Bisection Method Calculator
Find roots of equations using the bisection (binary search) method.
Function f(x)
Interval [a, b]
Parameters
Bisection Algorithm
- Check f(a) * f(b) less than 0
- Compute midpoint c = (a + b) / 2
- If f(a) * f(c) less than 0, set b = c
- Otherwise, set a = c
- Repeat until |b - a| less than tolerance
Root
1.4142456055
Converged in 14 iterations
Iteration History
| n | a | b | c | f(c) |
|---|---|---|---|---|
| 1 | 1.000000 | 2.000000 | 1.500000 | 2.500e-1 |
| 2 | 1.000000 | 1.500000 | 1.250000 | -4.375e-1 |
| 3 | 1.250000 | 1.500000 | 1.375000 | -1.094e-1 |
| 4 | 1.375000 | 1.500000 | 1.437500 | 6.641e-2 |
| 5 | 1.375000 | 1.437500 | 1.406250 | -2.246e-2 |
| 6 | 1.406250 | 1.437500 | 1.421875 | 2.173e-2 |
| 7 | 1.406250 | 1.421875 | 1.414063 | -4.272e-4 |
| 8 | 1.414063 | 1.421875 | 1.417969 | 1.064e-2 |
| 9 | 1.414063 | 1.417969 | 1.416016 | 5.100e-3 |
| 10 | 1.414063 | 1.416016 | 1.415039 | 2.336e-3 |
| 11 | 1.414063 | 1.415039 | 1.414551 | 9.539e-4 |
| 12 | 1.414063 | 1.414551 | 1.414307 | 2.633e-4 |
| 13 | 1.414063 | 1.414307 | 1.414185 | -8.200e-5 |
| 14 | 1.414185 | 1.414307 | 1.414246 | 9.063e-5 |
About Bisection Method
- Always converges if root exists in interval
- Linear convergence rate
- Requires sign change (opposite signs at endpoints)
- Error halves with each iteration
- Simple but slower than Newton-Raphson
What Is the Bisection Method?
The bisection method is a root-finding algorithm that repeatedly halves an interval where a continuous function changes sign. If f(a) and f(b) have opposite signs, the Intermediate Value Theorem guarantees at least one root in [a,b]. The midpoint c = (a+b)/2 is evaluated; if f(c) has the same sign as f(a), the root lies in [c,b]; otherwise in [a,c]. The interval halves each iteration, converging to the root at a predictable linear rate.
This calculator supports five preset functions (x²−2, x³−x−1, x³−2, cos(x)−x, eˣ−3) and stops when the interval width falls below the tolerance or the maximum iteration count is reached. A table shows each iteration's bounds, midpoint, and function value.
Bisection Method Algorithm
Bisection Iteration
Where:
- a= Lower bound of the search interval
- b= Upper bound — must have f(a) × f(b) < 0
- tolerance= The desired precision — stopping when interval width < tolerance
How to Use
- Select function: Choose from the preset list of continuous functions.
- Set bounds: Enter a and b such that f(a)×f(b) < 0 (sign change required).
- Adjust tolerance and max iterations: Smaller tolerance = more precision but more steps.
- View iterations: A table shows each step's midpoint and f(mid) values.
Applications
The bisection method is the most robust root-finding algorithm — it never fails for a continuous function with a sign change. In engineering, it's used to solve nonlinear equations like finding the natural frequency of vibrating systems. In finance, it computes the internal rate of return (IRR) by solving NPV(r) = 0. In computer graphics, it finds ray-surface intersection points for rendering.
Worked Examples
Find √2 using x² − 2
Problem:
Use bisection on [1, 2] with f(x) = x² − 2.
Solution Steps:
- 1f(1) = −1 (negative), f(2) = 2 (positive) — sign change exists.
- 2Iter 1: mid=(1+2)/2=1.5, f(1.5)=0.25. Since f(1)×f(1.5)<0, b=1.5.
- 3After ~14 iterations: interval converges to ~1.4142.
Result:
√2 ≈ 1.4142 within 0.0001 tolerance.
Tips & Best Practices
- ✓Choose a and b such that f(a)×f(b) < 0 — without the sign change, the method cannot start.
- ✓Smaller tolerance gives more accurate results but requires more iterations; 1e−4 is sufficient for most applications.
- ✓The bisection method is guaranteed to converge for any continuous function, unlike Newton's method which can diverge.
- ✓If f(mid)=0 exactly, the algorithm stops immediately — you've found an exact root.
Frequently Asked Questions
Sources & References
Last updated: 2026-06-06
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
Formula Source: Handbook of Mathematical Functions
by Abramowitz & Stegun