1.1. What are Root Finding Problems?

Root-finding problems are problems where you are given a continuous function, \(f(x)\) and have to find values \(r\) where: \(f(r) = 0\)

Example 1

Formulate finding the cube root of \(2\) as a root-finding problem

Solution

1. Get the equation that expresses the relation:

   \(x^3 = 2\)

2. Factor the equation to make one side equal to \(0\):

   \(x^{3} - 2 = 0\)

3. Now that the equation is equal to \(0\) convert it to a function \(f(x)\):

   \(f(x) = x^{3} - 2\)

4. Find a value \(r\) such that \(f(r) = 0\)

Vóila, you now have a continuous function \(f(x)\) that express the relation you want and by finding \(r\) you solve your problem.

Numerical Methods are primarily concerned with step \(4\), hence the name root-finding problems.