Gaussian Quadrature Explained: Example Included

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Discussion Overview

The discussion centers around the concept of Gaussian quadrature, particularly its definition, application in integration, and the use of various polynomials for approximation. Participants seek clarification on the topic and request examples to aid understanding.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests an explanation of Gaussian quadrature and mentions difficulty understanding existing resources.
  • Another participant suggests that Gaussian quadrature relates to solving integrals.
  • A different participant notes that Gaussian quadrature utilizes Legendre polynomials for function approximation but expresses uncertainty about the practical application.
  • Another contribution mentions that Gaussian integration can involve Hermite, Legendre, and Laguerre polynomials, and describes the general form of the integral involving a weighting function.
  • A participant recommends the Numerical Recipes website for further explanation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and uncertainty regarding the specifics of Gaussian quadrature, with no consensus on a clear explanation or method of application.

Contextual Notes

Some participants reference different types of polynomials and weighting functions without fully resolving how they interrelate or the implications for specific applications of Gaussian quadrature.

skiboka33
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Anyone care to explain the concept of gaussian quatrature? I've tried some websites but they're a little over my head. An example would be appreciated, thanks!
 
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i think it just means solving using integration.
 
Well I know its a system of Legendre polynomials used to approximate functions, but I'm not sure of how to actually do it, if anyone knows I'd appreciate an explanation.
 
Gaussian integration involves Hermite polynomials, Legendre polynomials, Laugarre polynomials amongst others. In general, the integral of a function [tex]\int f(x) g dx[/tex] where [tex]g[/tex] is a weighting function, which can be [tex]e^{-x}, e^{-x^2/2}[/tex] etc... can be written as [tex]\sum_{i} f(x_i) w_i[/tex]. The [tex]x_i[/tex] are the roots of the polynomials you wsh to use to fit and [tex]w[/tex] are weighting functions at those roots.


Look at the Numerical Recipies website and they explain it very well.

www.nr.com
 

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