Newton and Cotes formulas

The usual way of numerically integrate a function, is to find a simpler function which approximates the given function and then integrating the interpolationMathworldPlanetmath function. That is, if we want to find abf(x)𝑑x, we find an approximating function p(x) such that f(x) and p(x) be close (on some concept of distance) and then we say


The simplest approximation functions are polynomials. If we evaluate f(x) at some points x0,x1,,xn, we can use Lagrange’s interpolating polynomial to find a polynomial p(x) with degree n such that p(xj)=f(xj) for j=0,1,,n.

Newton and Cotes’ integration formulas are obtained when the x0,x1,,xn are sampled evenly over the interval, and then Lagrange interpolating polynomials are used to approximate the function.

The Newton and Cotes formulas for small values of n are given on the following table.

n p(x) Name
1 h2(f(x0)+f(x1)) Trapezoidal ruleMathworldPlanetmath
2 h3(f(x0)+4f(x1)+f(x2)) Simpson’s rule
3 3h8(f(x0)+3f(x1)+3f(x3)+f(x3)) Simpson’s 3/8 rule
4 2h45(7f(x0)+32f(x1)+12f(x2)+32f(x3)+7f(x4)) Milne’s rule

recalling that x0,x1,,xn are evenly spaced on [a,b].

Since the Simpson’s rule is actually the Newton and Cotes formula for n=2, the proof of Simpson’s rule illustrates this method.

Title Newton and Cotes formulas
Canonical name NewtonAndCotesFormulas
Date of creation 2013-03-22 14:50:28
Last modified on 2013-03-22 14:50:28
Owner drini (3)
Last modified by drini (3)
Numerical id 7
Author drini (3)
Entry type Definition
Classification msc 65D32
Synonym Newton-Cotes
Related topic SimpsonsRule
Related topic CodeForSimpsonsRule