Legendre-Gauss Quadrature formula

Legendre-Gauss Quadratude formiula is the approximation of the integral

(1) $\int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{N}w_{i}f(x_{i}).$

with special choice of nodes $x_{i}$ and weights $w_{i}$, characterised in that, if the finction $f$ is polynomial of order smallet than $2N$, then the exact equality takes place in equation (1).

Legendre-Gauss quadratude formula is special case of Gaussian quadratures of more general kind, which allow efficient approximation of a function with known asumptiotic behavior at the edges of the interval of integration.

Nodes and weights

Nodes $x_{i}$ in equation (1) are zeros of the Polynomial of Legendre $P_{N}$:

(2) $P_{N}(x_{i})=0$

(3) $-1<x_{1}<x_{2}<...<x_{N}<1$

Weight $w_{i}$ in equaiton (1) can be expressed with

(4) $w_{i}={\frac {2}{\left(1-x_{i}^{2}\right)(P'_{N}(x_{i}))^{2}}}$

There is no straightforward espression for the nodes $x_{i}$; they can be approximated with many decimal digits through only few iterations, solving numerically equation (2) with initial approach

(5) $x_{i}\approx \cos \left(\pi {\frac {1/2+i}{N}}\right)$

These formulas are described in the books

Example

Fig.1. Example of estimate of precision: Logarithm of residual versus number $N$ of terms in the right hand side of equation (1) for various integrands $f(x)$.

Extension to other interval

is straightforward. Should I copypast the obvious formulas here?