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w i are quadrature weights, and; x i are the roots of the nth Legendre polynomial. This choice of quadrature weights w i and quadrature nodes x i is the unique choice that allows the quadrature rule to integrate degree 2n − 1 polynomials exactly. Many algorithms have been developed for computing Gauss–Legendre quadrature rules.
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.
Moreover, this method is L-stable if and only if equals one of the roots of the polynomial +, i.e. if =. Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with x = 1 / 4 {\displaystyle x=1/4} .
The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π . However, it has some drawbacks (for example, it is computer memory -intensive) and therefore all record-breaking calculations for many years have used other ...
Special examples are the Gaussian quadrature for polynomials and the Discrete Fourier Transform for plane waves. It should be stressed that the grid points and weights, x i , w i {\displaystyle x_{i},w_{i}} are a function of the basis and the number N {\displaystyle N} .