How to vectorize integral2?

We can use 'integral' for twice only when these two variables are independent to each other. And then the option 'Arrayvalued' could be used then. But what if we can't seperate them? For instance:

fun = @(n,x,y) sin(n*x-y).*sqrt(x-n*y);
for i=1:1e3
    integral2(@(x,y) fun(i,x,y),0,2*pi,0,2*pi);
end

Is there anaway to speed it up by vectorizing it?

Answer 1

[Walter Roberson]

No, there is no way to speed that up by vectorizing.

In general, if you have multiple functions to calculate at the same time, then they might well have different convergence properties. integral2() would have no way of splitting up a vector-valued function into its different components, so if it did support vector-valued functions, it would end up having to evaluate every segment according to the worst convergence over that segment.

[Walter Roberson]

By the way, under the assumption that n is a positive integer, there does turn out to be a closed form solution

-pi^(1/2)*(-n^3*(fresnels(2/n^(1/2))-fresnels((-2*n+2)^(1/2)/n^(1/2)*2^(1/2)))* ...
 cos((2*n^2-2)*pi/n)-n^3*(fresnelc(2/n^(1/2))-fresnelc((-2*n+2)^(1/2)/n^(1/2)* ...
2^(1/2)))*sin((2*n^2-2)*pi/n)+1i*n^3*fresnels(2)+fresnels(2*1i*n)+...
 fresnels(2*n^(1/2))-fresnels((-2*n+2)^(1/2)*2^(1/2)*n^(1/2)))/n^(3/2)*2^(1/2)/(2*n^2-2)

[祥宇 崔]

Thanks a lot!