Hello Melanie,
I understand that you would like to numerically integrate the given function using (4 X 4) Gaussian quadrature points while reducing the dimension of the integral.
To achieve this, I suggest you try following steps.
- Define the function you want to integrate.
- Choose number of quadrature points (4 in your case).
- Define limits.
- Define the quadrature points and their respective weights, to know more about how to calculate evaluation points and weights, I recommend you to refer the following documentation link – ( Gauss-Laguerre Quadrature Evaluation Points and Weights - MATLAB & Simulink Example (mathworks.com))
- Perform the integration using nested loops.
- To perform the integral using nested loop, please refer to the following example code
for i = 1:n
for j = 1:n
for k = 1:n
% Map the quadrature points to the integration limits
x_mapped = ((x_b - x_a) * x(i) + (x_b + x_a)) / 2;
y_mapped = ((y_b(x_mapped) - y_a(x_mapped)) * x(j) + (y_b(x_mapped) + y_a(x_mapped))) / 2;
z_mapped = ((z_b - z_a(x_mapped, y_mapped)) * x(k) + (z_b + z_a(x_mapped, y_mapped))) / 2;
% Evaluate the function at the mapped quadrature points
f_eval = f(x_mapped, y_mapped, z_mapped);
% Add the weighted function evaluation to the sum
integral_sum = integral_sum + w(i) * w(j) * w(k) * f_eval;
end
end
end
I hope this helps.
