Root Finding

Bisection Method

Taking advantage of IVT to be confident that there is a $0$ within two intervals. However, this method can be very slow.

Example: Find the root of $f(x)=x-e^{-x}$.

1. Choose the interval $(0,1)$, which works since $f(0)=-1<0$ and $f(1)>0$.
   1. We check the midpoint $x=0.5$ and find that $f(0.5)\approx -11$. Since $f(1)>0$, so we know the root lies between $(0.5,1)$.
2. Repeat this process as many times as needed, setting the new bounds.

Newton’s Method

Super powerful method presented by Isaac Newton.

We use the iterative formula, given by: $x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$

Some Exceptions to Newton's Method

Doesn’t work when:

• $f^{\prime }(x)$ isn’t continuous at the root
• $f^{\prime }(x)=0$ at the root
• $f^{\prime \prime }(x)$ is infinte at the root