Bisection Method Root Calculator
Locate a root of a cubic inside a bracketing interval with the bisection method — guaranteed convergence when the endpoints differ in sign.
How it works
Repeatedly halve [lo, hi], keeping the half whose endpoints straddle zero · requires f(lo)·f(hi) < 0
Bisection relies on the intermediate value theorem: if a continuous function changes sign across an interval, it must cross zero somewhere inside. Each step halves the interval and keeps the half that still brackets the root, so the uncertainty shrinks by a factor of two every iteration. It is slower than Newton's method but cannot diverge — as long as you start with a genuine sign change.
Worked example
To solve x² − 2 = 0 on [0, 2]: f(0) = −2 and f(2) = 2 bracket the root. After each halving the interval narrows — 1, 1.5, 1.25, 1.375, 1.4375 … — closing in on √2 ≈ 1.41421 after around 40 steps to full precision.
Sources: Bisection method — Wikipedia
Frequently asked questions
Why must f(lo) and f(hi) have opposite signs?
A sign change guarantees a continuous function crosses zero between the two points. Without it there may be no root in the interval (or an even number of them), and halving cannot reliably corner one — so this tool asks you to fix the bracket first.
How many iterations does bisection need?
The interval width halves each step, so to reach a tolerance ε from a width w you need about log₂(w/ε) iterations — roughly 3.3 steps per extra decimal digit. It is steady but slower than Newton's method.
What if there are several roots in the interval?
Bisection converges to one root, but which one is not guaranteed when the bracket contains several sign changes. Narrow the interval around a single crossing to target a specific root.
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This calculator is for educational purposes. Double-check important results before acting on them.