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.
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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 ๏ผปโ1, 1๏ผฝ. 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 | โ |
| 2 | -0.9491079123 | 0.0630920926 | 0.1294849662 |
| 3 | -0.8648644234 | 0.1047900103 | โ |
| 4 | -0.7415311856 | 0.1406532597 | 0.2797053915 |
| 5 | -0.5860872355 | 0.1690047266 | โ |
| 6 | -0.4058451514 | 0.1903505781 | 0.3818300505 |
| 7 | -0.207784955 | 0.2044329401 | โ |
| 8 | 0 | 0.2094821411 | 0.4179591837 |
| 9 | 0.207784955 | 0.2044329401 | โ |
| 10 | 0.4058451514 | 0.1903505781 | 0.3818300505 |
| 11 | 0.5860872355 | 0.1690047266 | โ |
| 12 | 0.7415311856 | 0.1406532597 | 0.2797053915 |
| 13 | 0.8648644234 | 0.1047900103 | โ |
| 14 | 0.9491079123 | 0.0630920926 | 0.1294849662 |
| 15 | 0.9914553711 | 0.022935322 | โ |
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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