Gauss–Lobatto Endpoint Exactness

Worked Example

Let's calculate the 3-node Gauss-Lobatto integral of f(x) = x⁴ on the interval [-1, 1].

The 3 nodes are at -1, 0, and 1, with corresponding weights 1/3, 4/3, and 1/3.

1. At x = -1, f(-1) = 1. Weight is 1/3, contribution = 1/3.
2. At x = 0, f(0) = 0. Weight is 4/3, contribution = 0.
3. At x = 1, f(1) = 1. Weight is 1/3, contribution = 1/3.
4. The quadrature sum is 1/3 + 0 + 1/3 = 2/3.
5. The exact integral of x⁴ on [-1, 1] is [x⁵/5] from -1 to 1 = 1/5 - (-1/5) = 2/5 = 0.4.
6. The absolute error is |2/3 - 2/5| = 4/15 ≈ 0.2667.

Because the 3-node rule has a degree of exactness of 3, it cannot perfectly integrate the 4th-degree polynomial x⁴.

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.

Limitations

The primary limitation of any finite quadrature rule is its bounded degree of exactness. A Gauss-Lobatto rule with n nodes perfectly integrates polynomials up to degree 2n-3. For higher-degree polynomials or non-polynomial functions, an inherent truncation error remains.

Additionally, when integrating on an interval of zero width (Half width = 0), the true integral is trivially 0, but numerical representations must handle the scaling appropriately to prevent 0⁰ anomalies or undefined node generation.

References

For a comprehensive treatment of numerical integration and specific parameters for Gaussian and Gauss-Lobatto quadrature rules, see the DLMF reference on orthogonal polynomials and quadrature: NIST Digital Library of Mathematical Functions, Section 3.5.