Newton–Cotes formulas

Description

It is assumed that the value of a function ƒ defined on [a, b] is known at equally spaced points x_i, for i = 0, …, n, where x₀ = a and x_n = b. There are two types of Newton–Cotes formulae, the "closed" type which uses the function value at all points, and the "open" type which does not use the function values at the endpoints. The closed Newton–Cotes formula of degree n is stated as

where x_i = h i + x₀, with h (called the step size) equal to (x_n − x₀) / n = (b − a) / n. The w_i are called weights.

As can be seen in the following derivation the weights are derived from the Lagrange basis polynomials. They depend only on the x_i and not on the function ƒ. Let L(x) be the interpolation polynomial in the Lagrange form for the given data points (x₀, ƒ(x₀) ), …, (x_n, ƒ(x_n) ), then

The open Newton–Cotes formula of degree n is stated as

The weights are found in a manner similar to the closed formula.

Instability for high degree

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. This allows building numerically stable formulas even for high degrees.

Closed Newton–Cotes formulae

This table lists some of the Newton–Cotes formulae of the closed type. The notation f i is a shorthand for f ( x i ) , with x_i = a + i (b − a) / n, and n the degree.

Boole's rule is sometimes mistakenly called Bode's rule, as a result of the propagation of a typographical error in Abramowitz and Stegun, an early reference book.

The exponent of the segment size b − a in the error term shows the rate at which the approximation error decreases. The derivative of ƒ in the error term shows which polynomials can be integrated exactly (i.e., with error equal to zero). Note that the derivative of ƒ in the error term increases by 2 for every other rule. The number ξ is between a and b.

Open Newton–Cotes formulae

This table lists some of the Newton–Cotes formulae of the open type. Again, ƒ_i is a shorthand for ƒ(x_i), with x_i = a + i (b − a) / n, and n the degree.

Composite rules

For the Newton–Cotes rules to be accurate, the step size h needs to be small, which means that the interval of integration [ a , b ] must be small itself, which is not true most of the time. For this reason, one usually performs numerical integration by splitting [ a , b ] into smaller subintervals, applying a Newton–Cotes rule on each subinterval, and adding up the results. This is called a composite rule. See Numerical integration.