Root Finding

Root finding refers to the general problem of searching for a solution of an equation $F(x)=0$ for some function $F(x)$. This is a very general problem and it comes up a lot in mathematics! For example, if we want to optimize a function $f(x)$ then we need to find critical points and therefore solve the equation $f'(x)=0$.

There are few examples where there exist exact methods for finding solutions. For example, the quadratic formula

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

gives us an exact method for finding roots of the equation

$$ ax^2 + bx + c = 0 $$

There is a general formula to solve a cubic equation and even a quartic (degree 4) equation (but the formula is too complicated to be useful).

But there does not exist a formula for a quintic (degree 5) polynomial. And there are many more examples of equations with no known method to solve them exactly.

What can we do? Use numerical methods to find approximate solutions.