What is quadrature points?
What is quadrature points?
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. ( See numerical integration for more on quadrature rules.)
What is quadrature in numerical integration?
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; others take quadrature to include higher-dimensional integration.
How does quadrature integration work?
For a function of one independent variable, the basic idea of a quadrature rule is to replace the definite integral by a sum of the integrand evaluated at certain points (called quadrature points ) multiplied by a number (called quadrature weights ).
What is a quadrature in physics?
Quadrature (astronomy), the position of a body (moon or planet) such that its elongation is 90° or 270°; i.e. the body-earth-sun angle is 90°
What is quadrature in quantum optics?
Operators given by. and. are called the quadratures and they represent the real and imaginary parts of the complex amplitude represented by . The commutation relation between the two quadratures can easily be calculated: This looks very similar to the commutation relation of the position and momentum operator.
What is the quadrature rule for integrable functions?
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. Instead, if the integrand can be written as
What is adaptive quadrature?
Adaptive quadrature involves careful selection of the points where f(x) is sam- pled. We want to evaluate the function at as few points as possible while approx- imating the integral to within some specified accuracy. A fundamental additive property of a definite integral is the basis for adaptive quadrature.
How do you find the integral of a Gaussian quadrature?
The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 ) = 2 / 3 {displaystyle y(-{sqrt {scriptstyle 1/3}})+y({sqrt {scriptstyle 1/3}})=2/3} . Such a result is exact, since the green region has the same area as the sum of the red regions.
What is the Gauss-Legendre quadrature rule?
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.
