Legendre-Gauss Quadrature formula

The Legendre-Gauss Quadratude 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 order smaller than $2N$, then the exact equality takes place in equation (1).

The Legendre-Gauss quadratude 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 for an integrand, which is holomorphic function at 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 equation (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 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

For an integrand, which is polynomial of $M$th power, the Legendre-Gauss quadrature with $2M\!-\!1$ nodes gives the exact expression.

For an integrand, which is holomorphic function a the path of integration, the error of approximation reduces as exponential function of number of points of integration.

For an ntegrand with singulatiries at the interval of approximation, the formula still can be used, but the approximation is poor.
If the singularity of a holomorphic integrand is inside the range of integration, it can be avoided, deforming the contour of integration.
If the singularity is at the edge of the interval, one may consider to use another Gaussian quadrature, more sutable for the specific function. Alternatively, one may consider to change the variable of integration, making the integral 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?