Gauss-Legendre nodes and weights

Compute Gauss-Legendre quadrature nodes and weights on an interval \([a,~b]\).

Usage

gauss_legendre(n, a = -1, b = 1)

Arguments

n

Number of quadrature nodes (n >= 1).

a

Left endpoint of the integration interval.

b

Right endpoint of the integration interval (b > a).

Value

A list with components:

nodes

Quadrature abscissae \(x_i\) in \([a,~b]\).

weights

Quadrature weights \(w_i\) such that \(\int_a^b f(x)\,dx \approx \sum_{i=1}^n w_i\,f(x_i).\)

Details

Gauss-Legendre quadrature provides exact integration for polynomials of degree up to \(2n-1\) using n nodes and weights chosen as the roots of the Legendre polynomial \(P_n(x)\) on the canonical interval \([-1,1]\). For a general interval \([a,b]\) the canonical nodes are shifted and rescaled onto \([a,b]\), and the weights are scaled by \((b-a)/2\).

This wrapper performs basic argument validation and calls the C++ routine to obtain nodes and weights with high accuracy for moderate n.

References

Davis, P. J., & Rabinowitz, P. (2007). Methods of Numerical Integration (2nd ed.).