Fast computation of an equation.

Given two matrices:

We want a the fastest way to compute the vector \phi.... (j=1 to m).

Is there a way to use parallel computation, any trick?

[darova]

What does it mean ?

[Walter Roberson]

As then is a vector and would be the ith element of vector

If you arrange Y and Z as 2D array

so that varying i in is traveling along rows, then from R2019a onwards,

sum((x-Y).*(Z-x), 2)

[Bruno Luong]

+1 on Walter's solution. You probably want to write

"then from R2016b onwards".

I probably organize data in transpose way, but this does not have huge impact.

Not sure if the computation is to do once or multiple times with Y/Z fix or x fix.

[David Ernesto Caro]

function Z = quad_trans(X,W,V)
     % X is a matrix n by s
     % W and V are matrices m by n
     N = size(X,2);  
     Z=zeros(size(W,1),N);
     for i=1:N
           Z(:,i)=sum((X'- W).*(V-X'),2);
     end
end

Well, I modified a little bit the statement.

Thanks for the observation Bruno Luong.

Walter, We currently are computing the solution that way. But I think we are not taking advantage of parallelism or other faster matlab characteristics.

[Walter Roberson]

Bruno is right, I should have said R2016b onwards.

[Walter Roberson]

I see that you extended by a dimension. The code remains the same provided that you arrange the so that the k run along the third dimension, so being MATLAB's x(1,i,k) . The result would be m (length of j) by 1 by s (length of k), which you might want to squeeze() to a 2D array.

[David Ernesto Caro]

Thanks Walter! Is working.

[Bruno Luong]

what are typically m,n and s?

If you have X as matrix the best method might be different, see my answer bellow.

Answer 1

[Bruno Luong]

The efficientcy of the bellow method depend on m, n, s

% Generate dummy test data
m = 1000;
n = 1000;
s = 1000;
W = rand(m,n);
V = rand(m,n);
X = rand(n,s);

% Walter method
tic
Xp = permute(X,[3 1 2]);
Z1 = sum((Xp-W).*(V-Xp),2);
Z1 = reshape(Z1,[m s]).';
toc % Elapsed time is 2.816333 seconds.

% My new method
tic
Z2 = -sum(X.^2,1)+(W+V)*X-sum(W.*V,2);
Z2 = Z2.';
toc % Elapsed time is 0.040274 seconds.

% Check correctness
norm(Z1-Z2,'fro')/norm(Z1,'fro') % 4.6622e-15

[David Ernesto Caro]

Amazing! Thanks!