File:GaulegExample.png

${\displaystyle \int _{-1}^{1}f(x){\rm {d}}x}$ with the ${\displaystyle N}$-point Legendre-Gaussian quadrature formula versus number ${\displaystyle N}$ for the following functions:

• ${\displaystyle f(x)={\sqrt {1-x^{2}}}}$ (red)
• ${\displaystyle f(x)={\frac {1}{1+x^{2}}}}$ (green)
• ${\displaystyle f(x)={\frac {1}{3+x}}}$ (blue)
• ${\displaystyle f(x)=x^{16}}$ (black)

In the last case, at ${\displaystyle N>8}$ the residual would be zero; practically, it is determined by the precision of arithmetic used to perform the evaluation. In this example, long double variables were used.

The logarithm of modulus of residual is plotted versus number of points in the quadrature formula. If the residual is positive, the datum is represented with circle; if the residual is negative, the datum is represented with dot. Out of grid, at the bottom of the graphic, the precision of calculus is not sufficient.

The following code is used to generate the figure:

GauLegExample/code

The figure is intented to be used for article Legendre-Gauss Quadrature formula.

{{Image|GaulegExample.png|right|350px|Add image caption here.}}