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.
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.
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.
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.
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.