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Gauss–Legendre quadrature is optimal in a very narrow sense for computing integrals of a function f over [−1, 1], since no other quadrature rule integrates all degree 2n − 1 polynomials exactly when using n sample points. However, this measure of accuracy is not generally a very useful one---polynomials are very simple to integrate and ...
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 ...
Gauss–Legendre methods are implicit Runge–Kutta methods. 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.
The Gauss–Legendre methods use the points of Gauss–Legendre quadrature as collocation points. The Gauss–Legendre method based on s points has order 2s. [2] All Gauss–Legendre methods are A-stable. [3] In fact, one can show that the order of a collocation method corresponds to the order of the quadrature rule that one would get using the ...
This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and group theory, and acquire profound physical and geometrical meaning.
These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four ... For example, Lobatto IIID family introduced in ...
Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with α = β = 0. Similarly, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes α = β = −0.5 (+0.5).
An enhancement to the Chebyshev pseudospectral method that uses a Clenshaw–Curtis quadrature was developed. [18] The LPM uses Lagrange polynomials for the approximations, and Legendre–Gauss–Lobatto (LGL) points for the orthogonal collocation. A costate estimation procedure for the Legendre pseudospectral method was also developed. [19]