Gauss-Jacobi Quadrature Calculator
Enter an integrand f(x), the order n, and the weight exponents α and β to evaluate ∫₋₁¹ f(x)(1−x)^α(1+x)^β dx by Gauss-Jacobi quadrature. Nodes and weights are computed with the Golub-Welsch method (eigendecomposition of a tridiagonal matrix).
Input
Enter the integrand f(x), the order n, and the weight exponents α and β to evaluate ∫₋₁¹ f(x)(1−x)^α(1+x)^β dx by Gauss-Jacobi quadrature.
e.g. cos(x), 1/(2-x), x^2 + 1. See the method for available functions, constants and operators. The weight (1−x)^α(1+x)^β is applied automatically.
Number of nodes (1–256)
Exponent of the weight (1−x)^α (α ＞ −1)
Exponent of the weight (1+x)^β (β ＞ −1)
Result
Integral ∫ f(x)(1−x)^α(1+x)^β dx
1.3824596874
Weight (1−x)^(α=0.5) (1+x)^(β=0.5) on [−1, 1]
Order n
8
Number of nodes
8
Zeroth moment μ₀
1.57079633
Nodes, weights and function values
Shows the nodes x_i on [−1, 1], the quadrature weights w_i (which include (1−x)^α(1+x)^β), and the function values f(x_i).
| # | Node x_i | Weight w_i | f(x_i) |
|---|---|---|---|
| 1 | -0.93969262 | 0.04083295 | 0.59003622 |
| 2 | -0.76604444 | 0.1442256 | 0.72065865 |
| 3 | -0.5 | 0.26179939 | 0.87758256 |
| 4 | -0.17364818 | 0.33854023 | 0.984961 |
| 5 | 0.17364818 | 0.33854023 | 0.984961 |
| 6 | 0.5 | 0.26179939 | 0.87758256 |
| 7 | 0.76604444 | 0.1442256 | 0.72065865 |
| 8 | 0.93969262 | 0.04083295 | 0.59003622 |
How it works
- Gauss-Jacobi quadrature approximates the weighted definite integral ∫₋₁¹ f(x)(1−x)^α(1+x)^β dx over [−1, 1] as Σ w_i f(x_i) using n nodes x_i and weights w_i. The weight is integrable only when α > −1 and β > −1.
- The nodes x_i are the zeros of the Jacobi polynomial P_n^{(α,β)}(x) and the w_i are the corresponding quadrature weights. This tool builds the symmetric tridiagonal Jacobi matrix from the three-term recurrence coefficients and applies the Golub-Welsch method: the eigenvalues give the nodes, and the squared first components of the eigenvectors give the weights. The eigendecomposition uses implicit-shift QL iteration.
- An n-point rule is exact for any polynomial f(x) of degree up to 2n−1. For integrals whose weight introduces endpoint singularities through α and β, this rule is far more accurate than equally spaced formulas.
- Special cases: α=β=0 reduces to Gauss-Legendre, α=β=−1/2 to Chebyshev-Gauss (first kind), and α=β=1/2 to Chebyshev-Gauss (second kind).
- The integrand f(x) is evaluated safely by a recursive-descent parser (no eval). It supports functions such as sin, cos, tan, exp, log, ln, sqrt, abs, the constants pi and e, the four arithmetic operators, exponentiation ^, and implicit multiplication (e.g. 2x, 3(x+1)). The only variable is x.
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Gauss-Jacobi Quadrature Calculator