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The Gauss-Legendre methods are implicit, so in general they cannot be applied exactly. Instead one makes an educated guess of , and then uses Newton's method to converge arbitrarily close to the true solution. Below is a Matlab function which implements the Gauss-Legendre method of order four.
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 ...
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.
The Gauss–Legendre method with s stages has order 2s, so its stability function is the Padé approximant with m = n = s. It follows that the method is A-stable. [34] This shows that A-stable Runge–Kutta can have arbitrarily high order. In contrast, the order of A-stable linear multistep methods cannot exceed two. [35]
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 ...
The Gauss–Legendre method of order six has Butcher tableau: ... Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary differential equations II: Stiff and ...
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 ...
This is the Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who described it in 1768. The Euler method is an example of an explicit method. This means that the new value y n+1 is defined in terms of things that are already known, like y n.