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Bisection Method Calculator

Find roots of equations using the bisection (binary search) method.

Function f(x)

Interval [a, b]

Parameters

Bisection Algorithm

  1. Check f(a) * f(b) less than 0
  2. Compute midpoint c = (a + b) / 2
  3. If f(a) * f(c) less than 0, set b = c
  4. Otherwise, set a = c
  5. Repeat until |b - a| less than tolerance

Root

1.4142456055

Converged in 14 iterations

f(root)
9.0633e-5
Iterations
14

Iteration History

nabcf(c)
11.0000002.0000001.5000002.500e-1
21.0000001.5000001.250000-4.375e-1
31.2500001.5000001.375000-1.094e-1
41.3750001.5000001.4375006.641e-2
51.3750001.4375001.406250-2.246e-2
61.4062501.4375001.4218752.173e-2
71.4062501.4218751.414063-4.272e-4
81.4140631.4218751.4179691.064e-2
91.4140631.4179691.4160165.100e-3
101.4140631.4160161.4150392.336e-3
111.4140631.4150391.4145519.539e-4
121.4140631.4145511.4143072.633e-4
131.4140631.4143071.414185-8.200e-5
141.4141851.4143071.4142469.063e-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

mid = (a + b) / 2 if f(mid) × f(a) > 0: a = mid else: b = mid Repeat until |b − a| < tolerance

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

  1. Select function: Choose from the preset list of continuous functions.
  2. Set bounds: Enter a and b such that f(a)×f(b) < 0 (sign change required).
  3. Adjust tolerance and max iterations: Smaller tolerance = more precision but more steps.
  4. 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:

  1. 1f(1) = −1 (negative), f(2) = 2 (positive) — sign change exists.
  2. 2Iter 1: mid=(1+2)/2=1.5, f(1.5)=0.25. Since f(1)×f(1.5)<0, b=1.5.
  3. 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.

Related Calculators

Newton-Raphson Calculator

Faster root-finding method

Secant Method

Derivative-free alternative

False Position Method

Weighted interval method

Bisection Method (Advanced)

Custom function bisection

Frequently Asked Questions

The Intermediate Value Theorem guarantees a root exists only when the function crosses the x-axis. If both are positive or both negative, there's no guarantee — the function might not cross zero, or might cross an even number of times. The bisection method requires this sign change to converge.
Each iteration halves the interval, so after n iterations the error is at most (b−a)/2^n. To get within ε, you need n > log₂((b−a)/ε) iterations. This is linear convergence — slower than Newton's method but guaranteed to work.

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.

Source

Formula Source: Handbook of Mathematical Functions

by Abramowitz & Stegun

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.