Gauss-Lobatto Nodes and Weights
Enter the number of points n to list the Gauss-Lobatto quadrature nodes and weights, including the endpoints at minus one and plus one.
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Input
Enter the number of points n for Gauss-Lobatto quadrature to display the nodes and weights, including both endpoints at plus and minus one.
An integer from 2 to 128 (total points including the two endpoints)
Result
Total number of nodes
5
Nodes and weights on the standard interval from minus one to one
Exact polynomial degree
7
Endpoint weight
0.1
Nodes and weights
Nodes are listed in ascending order. The first and last rows are the endpoints at plus and minus one.
| Index | Node x | Weight w | Type |
|---|---|---|---|
| 1 | -1 | 0.1 | Endpoint |
| 2 | -0.654653671 | 0.544444444 | Interior |
| 3 | 0 | 0.711111111 | Interior |
| 4 | 0.654653671 | 0.544444444 | Interior |
| 5 | 1 | 0.1 | Endpoint |
How it works
- Gauss-Lobatto quadrature is a numerical integration rule that always includes the two endpoints of the standard interval, where x equals minus one and plus one, among its nodes. This makes it handy when joining sub-intervals or imposing boundary conditions.
- For n points there are 2 endpoint nodes plus n minus 2 interior nodes. The interior nodes are the zeros of the derivative of the Legendre polynomial of degree n minus 1, found here by Newton iteration.
- The endpoint weight equals 2 divided by n times the quantity n minus 1. The interior weights follow a similar formula, and all weights add up to exactly 2.
- The rule integrates polynomials up to degree 2n minus 3 exactly. Its exactness degree is lower than Gauss-Legendre (up to 2n minus 1) for the same point count, but it gains the benefit of including the endpoints.
- The listed nodes and weights are for the standard interval. To use them on a general interval, map them with a linear transform and multiply each weight by half the interval width.
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