3. Regula Falsi

By now, you should be convinced that the Bisection Method actually works, (provided you start with an appropriate interval). You might, however, still have an uneasy feeling about our ``approximation'' $c$ of the root. We haven't really used any information about our function f in order the determine this ``approximation'', so we shouldn't be too surprised when it turns out to be a very bad one. Can't we do better? After all, we evaluate our function f in the two endpoints of our interval, but all we ever use of this information is the sign of the function in those points, so we are throwing away information. We are now going to adapt the Bisection Method in such a way that this extra information can be put to good use.

Imagine that we try to find the root of a first-degree function by using Bisection, that is, we try to solve $f(x)=0$, with $f(x) = ax + b$, by using the Bisection Method. (If you don't have such a good imagination, you can scroll a little up and see it in reality.) So we get narrower and narrower intervals, leading to better and better approximations... But this doesn't make sense, because we could find the root immediately as soon as we have the function values in the two endpoints of our starting interval.

Of course, in the general case our function wouldn't be a straight line. But when you look on a sufficient small interval, most functions that are of practical importance behave more or less like a line. So what's stopping us from drawing a line through the endpoints of our interval, finding the place were this line intersects the x-axis and using this point as our approximation $c$ of the root of $f$. Apparently, nothing. This modified version of the Bisection Method is often called Regula Falsi, which means "Wrong Line". We hope, of course, that the line is not too wrong.

It's time for some action again! Now see how Mr Jansen solves the same problem...

When you have tried Regula Falsi a few times on different functions, you will have noticed that, although the approximations of the root were quite good, they were almost always at the same side of the real root, so one point of the interval stayed the same all the time, and the interval as a whole shrunk only very slowly. This is a very common fenomenon with Regula Falsi, and one of the reasons that it is almost never applied in the way you just saw. I will explain the cause of this phenomenon in the next paragraph, but you can try to find it yourself first. A function that shows the phenomenon very good is $e^x-{1\over 2}$. Try it!OK, here's the solution. When the interval gets smaller and smaller, you finally arrive at the point where the second derivative $f''$ of $f$ has the same sign all over the interval. In that case, the function $f$ is either convex ($f''(x) < 0$) or concave ($f''(x) > 0$) on the interval. So any line you draw between two point on the graph of f will always lay entirely at one side of the graph, and therefore, all our approximations also lay at one side of the graph. The reason that $e^x-{1\over 2}$ shows this phenomenon so well is because this function is concave on the whole real axis; therefore, the approximations of Regula Falsi will always lay at the left of the real root.

Because of this phenomenon, Regula Falsi is almost never used "as is". However, it is quite often combined with other methods. It could, for example, be combined with the Bisection Method. Another popular combination is Regula Falsi together with the Secant Method. The Secant Method is at the time not discussed in this text; it is a modified version of Newton's Method, which comes next.