Bisection Method

Updated 21 Sep 2026

•
What it is
- •
  a Bracketing Methods algortihm that repeatedly chooses the interval midpoint
•
Formula
- •
  $c=(a+b)/2$
•
Procedure
- •
  evaluate $f(c)$
- •
  stop if it is a root
- •
  retain the half-interval with opposite endpoint signs
•
Error bound
- •
  after $n$ interval halvings, bracket width is $(b_0-a_0)/2^n$
- •
  the midpoint of that resulting bracket is at most $(b_0-a_0)/2^{n+1}$ from an enclosed root
•
Example
- •
  $f(x)=x^2-2$ , initial bracket $1,2$
- •
  $c_1=1.5$ , $f(c_1)=0.25$ → retain $1,1.5$
- •
  $c_2=1.25$ , $f(c_2)=-0.4375$ → retain $1.25,1.5$
- •
  $c_3=1.375$ , $f(c_3)=-0.109375$ → retain $1.375,1.5$
•
Tradeoff
- •
  predictable bracket reduction, but it does not use function magnitudes to choose a potentially better candidate
•

Linked references 3