Introduction to Gaussian Integration Implementation in MATLAB with Source Code

In the following paragraphs, we will introduce the MATLAB implementation of Gaussian integration along with its source code.

Gaussian integration is a numerical integration method used to compute definite integrals of specific functions over given intervals. It utilizes the roots and coefficients of Gauss-Legendre polynomials, enabling precise calculation of polynomial functions and certain non-polynomial functions. This method typically offers greater accuracy and faster computation compared to other numerical integration approaches.

In MATLAB, we can implement Gaussian integration using either built-in functions or by manually coding the algorithm. While built-in functions simplify implementation, manual coding provides deeper understanding of algorithmic details and implementation processes. The following code segment demonstrates manual implementation of Gaussian integration with explanations for each computational step.

% Gaussian quadrature implementation for n=5 points n = 5; % number of quadrature points x = linspace(-1,1,n); % calculate Legendre polynomial roots (approximated) w = zeros(1,n); % preallocate weight coefficients array for k = 1:n % Construct Lagrange basis polynomial by excluding current root p = poly(x([1:k-1 k+1:end])); % Calculate weight using standard Gaussian quadrature formula w(k) = 2/((1-x(k)^2)*polyval(p,x(k))^2); end % Compute integral: weighted sum of function evaluations I = sum(w.*f(x)); % where f(x) is the target function

The code above demonstrates the complete workflow: first defining the integration interval and target function, then computing roots of Gauss-Legendre polynomials (approximated via linear spacing). Using these roots, we construct orthogonal polynomials and calculate corresponding weight coefficients through polynomial evaluation. Finally, the integral is computed as the weighted summation of function values at quadrature points. This implementation clarifies both the theoretical foundation and practical execution of Gaussian integration algorithms.