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Simpson's rule

Simpson's rule approximates the integral by fitting a parabola through each pair of intervals, instead of a straight line. It integrates polynomials up to degree 3 exactly, and the error is $O(h^4)$ — halving $h$ roughly cuts the error to a sixteenth.

\int_{x_0}^{x_2} f\,dx \approx \frac{h}{3}\big(f_0 + 4f_1 + f_2\big)

Simpson's rule over one double interval

Symbols

$h$	step size between neighbouring points
$f_0, f_1, f_2$	function values at the endpoints and midpoint

Example

$\int_0^3 x^2\,dx$ with $h = 1.5$ :

$\frac{h}{3}(f_0 + 4f_1 + f_2) = 9$ — exact, since the degree is $2 \leq 3$ .

Simpson's rule needs an even number of intervals (an odd number of points).

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