How to vectorize a function for integral2? - matlab

Question

I want to evaluate a double integral of the form
$$\int_{-\infty}^a \int_{-\infty}^b \sum_{i,j}^K a_ia_jx^iy^j\exp(-x^2 - y^2 + xy)dx dy $$
where $a_i$ and $a_j$ are constants. Since the integral is linear, I can interchange summation and integration, but in this case I have to evaluate $K^2$ integrals and it takes too long. In that case I do the following:
for i = 1:K
for j = 1:K
fun = #(x,y) x.^i.*y.^j.*exp(-2.*(x.^2 + y.^2 - 2.*x.*y))
part(i,j) = alpha(i)*alpha(j)*integral2(fun,-inf,a,-inf,b)
end
end
It takes too long, so I want to evaluate only one integral, but I don't know how to vectorize $\sum_{i,j}^K a_ia_jx^iy^j\exp(-x^2 - y^2 + xy)$, namely, how to supply it to integral2. I would be very grateful for any help.

Answer 1

It looks like you'll need to have i and j be third and fourth dimensions, in order for there to be a chance that the code will work.
I also don't have integral2 (I use octave, integral2 is a new matlab function that octave doesn't yet have), so I can't test it, but I'd think something like this might work:
alphaset=zeros(1,1,K,K);
alphaset(1,1,1:K,1:K)=alpha(1:K)'*alpha(1:K);
i_set=zeros(1,1,K,1);
j_set=zeros(1,1,1,K);
i_set(:)=1:K;
j_set(:)=1:K;
fun=#(x,y) x.^i_set.*y.^j_set.*exp(-2.*(x.^2 + y.^2 - 2.*x.*y));
part = squeeze(alphaset.*integral2(fun,-inf,a,-inf,b));
As I said, I can't promise that it'll work, because I don't know how integral2 works. But if you replace the integral2 with simply "sum(sum(fun([1,2,4],[3,-1,2])))", then it works as intended for that operation (that is, it sums over the x and y values, and the result is a matrix over the set of indices).

Answer 2

If you just want to improve speed, you may try parfor.
Let $X=(x,x^2,\cdots,x^K)$, $Y=(y,y^2,\cdots,y^K)$, $A=(a_{ij})$ be a matrix with $a_{ij}=a_{i}a_{j}$, then
$$\sum_{i,j}^K a_{i}a_{j}x^iy^j=XAY^{T}$$
I don't have integral2 function on my matlab, so I didn't test if it will improve the speed a lot.
Also, I think you need to use syms x and y, after you compute the $$XAY^{T}$$, then use matlabFunction to convert symbolic expression to function handle. Here it is my test code: syms x y; X=[x,x^2]; Y=[y,y^2]; Z=X*Y'; fun =matlabFunction(Z); ff=#(x,y) x^2+y^2; gg=fun(x,y).*ff(x,y);