sum of k combination of vectors from n vectors

Question

I want sum of two vectors at time from set of n vectors for eg;

A1=[1 2 3] 
A2=[2 3 4] 
A3=[3 4 5]
.
.
.
An=[6 6 9]

I want sum of (Ai + Aj) for all values of i and j. so if n=10 then I need all combinations i.e. 10*9/2

Answer 1

Here's a way to compute it manually, assuming the set of n vectors is stored in a matrix A, row by row:

1. Obtain all possible pairs of indices (see this question for possible answers). For instance:

   [idx2, idx1] = find(ones(N, N));
   

   The corresponding pairs are given by:

   pairs = [idx1(:), idx2(:)];
   

   Alternatively, if you're not interested in repetitions (e.g. you don't want the sum A₁+A₁, etc.), you can use nchoosek:

   pairs = nchoosek(1:N, 2)
   idx1 = pairs(:, 1);
   idx2 = pairs(:, 2);
   

2. Use each pair of indices to sum the corresponding rows in A:

   sums = A(idx1(:), :) + A(idx2(:), :);
   

   Alternatively, if you want the total sum of elements for each pair of A_i and A_j, you can do sum(A(idx1(:), :) + A(idx2(:), :), 2) instead.

Example

Here's an example for N = 3:

A = [1 2 3; 2 3 4; 3 4 5];
N = size(A, 1);
[idx2, idx1] = find(ones(N, N));
pairs = [idx1(:), idx2(:)];
sums = A(idx1(:), :) + A(idx2(:), :);

The result is:

pairs =
     1     1
     1     2
     1     3
     2     1
     2     2
     2     3
     3     1
     3     2
     3     3

sums =      
     2     4     6
     3     5     7
     4     6     8
     3     5     7
     4     6     8
     5     7     9
     4     6     8
     5     7     9
     6     8    10

Answer 2

Have a look at pdist

pdist(X) computes the Euclidean distance between pairs of objects in m-by-n data matrix X. Rows of X correspond to observations, and columns correspond to variables.

And define your own custom metric which will just be a function that sums two vectors (although I have a feeling that @plus will work in your case i.e. pdist(X, @plus))

Answer 3

Lets have a try, as I am not sure whether you want a list of vectors as output or a list of sums I will give you both.

A1=[1 2 3] 
A2=[2 3 4] 
A3=[3 4 5] 
An=[6 6 9]

First of all make sure everything is put together in a matrix (Can be automated if required, but I hope you can get this matrix as input)

A = [A1;A2;A3;An]

Now we can just use a small loop to simply work on the combinations:

n = size(A,1);
m = size(A,2);
nr_comb = (n*(n-1))/2;
pair = zeros(nr_comb,2);
result = zeros(nr_comb,m);
count = 0;
for i = 1:n-1;
   for j = i+1:n
       count = count +1;
       pair(count,:) = [i j];
       result(count,:) = A(i,:) + A(j,:);
   end
end

Now assuming you actually want the sums of the combinations of vectors you can easily get them like so:

sumresult = sum(result')

It should not be too hard to add the symmetric variation, or the case where you combine a vector with itself, but given the number of combinations that you expect, this should be what you are looking for.