Hi Stephan,
I understand that you intend to perform Gaussian quadrature for arbitrary weight functions in MATLAB. I recommend the following steps to achieve the desired results:
Define the weight function and desired number of nodes:
weight_function = @(x) exp(-x); % example weight function
n = 5; % number of nodes
Generate the quadrature nodes and weights:
You can use the Golub-Welsch algorithm to find the nodes and weights for any weight function. This method involves computing the eigenvalues and eigenvectors of a Jacobi matrix constructed from the moments of the weight function.
function [nodes, weights] = custom_gaussian_quadrature(weight_function, n)
% Define a suitable interval for integration, e.g., [a, b]
a = -1; % start of interval
b = 1; % end of interval
% Compute the moments of the weight function
moments = zeros(1, 2*n);
for k = 1:2*n
moments(k) = integral(@(x) (x.^(k-1)) .* weight_function(x), a, b);
end
% Construct the Jacobi matrix
J = zeros(n);
for i = 1:n
J(i,i) = moments(2*i-1) / moments(2*i-2); % diagonal elements
if i < n
J(i,i+1) = sqrt(moments(2*i) * moments(2*i-1)) / moments(2*i-2); % off-diagonal elements
J(i+1,i) = J(i,i+1); % symmetric matrix
end
end
% Compute the eigenvalues and eigenvectors of the Jacobi matrix
[V, D] = eig(J);
nodes = diag(D); % nodes are the eigenvalues
weights = moments(1) * V(1,:)'.^2; % weights are the square of the first element of eigenvectors
end
% Use the function to compute nodes and weights
[nodes, weights] = custom_gaussian_quadrature(weight_function, n);
Use the nodes and weights for integration:
integral_value = sum(weights .* f(nodes)); % where f is the function to be integrated
This approach gives you the Gaussian quadrature nodes and weights for any arbitrary weight function. Adjust the interval [a, b] as per the specific requirement.
Hope this helps.
Regards,
Nipun
