Introduction to Gaussian Integration Implementation in MATLAB with Source Code

Gaussian integration is an efficient numerical integration method particularly suitable for calculating definite integrals. By selecting appropriate integration nodes and corresponding weight coefficients, it achieves high-precision integration results with relatively low computational cost. In MATLAB implementation, Gaussian integration typically requires pre-determining integration nodes and weights, followed by weighted summation calculations using these parameters.

Fundamental Principles of Gaussian Integration Gaussian integration (also known as Gauss-Legendre integration) core concept involves selecting optimal nodes and corresponding weights within the integration interval to enable exact integration of polynomial functions. Nodes and weights are typically determined through the roots of Gauss-Legendre polynomials. The integration formula can be expressed as:

[ int_{-1}^{1} f(x) , dx approx sum_{i=1}^{n} w_i f(x_i) ]

where (x_i) represents integration nodes, (w_i) denotes corresponding weights, and (n) indicates the number of integration points.

Key Implementation Steps in MATLAB 1. Determining Integration Nodes and Weights: Utilize built-in functions like `lgwt` (Legendre-Gauss Weights and Nodes) or custom calculations to obtain Gaussian quadrature parameters Example code approach: [nodes, weights] = lgwt(n, -1, 1) for n-point quadrature 2. Adapting Integration Intervals: For general intervals [a,b] instead of [-1,1], apply linear transformation: x_transformed = (b-a)/2 * nodes + (a+b)/2 weights_transformed = (b-a)/2 * weights 3. Computing Weighted Summation: Evaluate the integrand function at transformed nodes, multiply by corresponding weights, and accumulate results MATLAB implementation: integral_result = sum(weights_transformed .* f(x_transformed))

Extension Concepts Adaptive Gaussian Integration: When integrand functions exhibit rapid variations in specific intervals, implement adaptive strategies by subdividing intervals and applying Gaussian quadrature on subintervals for enhanced accuracy Multidimensional Integration: For multivariate functions, employ tensor product forms of Gaussian integration, though computational cost increases significantly with dimensionality Error Analysis: Gaussian integration demonstrates rapid convergence for smooth functions but may perform poorly for singular functions. Consider combining with other numerical integration methods (like composite trapezoidal rule) for optimization

When implementing Gaussian integration in MATLAB, leverage its powerful matrix operations capabilities to enhance code efficiency and readability through vectorized computations and built-in polynomial functions.