Simpson's ruleSimpsonsRule2003-06-08 22:04:062006-06-15 15:20:09Theorem\usepackage{amssymb}
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%\usepackage{xypic}\emph{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: $x_0x_2$ 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
\[
I = \frac{h}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-3}) + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n))
\]