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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.
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 ...
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.
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind: ∫ − 1 + 1 f ( x ) 1 − x 2 d x {\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx}
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.
These are named after Rehuel Lobatto [7] as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. [8] All are implicit methods, have order 2s − 2 and they all have c 1 = 0 and c s = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding + points to an -point rule in such a way that the resulting rule is exact for polynomials of degree less than or equal to + (Laurie (1997, p. 1133); the corresponding Gauss rule is of order ).