gaussian quadrature with quadrature points

I need help writing a matlab code that numerically integrates the function using 4 × 4 Gaussian quadrature points over the given domain by reducing the dimension of the integral.

Answer 1

[SANKALP DEV]

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.