Simpson’s rule

Simpson’s rule is a method of (approximate) numerical definite integration (or quadrature). Simpson’s rule is based on a parabolic model of the function to be integrated (in contrast to the trapezoidal model of the trapezoidal rule). Thus, a minimum of three points and three function values are required. Here we take three equidistant points: $x_0{x}_2$ the interval endpoints, $x_1=(x_0+x_2)/2$ the midpoint, and let $h=|b-a|/2$ the distance between each. The definite integral is then approximated by:

$${\int }_{x_0}^{x_2}f(x)𝑑x\approx I=\frac{h}{3}(f(x_0)+4f(x_1)+f(x_2))$$

We can extend this to greater precision by breaking our target domain into $n$ equal-length fragments. The quadrature is then the weighted sum of the above formula for every pair of adjacent regions, which works out for even $n$ to

$$I=\frac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\mathrm{\cdots }+4f(x_{n-3})+2f(x_{n-2})+4f(x_{n-1})+f(x_n))$$

Title	Simpson’s rule
Canonical name	SimpsonsRule
Date of creation	2013-03-22 13:40:12
Last modified on	2013-03-22 13:40:12
Owner	drini (3)
Last modified by	drini (3)
Numerical id	9
Author	drini (3)
Entry type	Theorem
Classification	msc 28-00
Classification	msc 26A06
Classification	msc 41A55
Classification	msc 65D32
Related topic	LagrangeInterpolationFormula
Related topic	NewtonAndCotesFormulas
Related topic	Prismatoid