Describe the process of the bisection method for root finding.

The bisection method is a numerical algorithm used to find the root of a function.

To use the bisection method, we first need to have a function f(x) that we want to find the root of. We also need to have two initial guesses, a and b, such that f(a) and f(b) have opposite signs. This means that there is a root of the function between a and b.

The bisection method then works by repeatedly dividing the interval [a, b] in half and checking which half contains the root. We start by finding the midpoint c = (a + b)/2. We then check the sign of f(c). If f(c) has the same sign as f(a), then the root must be in the interval [c, b]. Otherwise, the root must be in the interval [a, c]. We then repeat this process with the new interval until we reach a desired level of accuracy.

The bisection method is guaranteed to converge to a root of the function as long as the function is continuous and changes sign on the interval [a, b]. However, the method can be slow to converge, especially if the function is very steep or has multiple roots.

The algorithm can be summarised as follows:

1. Choose initial guesses a and b such that f(a) and f(b) have opposite signs.
2. Set an error tolerance, e.g. 0.0001.
3. While the interval [a, b] is larger than the error tolerance:
a. Find the midpoint c = (a + b)/2.
b. If f(c) has the same sign as f(a), set a = c. Otherwise, set b = c.
4. Return the midpoint c as the approximate root of the function.