Friday, 27 May 2011

Tutorial: Midpoint rule

In a previous post I spoke about the trapezium rule for integrating functions numerically. The trapezium rule could find the value of an integral with second order accuracy. In this post I will present an alternative method, the midpoint rule. The geometric interpretation of the midpoint rule appears to be more crude than the trapezium rule. Nonetheless, it turns out that the midpoint rule also has second order accuracy.

Like the trapezium rule, we divide our interval [a;b] into N equal sections of width h = (ba) / N. The area of each section is now approximated by the area of the rectangle of width h and whose height is given by the function value at the centre of the section. The centre of the nth section is given by
xn + 1 / 2 = a + (n + 1 / 2)h.

Here n = 0,1,...,N − 1. As before, we define fn + 1 / 2 = f(xn + 1 / 2). The area of the nth rectangle is then given by hfn + 1 / 2. And the value of the integral is given by

\int_a^b f(x)\;dx = h( f_{1/2} + f_{3/2} + \ldots + f_{N-1/2}) = h\sum_{k=0}^{N-1}f_{k + 1/2}

This is the final form for the midpoint rule. As with the trapezium rule, the accuracy of the integral can be increased by increasing N, i.e. decreasing the step size h. Again we can plot the error of the numerical integral against the number of steps on a logarithmic scale.

In figure the error of the integral

\int_0^1\sin(\pi x) dx = \frac{2}{\pi}

when evaluated with the midpoint and the trapezium rule is plotted. Clearly, the midpoint rule has a second order error, as has the trapezium rule. At first sight this is somewhat surprising since the midpoint rule makes no attempt to incorporate the change in the function value over the sections into the approximation.

To understand the accuracy of the approximation, we have to re-interpret the midpoint formula hfn + 1 / 2 as the area of a trapezoid that intersects the function graph at the midpoint an which has a slope identical to the derivative of the function at that point. It is a simple, but very useful, geometric fact that the area of a trapezoid is the same as the area of a rectangle whose height is the average of the trapezoid's height.

Looking at the error curves a bit more closely, we can observe that the error of the midpoint rule is always slightly smaller than that of the trapezium rule. In fact when we divide the two error curves in the region where they decrease with N^2, we get an almost constant error ratio of about 1/2. This will lead us to improve the two methods. But this will be the subject of another post.

No comments:

Post a Comment