Gaussian quadrature

Instead of evaluating the integrand at equally spaced nodes as in Newton-Cotes methods, Gaussian quadrature methods make a judicious choice of nodes so as to maximize the precision of the numerical integration relative to the number of integrand evaluations.

The common Gaussian quadrature methods are:

• Gauss-Legendre used to integrate a function \(f(x)\) over a closed and bounded interval \([a,b]\).
• Gauss-Laguerre used to integrate a function of the form \(f(x) e^{-x}\) over the positive x-axis \(\lbrace x \in \mathbb{R} : x > 0 \rbrace\).
• Gauss-Hermite used to integrate a function of the form \(f(x) e^{-x^2}\) over the entire x-axis, \(\lbrace x \in \mathbb{R} : -\infty < x < \infty \rbrace\).
• Gauss-Chebyshev First Kind used to integrate a function of the form \(\frac{f(x)}{\sqrt( 1-x^2 )}\) over the interval \([-1,1]\).
• Gauss-Chebyshev Second Kind used to integrate a function of the form \(f(x) * \sqrt{ 1-x^2 }\) over the interval \([-1,1]\).