What is Gauss-Legendre quadrature?

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What is Gauss-Legendre quadrature?​

Gaussian quadrature. The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as where g(x) is well-approximated by a low-degree polynomial, then alternative nodes and weights will usually give more accurate quadrature rules.

What is Gaussian quadrature rule?​

Gaussian quadrature. In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.
How do you integrate using Gauss quadrature?
Use Gauss quadrature and integrate f(x) = 1 0.3 − x4 between 1 and 0.8. Compare the value you obtain with the exact value. In applying Gauss quadrature the limits of integration have to be –1 and + 1. In this example the lower limit is a = 1 and the upper limit is b = 0.8.

What is the difference between trapezoidal rule and Gaussian rule?​

The trapezoidal rule returns the integral of the orange dashed line, equal to . The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to . Such a result is exact, since the green region has the same area as the sum of the red regions.

Why Gaussian quadratures are not suitable for engineering applications?​

When the function is known and smooth, the Gaussian quadratures usually have decisive advantages in efficiency. However, engineering data obtained from measurements are not always smooth or located right on the abscissas which are not uniformly spaced. Therefore, the Gaussian quadratures are not suitable for such cases.
Can the Gaussian integral be evaluated?
Gaussian integral. can be evaluated. The definite integral of an arbitrary Gaussian function is The Gaussian integral is encountered very often in physics and numerous generalizations of the integral are encountered in quantum field theory .

What is Gauss-Kronrod rule?​

Gauss–Kronrod rules. Gauss–Kronrod rules are extensions of Gauss quadrature rules generated by adding n + 1 points to an n -point rule in such a way that the resulting rule is of order 2n + 1. This allows for computing higher-order estimates while re-using the function values of a lower-order estimate.