Root Bracketing

Root bracketing is the process of enclosing a root of a real-valued continuous function in an interval. If neither endpoint is itself a root, a sign-changing bracket satisfies

The intermediate value theorem then guarantees that at least one root lies in .

A root-bracketing algorithm repeatedly replaces the current bracket by a smaller interval whose endpoint function values still have opposite sign. Bisection chooses the midpoint and discards half of the interval. The method of false position instead uses the x-intercept of a secant line. Brent's method and Ridders' method combine preservation of the bracket with faster interpolation steps (Press et al. 1992).

A function that only touches zero need not change sign, so methods based solely on sign-changing brackets can miss such a root.

Root bracketing uses an interval as an enclosure, but it is distinct from interval arithmetic, which propagates ranges through arithmetic operations. Interval-arithmetic methods can nevertheless provide rigorous bounds for root-finding computations (Jaulin et al. 2003).