The **Bisection Method** is a very well known root-finding algorithm that always comes at the very beginning of every book on Numerical Analysis.

The algorithmn searches for a root in the interval, in whose endpoints the **continous **problem function, , takes opposite signs:

The **Bisection Method** is based on Bolzano´s Intermediate Value Theorem, and it can give, also, an alternative proof for it. (Reference [1])

Here, it does not matter the function values: The algorithm does not use any other information from , other than its sign changes.

There are excelent pages on the internet that treat this topic, (references [1], [2] and [3] for example), and it would be useless to add more redundant and, for sure, worse presented summary about this numerical method.

I just want to take out something hidden inside the computer software, a small piece of math inside the programming area.

**THE LOST FUNCTION**

The core of the **Bisection Method** is the conditional **IF… THEN…** sentence, where the algorithm chooses in which half of the interval is the desired root.

This conditional evaluation can be converted into a **decision function**, Let´s say , that gives if the root is on the left half interval and , otherwise.

Now the iterative sequence can be written using this function, , where:

is the midpoint of the search interval:

is the decision function, and the , in the second member, is the Kronecker´s delta:

Then:

The endpoints of the search interval must be:

Then, they can be expressed with the aid of , as:

*Note:*

If , has only a single root in the interval , then it is possible to use an alternate version of the decision function:

with the advantage that then it is not necessary to evaluate in the main loop of the algorithm: this implies an improvement in performance.

**LENGTHS RATIO:**

We can name the ratio of lengths between the intervals , and , as the parameter:

If is expressed in function of :

The sequence formed by the distinct values of the decision function, , matches up with the **binary expansion ** (or the binary digits) of

Then the midpoint and the endpoints of the interval in function of , are:

**HACKING BISECTION METHOD:**

The connection between and can be used to convert the Bisection Method into a method for finding the binary expansion for a root of any function that holds the conditions that this method demands.

For example if we take:

, where

with and , where , then:

and the sequence of gives the binary digits of

**Archives:**

[a]-010910-Binary Bisection.nb

**References:**

[1]-Mohamed A. Khamsi, Helmut Knaust – SOSMATH – The Bisection Method

[2]-John H. Mathews – California State Univ. Fullerton – Module for The Bisection Method

[3]-Holistic Numerical Methods Institute- – Bisection Method: Nonlinear Equations