Legendre-Gauss Quadrature formula
Legendre-Gauss Quadratude formiula is the approximation of the integral
- (1) [math]\displaystyle{ \int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^N w_i f(x_i). }[/math]
with special choice of nodes [math]\displaystyle{ x_i }[/math] and weights [math]\displaystyle{ w_i }[/math], characterised in that, if the finction [math]\displaystyle{ f }[/math] is polynomial of order smallet than [math]\displaystyle{ 2N }[/math], 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 [math]\displaystyle{ x_i }[/math] in equation (1) are zeros of the Polunomial of Lehendre [math]\displaystyle{ P_N }[/math]:
- (2) [math]\displaystyle{ P_N(x_i)=0 }[/math]
- (3) [math]\displaystyle{ -1\lt x_1\lt x_2\lt ... \lt x_N \lt 1 }[/math]
Weight [math]\displaystyle{ w_i }[/math] in equaiton (1) can be expressed with
- (4) [math]\displaystyle{ w_i = \frac{2}{\left( 1-x_i^2 \right) (P'_N(x_i))^2} }[/math]
There is no straightforward espression for the nodes [math]\displaystyle{ x_i }[/math]; they can be approximated with many decimal digits through only few iterations, solving numerically equation (2) with initial approach
- (5) [math]\displaystyle{ x_i\approx \cos\left(\pi \frac{1/2 +i}{N}\right) }[/math]
These formulas are described in the books [1] [2]
Precision of the approximation
Example
Extension to other interval
is straightforward. Should I copypast the obvious formulas here?
References
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