Bisection Method

How the bisection method finds roots

Take a continuous function f on some interval [a, b] where f(a) · f(b) < 0. That sign change is the key: by the Intermediate Value Theorem, a root has to sit somewhere inside. From there the method just keeps checking the midpoint c = (a + b) / 2. If f(c) · f(a) < 0, the root is in [a, c], so you set b = c; if not, you set a = c. Each pass throws away half the interval.

Convergence is linear, which means the error after n iterations stays below (b − a) / 2^n. Try x² − 2 = 0 on [1, 2] and the midpoints come out as 1.5, then 1.25 (rejected once the sign check moves the bracket), then 1.375, 1.4375, and so on. Around 20 iterations get you six correct decimal digits of √2 ≈ 1.41421356. Yes, it is slower than Newton's method with its quadratic convergence, but it never asks for f' and it is mathematically guaranteed to converge as long as the starting bracket is valid.

Where bisection shows up in practice

It earns its keep whenever the derivative is missing, unstable, or costly to evaluate. It also makes a dependable fallback inside hybrid root-finders, where Brent's method pairs bisection with secant and inverse-quadratic steps to stay both safe and fast. You will run into it in industrial process control and PID tuning, when solving for implied volatility in option pricing, in binary-search variants across algorithms, and in calibration curves used in instrumentation.

FAQ

What if f(a) · f(b) > 0? Then the method has nowhere to start, since the bracket shows no guaranteed sign change. Maybe there is no root, maybe there is an even number of them, or maybe you simply picked the wrong interval. Plotting f first is the easiest way to spot good brackets.

How many iterations do I need? Hitting a tolerance ε takes n ≥ log₂((b − a) / ε) steps. On a unit interval with ε = 10⁻⁶ that works out to 20 iterations, and the function itself does not change that number.

Why use bisection over Newton's method? Newton is faster (quadratic against linear), but it can also diverge, oscillate, or stall when it hits f'(x) = 0. Bisection gives up that speed for a guarantee that never breaks, which is exactly what you want when reliability counts more than how many iterations it takes.