Legendre-Gauss Quadrature formula

The Legendre-Gauss Quadrature formula or Gauss-Legendre quadrature 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 function $f$ is polynomial of degree smaller than $2N$, then the exact equality takes place in equation (1).

The Legendre-Gauss quadrature formula is a special case of Gaussian quadratures which allow efficient approximation of a function with known asymptotic behavior at the edges of the interval of integration. This approximation is especially recommended if the integrand is holomorphic in a neighbourhood of integration interval.

Nodes and weights

The nodes $x_{i}$ in equation (1) are zeros of the Legendre polynomial $P_{N}$:

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

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

The weights $w_{i}$ in equation (1) can be expressed bu

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

There is no straightforward expression for the nodes $x_{i}$; they can be approximated to many decimal places 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

Precision of the approximation

If the integrand is a holomorphic function along the path of integration, then the error of approximation decreases exponentially in the number of nodes $N$. If the integrand is a polynomial of degree $2N-1$ or less, then Legendre-Gauss quadrature with $N$ nodes yields the exact answer.

If the integrand has a singularity, then the formula still can be used, but the approximation is poor. In that case, one may consider to use another Gaussian quadrature, more suitable for the specific function. Alternatively, it may be possible to deform the contour of integration to avoid the singularity if it is inside the integration interval. Another possibility is to do a substitution that makes the integrand regular.

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 $f(x)$.

In Fig.1, the decimal logarithm of the modulus of the residual of the appdoximation of integral with Gaussian quadrature is shown versus number of terms in the sum, for four examples of the integrand.

$f(x)=x^{16}$ (black)

$f(x)={\sqrt {1+x^{2}}}$ (red)

$f(x)=1/(1+x^{2})$ (green)

$f(x)=1/(3+x)$ (blue)

The first of these functions is integrated "exactly" at $N>8$, and the residual is determined by the rounding errors at the long double arithmetic. The second function (red) has branch points at the end of the interval; therefore, the approximation does not improve quickly at the increase of number terms in the sum. The last two functions are analytic within the range of integration; the residual decreases exponentially, and the precision of evaluation of the integral is limited only by the rounding errors.

Extension to other interval

is straightforward. Should I copypast the obvious formulas here?