Efficient way to compute a double integral within a double integral

Hi all,

I would like to know what is the easiest and most efficient way to solve the integration shown in the figure attached.

I tried the following code, but it doesn't work:

R1 = @(x1,x2) integral2(@(x3,x4) fun_g(x1,x2,x3,x4),l3,u3,l4,u4);
R = integral2(R1.*@fun_f,l1,u1,l2,u2);


function f = fun_f(x1,x2)
% here we define f function (it is a rather complex function)
end

function g = fun_g(x1,x2,x3,x4)
% here we define g function (it is a rather complex function)
end

Thanks in advance!

Answer 1

[Torsten]

function main
  l1 = ...;
  u1 = ...;
  l2 = ...;
  u2 = ...;
  value_outer_integral = integral2(@fun,l1,u1,l2,u2)
end

function value_inner_integral = fun(x1,x2)
  l3 = ...;
  u3 = ...;
  l4 = ...;
  u4 = ...;
  for i = 1:numel(x1)
    value_inner_integral(i) = fun_f(x1(i),x2(i))*integral2(@(x3,x4)driver_fun_g(x1(i),x2(i),x3,x4),l3,u3,l4,u4);
  end
end

function g = driver_fun_g(x1,x2,x3,x4)
  for i = 1:numel(x3)
    g(i) = fun_g(x1,x2,x3(i),x4(i));
  end
end

function f = fun_f(x1,x2)
  % return f(x1,x2) for scalar values of x1 and x2
end

function g = fun_g(x1,x2,x3,x4)
  % return g(x1,x2,x3,x4) for scalar values of x1, x2, x3 and x4
end

[Igor Dakic]

Thanks! Is it possible to avoid loops and vectorize parts for value_inner_integral(i) and g(i)?

[Torsten]

For "value_inner_integral", the answer is no.

For "driver_fun_g": If you can evaluate g for arrays of x1, x2 x3 and y4, the answer is yes.