Gauss–Hermite quadrature

Who / What

Gauss–Hermite quadrature is a numerical integration technique, specifically a form of Gaussian quadrature. It's used to approximate the definite integral of functions involving the exponential function with a squared term. The method relies on selecting specific points (xi) and weights (wi) to improve the accuracy of the approximation.

Background & History

The Gauss–Hermite quadrature is rooted in mathematical analysis, developing as a way to efficiently compute integrals that are difficult to solve analytically. It builds upon the principles of Gaussian quadrature, leveraging the properties of Hermite polynomials for optimal point selection. While not attributable to a single founder or specific founding date, its development occurred within the broader advancements of numerical integration techniques in the 19th and 20th centuries.

Why Notable

Gauss–Hermite quadrature is significant because it provides a highly accurate method for approximating integrals of the form ∫−∞+∞e−x2f(x)dx, which arise frequently in physics, statistics, and engineering. Its efficiency stems from the careful selection of integration points and weights, leading to faster convergence compared to simpler quadrature rules. This makes it a valuable tool when analytical solutions are unavailable or computationally expensive.

In the News

While not currently "in the news" in the sense of recent events, Gauss–Hermite quadrature remains relevant in scientific computing and numerical analysis. It is a foundational method used in various software packages for scientific simulations, data analysis, and statistical modeling. Its continued use reflects its enduring accuracy and efficiency in handling complex integrals.

Key Facts

Type: Numerical Integration Technique

Also known as: Gaussian Quadrature (specific form)

Founded / Born: Development occurred throughout the 19th & 20th centuries; no specific founding date.

Key dates: Development of Hermite polynomials (1840s) influenced its creation.

Geography: Not applicable; a mathematical method.

Affiliation: Numerical Analysis, Mathematical Physics, Statistics.

Links

[Wikipedia](https://en.wikipedia.org/wiki/Gauss%E2%80%93Hermite_quadrature)

Sources

• 🔗 Wikipedia

In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind: ∫ − ∞ + ∞ e − x 2 f ( x ) d x . {\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx.} In this case ∫ − ∞ + ∞ e − x 2 f ( x ) d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})} where n is the number of sample points used. The xi are the roots of the physicists' version of the Hermite polynomial Hn(x) (i = 1,2,...,n), and the associated weights wi are given by w i = 2 n − 1 n !