Legendre-Gauss Quadrature formula

Legendre-Gauss Quadratude formiula is the approximation of the integral

(1) $\underset{-1}{\overset{1}{\int }}f(x)dx\approx \sum _{i=1}^N{w}_{i}f(x_i).$

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 Polunomial of Lehendre $P_N$:

(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) $w_i=\frac{2}{(1-x_i^2)(P{\text{'}}_N(x_i){)}^2}$

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

File:GaulegExample.png

) versus number $N$ of terms in the right hand side of equation (1).

Extension to other interval

is straightforward. Should I copypast the obvious formulas here?