## Poor Mr Boole

Numerical integration is awfully useful. Most of us will remember the trapezoid rule that we learn at school, where the function we are integrating is substituted by a straight line and the integral as estimated as the are under the curve:

$I_2~=~\frac{1}{2}h\left(f_1-f_0\right)$

where h is the interval that we split the curve into and $f_1$ and $f_0$ are the values of the function on either side of the function.

One way to obtain better methods is to interpolate higher order polynomials, and the integrate those. It is possible to, for example, use three points and interpolate with a parabola, four points and interpolate with a cubic or five points and interpolate with a quartic. This latter method is often called Bode’s rule. But here comes the shocking news: it wasn’t Bode who came up with, but the English mathematician Boole. It became known as Bode’s rule because of a misprint in a maths textbook from the 70s. I guess the consolation prize is that Boole got a whole branch of logic named after him.

So does Boole’s rule make a better approximation than the just applying the trapezoidal rule? In this picture I will show sin(x) with x between 0 and $\pi/2$ fitted with 5 straight lines (green curve) and with the polynomial used in Boole’s rule (red curve):

The red (corresponding to Boole’s rule) is pretty much right on the sin curve. If we actually take the difference between the fits and the sine curve we obtain the error:

where blue is the error using trapezoidal rule and green sticking with Boole.

The moral of the story is that you can get the name wrong, so long as you get the Maths rights!