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Concept
Newton–Cotes formulas
Summary
In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called quadrature) based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes.
Newton–Cotes formulas can be useful if the value of the integrand at equally spaced points is given. If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are probably more suitable.
It is assumed that the value of a function f defined on is known at equally spaced points: . There are two classes of Newton–Cotes quadrature: they are called "closed" when and , i.e. they use the function values at the interval endpoints, and "open" when and , i.e. they do not use the function values at the endpoints. Newton–Cotes formulas using points can be defined (for both classes) as
where
for a closed formula, , with ,
for an open formula, , with .
The number h is called step size, are called weights.
The weights can be computed as the integral of Lagrange basis polynomials. They depend only on and not on the function f.
Let be the interpolation polynomial in the Lagrange form for the given data points , then
A Newton–Cotes formula of any degree n can be constructed. However, for large n a Newton–Cotes rule can sometimes suffer from catastrophic Runge's phenomenon where the error grows exponentially for large n. Methods such as Gaussian quadrature and Clenshaw–Curtis quadrature with unequally spaced points (clustered at the endpoints of the integration interval) are stable and much more accurate, and are normally preferred to Newton–Cotes. If these methods cannot be used, because the integrand is only given at the fixed equidistributed grid, then Runge's phenomenon can be avoided by using a composite rule, as explained below.
Alternatively, stable Newton–Cotes formulas can be constructed using least-squares approximation instead of interpolation.
Official source
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