Gaussian Quadrature Explained: Example Included

In summary, Gaussian quadrature is a method of solving integrals using a weighted sum of function values at specific points. These points are the roots of certain polynomials, such as Hermite or Legendre polynomials, and the weights are determined by a weighting function. This method can be used to approximate integrals of various functions. More information and examples can be found on the Numerical Recipies website.
  • #1
skiboka33
59
0
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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  • #2
i think it just means solving using integration.
 
  • #3
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 explination.
 
  • #4
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
 

1. What is Gaussian quadrature?

Gaussian quadrature is a numerical method for approximating the definite integral of a function. It involves evaluating the function at specific points called "nodes" and using corresponding weights to calculate the integral.

2. How does Gaussian quadrature work?

Gaussian quadrature works by finding the optimal nodes and weights that will give the most accurate approximation of the integral for a given function. This is achieved by using a mathematical algorithm to determine the nodes and weights based on the properties of the function.

3. What is an example of Gaussian quadrature?

An example of Gaussian quadrature would be using it to approximate the integral of a polynomial function, such as f(x) = x^2 + 3x + 1, over a specific interval.

4. What are the advantages of using Gaussian quadrature?

Gaussian quadrature provides a more accurate approximation of integrals compared to other numerical methods such as the trapezoidal rule or Simpson's rule. It also requires fewer function evaluations, making it more efficient for complex integrals.

5. Are there any limitations to using Gaussian quadrature?

One limitation of Gaussian quadrature is that it is only applicable to functions that can be evaluated at specific points. It may also be less accurate for functions with highly oscillatory behavior or rapidly changing slopes.

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