Newton–Cotes formulas

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. 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 xi, for i = 0, ..., n, where x0 = a and xn = b. There are two types of Newton–Cotes formulas, 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 xi = h i + x0, with h (called the step size) equal to (xn − x0) / n = (b − a) / n. The wi 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 xi and not on the function f. Let L(x) be the interpolation polynomial in the Lagrange form for the given data points (x0, f(x0) ), …, (xn, f(xn) ), then The open Newton–Cotes formula of degree n is stated as