lpetrich
Contributor
Back to Gaussian integration. In general,
integral for x from xa to xb of wi(x)*f(x) = sum over i of wp(i) * f(x(i))
where one uses for the x(i) the roots of the polynomial for the integral weighting wi(x) and limits xa, ab, and finds the point weighting wp(i) from each x(i).
The most familiar kind is Gauss–Legendre quadrature with constant integral weighting.
There exist other types, like Chebyshev–Gauss quadrature and Gauss–Jacobi quadrature and Gauss–Laguerre quadrature and Gauss–Hermite quadrature with integral weighting appropriate for each one.
There are also variations of Gauss-Legendre integration with constant integral weighting and one or both limits of integration as points. For one limit, it is Gauss-Radau, and for both limits, it is Gauss-Lobatto. The points are at roots of polynomials related to Legendre polynomials.
integral for x from xa to xb of wi(x)*f(x) = sum over i of wp(i) * f(x(i))
where one uses for the x(i) the roots of the polynomial for the integral weighting wi(x) and limits xa, ab, and finds the point weighting wp(i) from each x(i).
The most familiar kind is Gauss–Legendre quadrature with constant integral weighting.
There exist other types, like Chebyshev–Gauss quadrature and Gauss–Jacobi quadrature and Gauss–Laguerre quadrature and Gauss–Hermite quadrature with integral weighting appropriate for each one.
There are also variations of Gauss-Legendre integration with constant integral weighting and one or both limits of integration as points. For one limit, it is Gauss-Radau, and for both limits, it is Gauss-Lobatto. The points are at roots of polynomials related to Legendre polynomials.