What is the bisection method?

1. **Initial Interval**: Start with an interval \([a, b]\) where the function \(f(x)\) changes sign, meaning \(f(a) \cdot f(b) < 0\). This indicates there is at least one root in the interval, according to the Intermediate Value Theorem.

2. **Compute Midpoint**: Calculate the midpoint \(m\) of the interval: \(m = \frac{a + b}{2}\).

3. **Evaluate Function**: Evaluate the function at the midpoint, \(f(m)\).

4. **Update Interval**:

- If \(f(m) = 0\), then \(m\) is the root, and the process stops.

- If \(f(a) \cdot f(m) < 0\), the root lies in the interval \([a, m]\), so update \(b\) to \(m\).

- If \(f(m) \cdot f(b) < 0\), the root lies in the interval \([m, b]\), so update \(a\) to \(m\).

5. **Repeat**: Repeat steps 2-4 until the interval is sufficiently small or the function value at the midpoint is close enough to zero.

The bisection method is straightforward and guarantees convergence as long as the function is continuous and the initial interval is chosen correctly. However, it can be slow compared to other methods because it only reduces the interval size by half each iteration.