# 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

 $I=\frac{h}{3}(f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+\cdots+4f(x_{n-3})+2f(x_{% n-2})+4f(x_{n-1})+f(x_{n}))$
 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