How do I vectorize the following the summation?

This is the code that I have. It's taking up way too much time so I was wondering if it is possible to 'vectorize' it, to save time. Here's the code:

Sigma=zeros(n,n);
G=zeros(n,n);
for i=1:m
Sigma=(y(i)-x(i,:)*bhat).^2*(x(i,:)).'*x(i,:)+Sigma;
G=*(x(i,:)).'*x(i,:)+G;
end

Where x is a mXn matrix, y is a mX1 and bhat is a nX1. I have already vectorized G but I can't get Sigma to work. My best try so far is the following.

 Sigma=x.'*(((y-x*bhat).')*(y-x*bhat))*x;
 G=x.'x;

I already realize why it is wrong, but I can't fix it. Thanks for the help in advance.

Answer 1

[David Goodmanson]

Hi Cache,

To obtain G, the sum over the row index i of matrix x leads to the product

G = x.'*x

as you determined.

Sigma involves the very same sum over the row index, only with an extra factor (y(i)-x(i,:)*bhat).^2. The index i is the row index for the quantity y-x*bhat, which is a column vector. This suggests the following:

A = (y-x*bhat).^2;
Sigma = x.'*diag(A)*x

[Andrei Bobrov]

+1

[Cache Ellsworth]

Hi David, This is pretty great. Thanks. The only problem is that my data is too large to make A a square matrix. ( It exceeds maximum array size preference. It would be 830479x830479) Is there another way to add the extra factor without making a large square matrix?

[David Goodmanson]

Hi Cache,

Yes there is. Since you are multiplying each row of x by a constant,

Sigma = x.'*(repmat(A,1,n).*x)    % A is column vector
% or
Sigma = x.'*(A.*x)   % with latest versions of Matlab (implicit expansion)

The first line assumes that you have enough space for another matrix the same size as x. If not, and if the second line doesn't work there is always for-loop in-place computation of A.*[each column of x] if necessary.