Bisection and root brackets

Evaluate the midpoint and retain the half-interval that still has a sign change. Repeated halving gives a predictable interval width. If the midpoint is an exact zero, the root has been found and further halving is unnecessary.

Continuity matters: a sign change across a pole need not indicate a root. An even-multiplicity root may have no sign change and escape this method. Report an interval or an approximation with an error bound rather than asserting an exact value from a rounded calculation.

Worked example

Bisect a root of x² − 2 in [1,2].

1. f(1) = −1 and f(2) = 2.
2. The midpoint is 1.5, where f = 0.25.
3. The sign change remains between 1 and 1.5.

Answer: New bracket [1,1.5]