(9) OPTIMIZATION: Bisection Method for Root Finding

An easy and powerful method!

This will be the last method we will discuss for univariate function minimum. This is also the more straightforward method to understand, so enjoy the calm before the storm (we will proceed with multivariate calculus in the following articles).

What is the root finding?

Root finding is the process of finding where a given function is zero. So root finding is solving the problem f(x)=0. If there exists a value of x such that f(x)= 0, then we say that x is the root of the function.

The bisection method for root finding is also called the interval halving method, binary search method, or dichotomy method.

To be possible to apply the bisection method, our function interval must comply with the intermediate value theorem or Bolzano theorem:

If a function f(x) is continuous in the interval [a, b] and f(a) . f(b) < 0, then a value c exists in the interval [a, b] for which f(c) = 0.

→ If f(a) multiplied by f(b) is less than zero, it implies that f(a) and f(b) have opposite signs. This indicates that one is positive and the other is negative, which implies that the function f(x) intersects the x-axis at some point. The value of x at which this…