mechanical quadrature

Seeks to obtain the best numerical estimate of an integral by picking optimal abscissas x_i at which to evaluate the function f(x). The fundamental theorem of Gaussian quadrature states that the optimal abscissas of the m-point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomial for the same interval and weighting function. Gaussian quadrature is optimal because it fits all polynomials up to degree 2m - 1 exactly. Slightly less optimal fits are obtained from Radau quadrature and Laguerre-Gauss quadrature.

Chebyshev-Gauss quadrature | Chebyshev quadrature | Chebyshev-Radau quadrature | fundamental theorem of Gaussian quadrature | Hermite-Gauss quadrature | Jacobi-Gauss quadrature | Laguerre-Gauss quadrature | Legendre-Gauss quadrature | Lobatto quadrature | Radau quadrature

Carl Friedrich Gauss