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This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1]. The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities ...
It is assumed that the value of a function f defined on [,] is known at + equally spaced points: < < <.There are two classes of Newton–Cotes quadrature: they are called "closed" when = and =, i.e. they use the function values at the interval endpoints, and "open" when > and <, i.e. they do not use the function values at the endpoints.
Carl Friedrich Gauss was the first to derive the Gauss–Legendre quadrature rule, doing so by a calculation with continued fractions in 1814. [4] He calculated the nodes and weights to 16 digits up to order n=7 by hand. Carl Gustav Jacob Jacobi discovered the connection between the quadrature rule and the orthogonal family of Legendre polynomials.
A Gaussian quadrature rule is typically more accurate than a Newton–Cotes rule that uses the same number of function evaluations, if the integrand is smooth (i.e., if it is sufficiently differentiable). Other quadrature methods with varying intervals include Clenshaw–Curtis quadrature (also called Fejér quadrature) methods, which do nest.
More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s. [1] All Gauss–Legendre methods are A-stable. [2] The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is:
One derivation replaces the integrand () by the quadratic polynomial (i.e. parabola) () that takes the same values as () at the end points and and the midpoint +, where = /.
One approach consists of using point sets from other quadrature rules. For example, taking independent and identically distributed realisations from ν {\displaystyle \nu } recovers a Bayesian approach to Monte Carlo , [ 3 ] whereas using certain deterministic point sets such as low-discrepancy sequences or lattices recovers a Bayesian ...
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind: