What is Gauss-Lobatto Quadrature?

Gauss-Lobatto quadrature is a numerical integration technique that uses a specific set of points, called Lobatto points, to approximate the definite integral of a function.

Key Features of Gauss-Lobatto Quadrature:

• Lobatto Points: These points are the roots of the Legendre polynomial and its first derivative.
• Accuracy: Gauss-Lobatto quadrature provides a high degree of accuracy for approximating integrals, especially when the integrand is smooth.
• Efficiency: Compared to other numerical integration methods, Gauss-Lobatto quadrature often requires fewer function evaluations to achieve a desired accuracy.

How It Works:

1. Choose the number of points (n): The number of points determines the order of the polynomial used for approximation.

2. Determine the Lobatto points (x_i): These points are the roots of the Legendre polynomial and its derivative, ensuring accurate representation of the function.

3. Calculate the weights (w_i): These weights are assigned to each Lobatto point and are determined by the specific polynomial used.

4. Approximate the integral: The integral of the function is approximated by a weighted sum of the function values at the Lobatto points:

   ∫[a, b] f(x) dx ≈ ∑[i=0 to n] w_i * f(x_i)

Practical Applications:

Gauss-Lobatto quadrature is commonly used in various fields, including:

• Scientific Computing: Solving differential equations, approximating solutions to complex systems.
• Engineering: Analyzing stress and strain distributions, modeling fluid flow.
• Finance: Pricing options and other financial derivatives.

Example:

Consider approximating the integral of the function f(x) = x^2 over the interval [0, 1] using Gauss-Lobatto quadrature with 3 points.

1. Lobatto points: The three Lobatto points for n=3 are: x_0 = 0, x_1 = 1/2, x_2 = 1.

2. Weights: The corresponding weights are: w_0 = 1/6, w_1 = 2/3, w_2 = 1/6.

3. Approximation: The integral is approximated as:

   ∫[0, 1] x^2 dx ≈ (1/6) * f(0) + (2/3) * f(1/2) + (1/6) * f(1)

   Plugging in the function values, we get:

   ∫[0, 1] x^2 dx ≈ (1/6) * 0 + (2/3) * (1/4) + (1/6) * 1 = 1/3

   The exact value of the integral is 1/3, demonstrating the accuracy of Gauss-Lobatto quadrature.