Gaussian quadrature
The Gaussian quadrature ( by Carl Friedrich Gauss ) is a method of numerical integration, at the given degree of freedom provides an optimum approximation of the integral. In this method, the function to be integrated is divided into, and a weight function, and is approximated by a particular polynomial with specially selected analysis points. This polynomial can be integrated exactly. The process is therefore of the form
For the weight function is that it is equal to zero greater, it has finitely many zeros and is integrable. is a continuous function. The integration range [a, b] is not restricted to finite intervals. Furthermore, as nodes or abscissas and the sizes referred to as weights.
The method was published in 1814 by Gauss., In its present form with orthogonal polynomials of Carl Gustav Jacobi in 1826.
Properties
To achieve optimum accuracy, the abscissa values have a Gaussian quadrature formula of degree n exactly the zeros of the nth orthogonal polynomial of degree n correspond. The polynomials, ..., have to be orthogonal with respect to the scalar product with weighted,
For the weights apply:
The Gaussian quadrature is exactly the same for polynomial function whose degree is maximum, the value of the integral. It can be shown that no quadrature formula exists that exactly integrates all the polynomials of degree. In this respect, the order of the quadrature method is optimal.
If the function is no longer a polynomial, however, sufficiently smooth, ie, It can be shown with n nodes for error of Gaußquadratur:
With
Application
The Gaussian quadrature is applied to the numerical integration. In this case, for a given weight function and a given degree n, which determines the accuracy of the numerical integration, calculated once the bases and weight values and tabulated. This can be followed for any numerical integration by simply summing of weighted function values .
This method is therefore potentially beneficial
For some special weight functions the values of the support points and weights are done tabulated.
Gauss - Legendre integration
Here is the best known form of the Gaussian integration on the interval, it is often referred to simply as Gaussian integration. It is true. The resulting orthogonal polynomials are the Legendre polynomials of the first kind, the extension to arbitrary intervals is done by a transformation of variables.
The interpolation points (also called Gauss points ) and weights of the Gauss - Legendre integration are:
Gauss - Chebyshev integration
A variant of the Gaussian integration on the interval with the weight function. The corresponding orthogonal polynomials are the Chebyshev polynomials, be the zeros and thus the points of the quadrature formula directly in analytical form:
While the weights depend only on the number of vertices
The extension to arbitrary intervals is carried out by a variable-transformation ( see below). If the integrand in the form, it can be converted into. For the numerical calculation of the integral is then approximated by the sum. By substituting the bases in analytical form is obtained
Which corresponds to n times the application of the control center over the interval 0 to pi. The error can be estimated for an appropriate value of t between 0 and about Pi
Gauss - Hermite integration
Gaussian integration on the interval. It is true. The resulting orthogonal polynomials are the Hermite polynomials. If the integrand in the form, it can be converted into. For the numerical calculation of the integral is then approximated by the sum.
Points and weights of the Gauss - Hermite integration:
Gauss - Laguerre integration
Gaussian integration on the interval. It is true. The resulting orthogonal polynomials are the Laguerre polynomials. If the integrand in the form, it can be converted into. For the numerical calculation of the integral is then approximated by the sum.
Points and weights of the Gauss - Laguerre integration:
Transformation of variables at the Gauss quadrature
An integral of [ A, B ] is fed back to an integral over [-1, 1], and before applying the method of the Gaussian quadrature. This transition can be done by using and and and application of integration by substitution as follows:
After applying the Gaussian quadrature approximation applies
Adaptive Gaussian Process
Since the error in Gaussian quadrature, as mentioned above, depends on the number of selected nodes and with a larger number of support points just the denominator can significantly increase, this suggests, to obtain better approximations larger. The idea is to an existing approximation, a better approximation, for example, to calculate, in order to consider the difference between the two approximations. If the estimated error exceeds a certain absolute default, the interval is split, so on and the quadrature can be done. However, the evaluation of a Gaussian quadrature is quite costly as to be calculated especially for generally new calibration points, so that lends itself to the Gaussian quadrature with Legendre polynomials, the adaptive Gauss - Kronrod quadrature.
Adaptive Gauss - Kronrod quadrature
The presented Kronrod modification, which exists only for the Gauss - Legendre quadrature, based on the use of the already selected nodes and the addition of new grid points. While the existence of optimal extensions for the Gauss formulas of Szego was occupied, headed Kronrod (1965 ) for the Gauss - Legendre formulas optimal points here, which ensure the degree of precision. When using the advanced node number is defined as calculated by approximation, is the error estimate:
This can then be compared to a to give the algorithm a termination criterion. The Kronrod nodes and weights of the Gauss -Legendre nodes and weights are recorded for in the following table. Gaussian nodes were marked with a (G).