Gauss–Lobatto Endpoint Exactness

Gauss–Lobatto quadrature is a numerical integration method that guarantees the endpoints of the integration interval are included as nodes. While standard Gauss-Legendre quadrature optimally selects all interior nodes to maximize the polynomial degree of exactness, Gauss-Lobatto specifies the endpoints and optimally selects the remaining interior nodes.

Experiment

We can verify this finite exactness directly using the interactive visualization.

1. Open the Gauss-Lobatto Endpoint Exactness module.
2. Set Nodes to 3.
3. Set Half width to 1 (this sets the interval to [-1, 1]).
4. Set Monomial degree to 4.
5. Observe that the Exact integral is exactly 0.4, but the Quadrature yields approximately 0.6667.
6. Now increase the Nodes to 4. The Quadrature matches the Exact integral perfectly, yielding an Absolute error of 0, because the 4-node rule has a degree of exactness of 5.