Overview
The Bisection Method keeps an interval whose endpoints have opposite signs and shrinks it around a root.
FoundationSource checked
Bisection Method
Finds a root by repeatedly halving an interval that contains it.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Beginner
- Family
- Root Finding
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 5
- Code example
- Not ready
- Visualization
- Ready
Core Concepts
numeric approximationroot estimateiterative updateroot findingconvergence checkerror control
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(1)
Average
O(log((b-a)/e))
Worst
O(log((b-a)/e))
Space
O(1)
When to Use
- Use it when a continuous function changes sign over a known interval.
- Use it when reliability matters more than fast local convergence.
- Use it to bracket roots before switching to a faster method.
When Not to Use
- Avoid it when no sign-changing bracket is available.
- Avoid it when the function is discontinuous over the interval.
- Avoid it when a derivative-based method is safe and much faster.
What It Does
- Maintains a bracketing interval
- Cuts the interval in half each step
Use Cases
- Reliable root finding
- Solving monotonic equations
- Finding numeric thresholds
Inputs
- Continuous function
- Interval with opposite signs at endpoints
Outputs
- Approximate root
Key Formulas (1)
Midpoint
mid = (low + high) / 2
mid = (low + high) / 2Splits the bracketing interval into two halves.
Pseudocode
- 1.Check that the interval brackets a root
- 2.Compute the midpoint
- 3.Evaluate the function at the midpoint
- 4.Keep the half that still brackets the root
- 5.Repeat until the interval is small enough
Canonical Example
- Input
- f(x) = x^2 - 9, interval = [0, 5]
- Output
- root near 3
- Explanation
- The interval brackets a sign change, and repeated halving narrows it around x = 3.
Implementation Notes
- Check f(left) and f(right) before the loop.
- Replace the endpoint whose sign matches the midpoint sign.
- Stop when the interval width, residual, or both meet the chosen tolerance.
Common Pitfalls
- Starting without verifying that endpoint signs differ.
- Stopping only on iteration count instead of interval width or residual.
- Losing the bracketing invariant after updating the wrong endpoint.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow an interval closing in on a root on a curve.
Step 1 of 5: Check that the interval brackets a root
interval-halvingNotes
Bisection is slower than Newton's method but preserves a root bracket when its assumptions hold.
Sources (1)
What each reference supports for this entry.
Numerical Recipes
BookReference text and implementation-oriented treatment for root finding, interpolation, integration, and ODE methods.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Alternative toNewton's Method
Optimization · Foundation
Bisection is slower but more robust when a sign bracket is available.
Compare with thisAlternativesGolden Section Search
Optimization · Core
Golden-section search optimizes a unimodal function, while bisection finds a bracketed root.
Compare with thisRelationships
View full relationship pageAlternative toNewton's Method
Bisection is slower but more robust when a sign bracket is available.
Alternative toSecant Method
Both find roots, but bisection requires bracketing and is more conservative.
Related Algorithms
Tags
Numerical MethodsRoot FindingIterativeClassic