Bisection Method

Numerical Analysis · Axiom Academy

A simple, reliable algorithm that uses the Intermediate Value Theorem to bracket and find roots. The Core Idea: Divide and Conquer The algorithm is beautifully simple: Start with an interval [a, b] where f(a) and f(b) have opposite signs Compute the midpoint c = (a + b) / 2 Replace a or b with c to maintain the sign change Repeat until the interval is small enough If f(a) and f(c) have opposite signs → root is in [a, c] If f(c) and f(b) have opposite signs → root is in [c, b] Watch It Work: Interactive Demo Click "Step" to see each iteration of bisection on f(x) = x³ - x - 2: Error Analysis: How Fast Does It Converge? Each iteration cuts the interval in half. This gives us linear convergence : After 10 iterations: error ≤ 1/2048 ≈ 0.00049 After 20 iterations: error ≤ 1/2097152 ≈ 0.00000048 After 30 iterations: error ≤ 1/2147483648 ≈ 4.7 × 10⁻¹⁰ Rule of thumb: About 3.3 iterations per decimal digit of accuracy. Guaranteed convergence - always works if sign change exists Simple to implement - no derivatives needed Predictable - know exactly how many iterations needed Robust - not sensitive to initial guess Slow - only linear convergence Needs bracket - must start with sign change One root only - finds at most one root per run Not for complex roots - only real roots When you need a guaranteed answer When the function is expensive to evaluate (you want few evaluations) As a starting point for faster methods When robustness matters more than speed

This is the written version of the interactive lesson above. See the full Numerical Analysis course.