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Bisection method

The bisection method finds a root of f(x)=0f(x) = 0 by starting with an interval [a,b][a, b] where ff changes sign, and repeatedly halving it. The method is robust and guaranteed to converge, but slow: each step only halves the uncertainty.

m=a+b2m = \frac{a+b}{2}the midpoint of the interval
f(a) f(b)<0f(a)\,f(b) < 0the requirement to start: a sign change in [a,b][a,b]

Symbols

a, ba,\ bthe endpoints of the interval
mmthe midpoint, new endpoint

Example

f(x)=x2−7f(x) = x^2 - 7 on [1,7][1, 7]: m=4m = 4, f(4)=9>0f(4) = 9 > 0.

The sign change is now in [1,4][1, 4], which becomes the new interval.

The number of steps to reach tolerance ε\varepsilon is n≥log⁡2 ⁣((b−a)/ε)n \geq \log_2\!\big((b-a)/\varepsilon\big).
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Part of Numerical Methods: Solving equations.