Simpson's Rule

Simpson’s rule aims to improve the approximation of the function in each interval by using higher order polynomials.

Simpson’s 1/3 rule connects points using a quadratic function. First, a 2nd order Lagrange polynomial interpolation is done. After some algebraic manipulation (see textbook), the area for each interval is:

$$\begin{array}{rl}I& \approx \frac{h}{3}[f(x_0)+4f(x_1)+f(x_2)]\\ & \approx \underset{\text{Width}}{\underbrace{(b-a)}}\underset{\text{Average height}}{\underbrace{\frac{f(x_0)+4f(x_1)+f(x_2)}{6}}}\end{array}$$

(Recall that $h=\frac{b-a}{2}$).

Composite Simpson’s 1/3 Rule

For multiple intervals, Simpson’s rule requires an even number of intervals or segments:

$$I\approx (b-a)\frac{f(x_0)+4{\sum }_{i=1,3,5}^{n-1}f(x_i)+2{\sum }_{j=2,4,6}^{n-2}f(x_j)+f(x_n)}{3n}$$

In a more readable form:

$$I=(b-a)\frac{f(x_0)+4\sum f(x_{odd})+2\sum f(x_{even})+f(x_n)}{3n}$$

Error/Accuracy

The error is:

$$E_a=-\frac{(b-a{)}^5}{180n^4}{\bar{f}}^4$$

where ${\bar{f}}^4$ is the average 4th derivative. Note that this is a function of $1/n^4$, so doubling the number of intervals $n$ would reduce the error by a factor of $16$ or $n^4$.