Numerical integration
In numerical mathematics called numerical integration (traditional quadrature referred to as numeric ) the approximate calculation of integrals. Often you can not solve integrals closed because the integrand no master function can be omitted or only discrete values , such as measurements, is given. Then one tries to determine approximate values.
For this purpose, the integral of a function is represented over the interval as the sum of the value of a quadrature formula and the error:
A quadrature formula, in general consists of a weighted sum of function values
The points are called nodes and the numbers weights. There are different approaches, such as support points and weights can be chosen so that the quadrature error is as small as possible.
A quadrature formula has degree of precision (or accuracy degrees) when it integrates all polynomials up to the maximum degree exactly, and the greatest possible natural number with this property is.
Just as the integral quadrature formulas are linear operators.
Interpolatory quadrature formula
An important class of quadrature formulas is given by the idea to approximate the function by an interpolation polynomial of degree and then integrate it. The weights are then obtained as the integrals of the Lagrange polynomials to the given sampling points. By construction, these quadrature formulas have at least the degree of precision. The quadrature formula is therefore
With weights
And the Lagrangepolynomen
If the limits of integration are reference points, it is called closed quadrature formulas, otherwise open. If the sampling points chosen equidistant, then there are, among others, the Newton-Cotes formulas. Among the closed Newton-Cotes formulas include the trapezoidal rule and Simpson's rule to the open part of the tangent trapezoid rule. The Newton-Cotes formulas for straight even have the degree of precision. At the open quadrature formulas also include the Gaussian quadrature formulas.
Error estimate
With the minimum interval will be referred to, which contains the sampling points and the interval. Furthermore, (n 1) is twice continuously differentiable on. According to the Interpolationsgüte of the interpolation there is a so true that:
By integration we obtain the error formula for numerical quadrature
If this is not for everyone, the quadrature error is equal to 0, since that is the case for all polynomials up to the degree of accuracy of such quadrature formulas is at least.
From this formula error the error estimate follows
If the function in the interval it does not change sign, ie if no reference point is in the interval, one can use the mean value theorem of integral calculus to derive the following representation for the remainder term:
With an intermediate point
Similar formulas for the quadrature error is obtained even for special distributions of the nodes in the interval, such as for the Newton-Cotes formulas and Gaussian quadrature formulas.
If the function is only continuous, then the above statements do not apply, the error can be very large.
Other quadrature formulas
The attempt to minimize the error order of the quadrature formula, leads to the Gaussian quadrature. These use the theory of orthogonal polynomials to obtain formulas that have the degree of accuracy, the number of function evaluations used is.
To minimize the number of function evaluations, the failure to control, while possible, is often used, the Rombergsche extrapolation. Here are the integral values of ever-shrinking ' stripes' extrapolated to a vanishing stripe width down.
Summed quadrature formulas
In order to approximate the integral even better, one divides the interval into adjacent subintervals. The sub-intervals do not have to have the same length. In each subinterval is used in the following to the same approximation for the individual surfaces and then adds the resulting approximations. Of particular interest are adaptive formulas, carry no further sub-division of an interval when the estimated error in the interval is less than a limit.
Monte - Carlo integration
A method which does not attempt to use an approximate formula for the function to be integrated is the Monte - Carlo integration. Illustratively stated, in this case, the integral is determined that random points in the integration interval (horizontal) will be generated. Then, an approximation of the integral is determined as the average of the function values of these posts
The advantage is the relatively simple implementation, as well as the relatively simple extensibility to multiple integrals. The computational effort is slightly higher compared to Quadrationsformeln.