Gauss-Kronrod Nodes and Weights (15-point)
List the nodes, Kronrod weights, and Gauss weights of the 15-point Gauss-Kronrod rule on the standard interval.
Input
This lists the nodes and weights of the Gauss-Kronrod (15, 7) quadrature rule. The 15-point Kronrod rule embeds the 7-point Gauss-Legendre rule, and their difference estimates the integration error.
The values are for the standard interval ［−1, 1］. Apply them to any interval by transforming the nodes and weights linearly.
Result
Kronrod nodes
15
of which 7 are also Gauss nodes
Kronrod points
15
Gauss points
7
Kronrod weight sum
2
Gauss weight sum
2
Nodes and weights
Nodes are in ascending order. Gauss weights appear only on rows that are also Gauss nodes.
| No. | Node | Kronrod weight | Gauss weight |
|---|---|---|---|
| 1 | -0.9914553711 | 0.022935322 | — |
| 2 | -0.9491079123 | 0.0630920926 | 0.1294849662 |
| 3 | -0.8648644234 | 0.1047900103 | — |
| 4 | -0.7415311856 | 0.1406532597 | 0.2797053915 |
| 5 | -0.5860872355 | 0.1690047266 | — |
| 6 | -0.4058451514 | 0.1903505781 | 0.3818300505 |
| 7 | -0.207784955 | 0.2044329401 | — |
| 8 | 0 | 0.2094821411 | 0.4179591837 |
| 9 | 0.207784955 | 0.2044329401 | — |
| 10 | 0.4058451514 | 0.1903505781 | 0.3818300505 |
| 11 | 0.5860872355 | 0.1690047266 | — |
| 12 | 0.7415311856 | 0.1406532597 | 0.2797053915 |
| 13 | 0.8648644234 | 0.1047900103 | — |
| 14 | 0.9491079123 | 0.0630920926 | 0.1294849662 |
| 15 | 0.9914553711 | 0.022935322 | — |
How it works
- The Gauss-Kronrod rule extends an n-point Gauss-Legendre rule with extra nodes to form a 2n+1 point rule. The 15-point Kronrod rule embeds the 7-point Gauss rule.
- This table gives the nodes and weights on the standard interval. To apply them to any interval, transform the nodes and weights linearly.
- The difference between the Kronrod and Gauss approximations estimates the integration error and underlies adaptive quadrature (for example QUADPACK QAGS).
- Nodes are symmetric about the center, and matching nodes share the same weight.
- Both the sum of the Kronrod weights and the sum of the Gauss weights equal the interval width (2 on the standard interval).
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Gauss-Kronrod Nodes and Weights (15-point)