Gauss-Jacobi Quadrature Nodes and Weights
Compute the nodes and weights of Gauss-Jacobi quadrature for the weight (1-x)^a (1+x)^b from the degree n and parameters a and b, shown in a table.
Input
Enter the degree n and weight parameters a and b to compute the Gauss-Jacobi quadrature nodes and weights for the weight (1-x)^a (1+x)^b.
Integer from 1 to 128 (the number of nodes)
Exponent of (1-x)^a. Must be greater than minus 1
Exponent of (1+x)^b. Must be greater than minus 1
Result
Degree n
5
Weight (1-x)^a (1+x)^b with a＝0, b＝0
Number of nodes
5
Sum of weights mu0
2
Weight function
(1-x)^a (1+x)^b
Nodes and weights
Nodes x_i are sorted ascending within the interval from minus 1 to 1, with the matching quadrature weights w_i.
| Index i | Node x_i | Weight w_i |
|---|---|---|
| 1 | -0.9061798459 | 0.2369268851 |
| 2 | -0.5384693101 | 0.4786286705 |
| 3 | -2.2967128660e-17 | 0.5688888889 |
| 4 | 0.5384693101 | 0.4786286705 |
| 5 | 0.9061798459 | 0.2369268851 |
How it works
- Computes the nodes and weights of degree n Gauss-Jacobi quadrature for the weight (1-x)^a (1+x)^b on the interval from minus 1 to 1.
- Nodes and weights are obtained by the Golub-Welsch method. The three-term recurrence coefficients of the Jacobi polynomials form a symmetric tridiagonal matrix whose eigenvalues are the nodes and whose first eigenvector components give the weights.
- The eigendecomposition uses the implicitly shifted QL iteration for symmetric tridiagonal matrices.
- The zeroth moment mu0 equals the sum of the weights, given by 2^(a+b+1) times Gamma(a+1)Gamma(b+1)/Gamma(a+b+2).
- Both a and b must be greater than minus 1 so the weight is integrable. When a equals b equals 0 the rule reduces to Gauss-Legendre quadrature.
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Gauss-Jacobi Quadrature Nodes and Weights