Bisection Method Calculator

By the intermediate value theorem, a continuous f with f(a) and f(b) of opposite sign has AT LEAST one root in (a, b); bisection exploits exactly this guarantee and nothing stronger. A single sign change does not prove the root is unique — uniqueness requires further conditions on f, such as strict monotonicity on the interval.

Convergence is linear and unconditional: the bracket width halves every iteration regardless of f, gaining roughly one bit of accuracy per step — slower than Newton or secant near a simple root, but never divergent.

Same-sign endpoints do NOT prove there is no root: a bracket containing an even number of roots has f(a) and f(b) of the same sign and is rejected before iterating, because the sign test alone cannot see roots that cancel in pairs.

A sign change can also occur across a pole rather than a root — for example 1/(x − 1) changes sign at x = 1 without being zero there — so the residual f(x*) should always be checked alongside the bracket.

Each iteration stops on whichever of three conditions is met first: f(m) is exactly zero, the normalised residual test |f(m)|/fScale ≤ atol + rtol, or the bracket-width test in x.

The residual is normalised by the characteristic magnitude of f on the initial bracket, so multiplying the equation by a non-zero constant — which does not change the root problem — does not change the verdict, the iteration count or the error bound.

An endpoint is reported as an exact starting root only when f(a) or f(b) evaluates to exactly zero; an endpoint that merely satisfies the residual test is not treated as one, and the ordinary bracket iteration runs instead.

When the run stops at the iteration or evaluation limit, the estimate is the last midpoint, which is an endpoint of the updated final bracket. The valid bound is then the full bracket width, not half of it — the page reports the engine's bound rather than assuming the converged case.