NewtonCotesi

Newton-Cotes formulas, sometimes written as NewtonCotes, are a family of numerical integration rules used to approximate definite integrals. They are named after Isaac Newton and Roger Cotes, who developed and published the method in the early 18th century. The idea is to approximate the integrand by an interpolating polynomial that matches the function values at a set of equally spaced points in the interval [a, b], and then integrate that polynomial exactly over the same interval.

The formulas come in two main flavors: closed and open. Closed Newton-Cotes rules include the endpoints of the interval, while open rules do not. Examples of closed rules include the trapezoidal rule (n = 1, using two endpoints), Simpson’s rule (n = 2, using three points), Simpson’s 3/8 rule (n = 3), and Boole’s rule (n = 4). Open rules, which use interior points only, are less common but useful in certain contexts.

For longer intervals or higher accuracy, the interval [a, b] is often subdivided into subintervals of equal width, and the chosen Newton-Cotes rule is applied on each subinterval (composite Newton-Cotes). The results are then summed to yield the final approximation. The quality of the approximation improves with higher-degree rules and with smaller subinterval widths, but high-degree rules can be numerically unstable and ill-conditioned for some functions.

In practice, Newton-Cotes methods are most effective for smooth functions and moderate degrees. They are one of several quadrature families used in numerical analysis, alongside methods such as Gauss-type quadrature, which may offer higher efficiency for certain problems.