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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.
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 ...
If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas. 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 ...
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 ).
1 Example with change of variable. 2 References. ... Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the ...
In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: + (). In this case
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind:
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 ...