Bisection Method Calculator
Find a root of f(x)=0 by bisection on a bracketing interval with convergence reporting.
Inputs
Function with a sign change on [a, b].
Root
1.521379706799
Iterations
33
|f(root)|
3.2780 × 10⁻¹¹
Status
Converged
Step by step
Values used
f(x) = x^3 - x - 2; Left bound (a) = 1; Right bound (b) = 2; Tolerance = 0.0000; Max iterations = 100
Bisection step
mid = (a + b) / 2; if f(a)·f(mid) < 0 then b = mid, else a = mid
Error bound
error ≤ (b - a) / 2^n after n iterations
Root
= 1.521379706799
Iterations
= 33
|f(root)|
= 0.0000
Status
= Converged
How it works
The bisection method finds a root by repeatedly halving a bracketing interval [a, b] where f(a) and f(b) have opposite signs. By the Intermediate Value Theorem, a root must exist in the interval. Each iteration halves the interval, guaranteeing convergence at a linear rate of about 1 bit per iteration.
Formulas
Bisection step
mid = (a + b) / 2; if f(a)·f(mid) < 0 then b = mid, else a = mid
- a
- Left bound
- b
- Right bound
- mid
- Midpoint
Error bound
error ≤ (b - a) / 2^n after n iterations
- n
- Iteration count
Frequently Asked Questions
Why do f(a) and f(b) need opposite signs?
The IVT guarantees a root only when the function crosses zero between a and b. If both have the same sign, the root may not exist in that interval.
How does bisection compare to Newton's method?
Bisection always converges but is slow (linear). Newton's method converges quadratically but can fail if the derivative is zero or the guess is bad.