Step 1 - Data and Initialization:

You have a definite integral to compute: ∫[a, b] f(x)dx, and you want a more accurate approximation than traditional numerical methods provide.

Step 2 - Choose the Number of Nodes and Weights:

Select the number of nodes (sample points) and their corresponding weights based on a specific Gaussian quadrature rule. Common rules include the Gauss-Legendre, Gauss-Chebyshev, or Gauss-Laguerre rules.

Step 3 - Transformation of the Integral:

If the integral is over a different interval or has singularities, perform a change of variables to transform it to a standard form ∫[-1, 1] g(t)dt, where t is a new variable, and g(t) is a suitable weight function.

Step 4 - Evaluation of the Integral:

Calculate the integral using the formula:

∫[a, b] f(x)dx ≈ ∑(i=1 to n) wi * f(xi)

Here, wi are the weights, and xi are the nodes specific to the selected Gaussian quadrature rule.

Step 5 - Output:

The result obtained using Gaussian quadrature provides an accurate approximation of the definite integral.

Gaussian quadrature is a highly accurate method for numerical integration and can provide very precise results for a wide range of functions.