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The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the third-degree polynomial y(x) = 7x 3 – 8x 2 – 3x + 3, the 2-point Gaussian quadrature rule even returns an exact result.
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature , in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature ; [ 1 ] others take "quadrature" to include higher-dimensional integration.
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 ...
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 ...
For example, implicit linear multistep methods include Adams-Moulton methods, and backward differentiation methods (BDF), whereas implicit Runge–Kutta methods [6] include diagonally implicit Runge–Kutta (DIRK), [7] [8] singly diagonally implicit Runge–Kutta (SDIRK), [9] and Gauss–Radau [10] (based on Gaussian quadrature [11]) numerical ...
In numerical analysis, Gauss–Hermite quadrature is a form of ... "A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related ...
Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞] Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature; Gauss–Kronrod rules; Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points