# How can I use the 'False Position Method' without being given an interval?

So I'm being given a task by my lecturer to write a program that uses the False Position Method to find the approximation of a function's root within an interval $\space [a,b] \space$.

We are given the following equations to use:

The equation of the line passing through $\space f(a) \space$, the $\space x \space$ axis and $\space f(b) \space$:

$$y = {{x(f(b) - f(a)) - af(b) + bf(a)}\over {b-a}}$$

The value of $\space x \space$ when it crosses the $\space x \space$ axis (when $\space y=0 \space$):

$$x_{0} = b - {{b-a}\over f(b)-f(a)}f(b)$$

The only problem is, we are only given the value $\space x_0 \space$ to use as an estimate of the root.

We have to use that estimate to find some formula/formulae such that NO MATTER WHAT FUNCTION WE USE (as long as it has real roots), it will always give use appropriate values of $\space a \space$ and $\space b \space$.

How do I do this?

I hope I'm making sense.

• Try $x_0,x_0+1$ maybe? – copper.hat May 10 '17 at 20:34

You can try with the interval $(x-h,x+h)$, starting with $h=1$ and $h=h/2$ each iteration. If $r\in(x-h,x+h)$ is the root and $f^\prime(r)\neq 0$ then it should converge. This can fail if $r$ is a multiple root, so you can stop if $h\leq\epsilon$ (for some $\epsilon$) and can try some steps of $h=h+1$.
If $f$ is like a black box, you must take extra care of exceptions, like tangential roots, discontinuinities, and asymptotes.