Newton–Cotes formulas
A Newton-Cotes formula (after Isaac Newton and Roger Cotes ) is a numerical quadrature formula for the approximate calculation of integrals. This formula is based on the idea to interpolate the function to be integrated by a polynomial and integrate this exactly as approximation. The nodes of the interpolation are chosen in equidistant.
Derivation
For the interpolation polynomial of degree n to integrate the support points
Equidistant with a constant distance selected such that they are symmetrical to center interval of the integration interval. Thus applies.
With (and thus ) one obtains intervals of length and thus and. These formulas are called closed Newton-Cotes formulas.
With (and thus ) one obtains open quadrature formulas:
- If you choose (and thus ), we obtain intervals of length and thus and. These formulas are called open Newton-Cotes formulas.
- If you choose (and thus ), we obtain intervals of length and thus and. These formulas are called Maclaurin formulas.
For the numerical integration we take the interpolation polynomial of the function at the given nodes. For this
Where the Lagrange polynomials. It follows
Definition
Then follows for the Newton-Cotes formula
With weights
The weights are symmetrical, ie.
Due to the special choice of the sampling points, the quadrature formulas integrating exactly when n is odd polynomials up to degree n, with n even, even up to degree n 1. Thus quadrature formulas with n even (ie an odd number of nodes ) which preferable with n odd. This property is also called the degree of accuracy of the quadrature formula.
Especially true for that and thus
If, as is the case with weights with different signs, there is a danger that swaying the rounding errors or cancellation occurs. Therefore, to favor with positive weights for numerical reasons quadrature formulas. Since for large n, the interpolation is useless, quadrature formulas are also not recommended with large n. If you want to achieve better approximations, the use of composite quadrature formulas is recommended.
Is the error ( procedural error ), which is made in the application of the quadrature formula. This always has in the particular choice of reference points in the form of
Where K is a constant independent of, p is the degree of accuracy and a well-known only in exceptional cases intermediate value. If he generally known, and thus calculate the integral exactly, could contradict the fact that you can not calculate exactly the most integrals.
Using the procedural error we obtain the error estimate:
The exact error is always less than / equal to this error estimate as well as the examples below indicate.
Completed Newton-Cotes formulas
The support points are valid for the integration interval [0,1]: For a general interval [a, b] are the interpolation points
For n = 8 for i = 2,4,6 and. For n = 10
Example:
Approximation using Simpson's rule ( n = 2). It is true and
Procedural errors: With obtained with
Error estimate:
Exact error:
Open Newton-Cotes formulas
The support points are valid for the integration interval [0,1]: For a general interval [a, b] are the interpolation points
For n = 5. For n = 6
From these formulas, only the rectangular rule is recommended. The formula where n = 1 has a higher cost in the same order as the rectangular rule, the higher the formulas have negative weights.
For example.
Approximation of the formula when n = 2 applies and
Procedural errors: With obtained with
Error estimate:
Exact error:
Maclaurin quadrature formulas
The support points are valid for the integration interval [0,1]: For a general interval [a, b] are the interpolation points
For n = 6. For n = 8
For example.
Approximation of the formula when n = 2 applies and
Procedural errors: With obtained with
Error estimate:
Exact error:
Summed Newton-Cotes formulas
From Grade 8 occur in many Newton-Cotes formulas for negative weights, which involves the danger of extinction with it. In addition, you can generally expect no convergence, since the polynomial is ill-conditioned. For larger areas of integration therefore you divided it into individual sub-intervals, and applies to each subinterval a formula to low-order.