Bisection Method Algorithm

The bisection method finds a root of a continuous function within a given interval by repeatedly halving the interval, checking the sign of the function at the midpoint against the signs at the interval's endpoints, and keeping whichever half still contains a sign change, guaranteeing a root lies within the shrinking interval as long as the function changes sign across the original endpoints. It converges linearly, slower than methods such as Newton-Raphson, but it is simple to implement and guaranteed to converge whenever the initial interval is valid, unlike faster methods that can diverge from a poor starting point. It is one of the oldest root-finding methods in numerical analysis, related to the ancient method of exhaustion.