thank you so much.

Answer 1

You can easily do this by meshgrid.
[X,Y] = meshgrid(1:m, 1:n);
pos = [0 0; X(:) Y(:)];
How the above code works is the following. meshgrid (in this case) creates a 2D grid of (X,Y) co-ordinates. X progresses horizontally while Y progresses vertically. As we can see in your for loops, m defines the horizontal boundaries while n denotes the vertical boundaries. By calling meshgrid(1:m, 1:n), I am creating a n x m grid for both X and Y, where each row of X progresses from 1 to m, while each column of Y progresses from 1 to n. Therefore, these will both be n x m matrices. Calling the above with m = 4 and n = 5 computes:
m = 4;
n = 5;
[X,Y] = meshgrid(1:m, 1:n)
X =
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
Y =
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
This almost follows the format you wish. You'll notice that by looking at the columns individually, this achieves what you want, but you want to stack all of the X and Y to be in a (n x m) + 1 x 2 matrix (1 to account for [0 0]). All we have to do now is take every column of X and Y and stack them on top of each other to create a single column for both. We can stack all of these together by doing X(:) and Y(:). X(:) will take every column of X and create a single column that stacks all of the columns together. The same is done for Y(:). As such, we first create pos by attaching [0 0] as the first row, and we then attach X(:) and Y(:) as columns to pos after, thus completing the construction of pos.
Let's do an example as a proof-of-concept. Suppose that we use the same values like we did before:
m = 4;
n = 5;
Using your for loop, we get:
pos =
0 0
1 1
1 2
1 3
1 4
1 5
2 1
2 2
2 3
2 4
2 5
3 1
3 2
3 3
3 4
3 5
4 1
4 2
4 3
4 4
4 5
Using the code I have written, we also get:
pos =
0 0
1 1
1 2
1 3
1 4
1 5
2 1
2 2
2 3
2 4
2 5
3 1
3 2
3 3
3 4
3 5
4 1
4 2
4 3
4 4
4 5
Minor Note
As you stated that m and n are going to be relatively large, I would recommend you clear X and Y from your workspace before you proceed. X and Y were only created to help you create pos. As you don't need them anymore, after you calculate pos, do:
clear X;
clear Y;