# Why Does Gaussian Integration Work?

• Faceless Master
In summary, Gaussian integration is a more accurate integration method compared to others because it uses a weighted sum of function values at specific points to approximate the integral. It works by breaking down the integral into smaller intervals and using a weighted sum of function values at specific points for each interval. The main difference between Gaussian integration and other methods is the use of weighted sums at specific points instead of equally spaced points. However, it is not suitable for all types of functions and there is a limit to the number of quadrature points that can be used, equal to the degree of the polynomial that can be exactly integrated using those points.
Faceless Master
Hi all,
why does Gaussian Integration in one dimension with n points integrate exactly with a polynomial of order 2n-1 ?

thanks

Because Gaussian integration, on n points, effectively approximates the integrand by a polynomial function passing through those n points. And that polynomial is a polynomial of degree n-1.

If the integrand itself is a polynomial of degree n-1, the "approximation" Gaussian integration uses would be the integrand and so would give an exact result.

## 1. Why is Gaussian integration more accurate than other integration methods?

Gaussian integration, also known as Gaussian quadrature, is more accurate because it uses a weighted sum of function values at specific points, also known as quadrature points, to approximate the integral. This method is designed to minimize the error by carefully choosing the quadrature points and their corresponding weights.

## 2. How does Gaussian integration work?

Gaussian integration works by breaking down the integral into smaller intervals and approximating each interval using a weighted sum of function values at specific points. The weights and points are chosen in such a way that the approximation is as accurate as possible. These smaller approximations are then summed to get the overall integral approximation.

## 3. What is the difference between Gaussian integration and other integration methods?

The main difference between Gaussian integration and other integration methods, such as the trapezoidal rule or Simpson's rule, is that Gaussian integration uses a weighted sum of function values at specific points, while the other methods use a weighted sum of function values at equally spaced points. This allows Gaussian integration to be more accurate for a wider range of functions.

## 4. Can Gaussian integration be used for all types of functions?

No, Gaussian integration is not suitable for all types of functions. It is designed to work best for functions that are relatively smooth and can be well-approximated by polynomials. If the function has sharp peaks or discontinuities, other methods may be more suitable.

## 5. Is there a limit to the number of quadrature points that can be used in Gaussian integration?

Yes, there is a limit to the number of quadrature points that can be used in Gaussian integration. The maximum number of points that can be used is equal to the degree of the polynomial that can be exactly integrated using those points. For example, with 3 quadrature points, a polynomial of degree 5 can be exactly integrated, but a polynomial of degree 6 cannot.

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