Gaussian quadrature

Gaussian quadrature is a numerical integration method used to approximate the definite integral of a function, particularly when the exact integral is difficult or impossible to obtain analytically. It is especially useful in the field of computational mathematics and various applications in physics and engineering. Gaussian quadrature approximates the integral of a function over a certain interval by a sum of the function's values at specified points within the interval, weighted appropriately. The method is named after the German mathematician and physicist Carl Friedrich Gauss.

Overview

The basic idea behind Gaussian quadrature is to approximate the integral

\[\int_a^b f(x) \, dx\]

by a sum of the form

\[\sum_{i=1}^n w_i f(x_i)\]

where \(n\) is the number of points (or nodes) at which the function is evaluated, \(x_i\) are the nodes, and \(w_i\) are the weights associated with each node. The choice of nodes and weights is what distinguishes one quadrature rule from another. In Gaussian quadrature, the nodes and weights are chosen to provide the exact result for polynomials of degree \(2n-1\) or less, which is a higher degree of accuracy compared to other quadrature methods for the same number of nodes.

Types of Gaussian Quadrature

There are several types of Gaussian quadrature, depending on the weight function and the interval of integration. The most common types include:

• Legendre Gaussian Quadrature: Used for integrating functions over the interval \([-1, 1]\) with a weight function of 1. The nodes are the roots of Legendre polynomials.
• Laguerre Gaussian Quadrature: Suitable for functions on the interval \([0, \infty)\) with a weight function of \(e^{-x}\). The nodes are the roots of Laguerre polynomials.
• Hermite Gaussian Quadrature: Used for functions on the interval \((-\infty, \infty)\) with a weight function of \(e^{-x^2}\). The nodes are the roots of Hermite polynomials.
• Chebyshev Gaussian Quadrature: There are two kinds, one for the first kind \(T_n\) and one for the second kind \(U_n\), used for functions on the interval \([-1, 1]\) with weight functions of \((1-x^2)^{-1/2}\) and \((1-x^2)^{1/2}\) respectively.

Application

Gaussian quadrature is widely used in applications where high precision numerical integration is required. This includes, but is not limited to, areas such as:

• Solving differential equations numerically
• Calculating areas, volumes, and expectations in statistics and probability
• Quantum mechanics and other areas of physics
• Engineering design and analysis

Advantages and Limitations

The main advantage of Gaussian quadrature over other numerical integration methods is its accuracy for polynomial functions. However, its effectiveness diminishes for functions that are highly oscillatory or have discontinuities within the interval of integration. Additionally, determining the appropriate nodes and weights can be computationally intensive for non-standard weight functions or intervals.

See Also

• Numerical integration
• Polynomial interpolation
• Legendre polynomials
• Laguerre polynomials
• Hermite polynomials
• Chebyshev polynomials