Bisection Method Calculator
Bisection halves the interval where f changes sign: x³ − x − 2 on [1, 2] narrows to the root x ≈ 1.5213804245.
Use x, ^ for powers, and sin, cos, tan, exp, ln, log, sqrt, abs, pi, e. 2x and xsin(x) are understood.
Stop when successive values or |f(x)| fall below this.
1.52138042449951
Full answer, then cut to fewer decimals (not rounded):
- 1.52138042449951
- 1.5213804244995
- 1.521380424499
- 1.52138042449
- 1.5213804244
- 1.521380424
- 1.52138042
- 1.5213804
- 1.521380
- 1.52138
- 1.5213
- 1.521
- 1.52
- 1.5
- f(root)
- 0.00000426583
- Iterations
- 20
- Iterations needed for this tolerance ⌈log₂((b−a)/tol)⌉
- 20
| n | c = (a + b)/2 | f(c) | a | b |
|---|---|---|---|---|
| 1 | 1.5 | -0.125 | 1 | 2 |
| 2 | 1.75 | 1.60938 | 1.5 | 2 |
| 3 | 1.625 | 0.666016 | 1.5 | 1.75 |
| 4 | 1.5625 | 0.252197 | 1.5 | 1.625 |
| 5 | 1.53125 | 0.0591125 | 1.5 | 1.5625 |
| 6 | 1.515625 | -0.0340538 | 1.5 | 1.53125 |
| 7 | 1.5234375 | 0.0122504 | 1.515625 | 1.53125 |
| 8 | 1.51953125 | -0.0109712 | 1.515625 | 1.5234375 |
| 9 | 1.521484375 | 0.000622176 | 1.51953125 | 1.5234375 |
| 10 | 1.5205078125 | -0.00517889 | 1.51953125 | 1.521484375 |
| 11 | 1.52099609375 | -0.00227944 | 1.520507813 | 1.521484375 |
| 12 | 1.52124023438 | -0.000828906 | 1.520996094 | 1.521484375 |
| 13 | 1.52136230469 | -0.000103433 | 1.521240234 | 1.521484375 |
| 14 | 1.52142333984 | 0.000259354 | 1.521362305 | 1.521484375 |
| 15 | 1.52139282227 | 0.0000779563 | 1.521362305 | 1.52142334 |
| 16 | 1.52137756348 | -0.0000127395 | 1.521362305 | 1.521392822 |
| 17 | 1.52138519287 | 0.0000326082 | 1.521377563 | 1.521392822 |
| 18 | 1.52138137817 | 0.00000993428 | 1.521377563 | 1.521385193 |
| 19 | 1.52137947083 | -0.00000140261 | 1.521377563 | 1.521381378 |
| 20 | 1.5213804245 | 0.00000426583 | 1.521379471 | 1.521381378 |
Step-by-step solution
- c = (a + b)/2; keep the half where f changes sign
- Root ≈ 1.5213804245 after 20 iterations
Bisection Method Formula
c = (a + b)/2; regula falsi c = (a·f(b) − b·f(a)) ÷ (f(b) − f(a))
keep the half where f changes sign
Worked Example
f(x) = x³ − x − 2 on [1, 2]:
- c = (a + b)/2; keep the half where f changes sign
- Root ≈ 1.5213804245 after 20 iterations
Iterations: 20 · Iterations needed for this tolerance ⌈log₂((b−a)/tol)⌉: 20
| n | c = (a + b)/2 | f(c) | a | b |
|---|---|---|---|---|
| 1 | 1.5 | -0.125 | 1 | 2 |
| 2 | 1.75 | 1.60938 | 1.5 | 2 |
| 3 | 1.625 | 0.666016 | 1.5 | 1.75 |
| 4 | 1.5625 | 0.252197 | 1.5 | 1.625 |
| 5 | 1.53125 | 0.0591125 | 1.5 | 1.5625 |
| 6 | 1.515625 | -0.0340538 | 1.5 | 1.53125 |
| 7 | 1.5234375 | 0.0122504 | 1.515625 | 1.53125 |
| 8 | 1.51953125 | -0.0109712 | 1.515625 | 1.5234375 |
| 9 | 1.521484375 | 0.000622176 | 1.51953125 | 1.5234375 |
| 10 | 1.5205078125 | -0.00517889 | 1.51953125 | 1.521484375 |
| 11 | 1.52099609375 | -0.00227944 | 1.520507813 | 1.521484375 |
| 12 | 1.52124023438 | -0.000828906 | 1.520996094 | 1.521484375 |
| 13 | 1.52136230469 | -0.000103433 | 1.521240234 | 1.521484375 |
| 14 | 1.52142333984 | 0.000259354 | 1.521362305 | 1.521484375 |
| 15 | 1.52139282227 | 0.0000779563 | 1.521362305 | 1.52142334 |
| 16 | 1.52137756348 | -0.0000127395 | 1.521362305 | 1.521392822 |
| 17 | 1.52138519287 | 0.0000326082 | 1.521377563 | 1.521392822 |
| 18 | 1.52138137817 | 0.00000993428 | 1.521377563 | 1.521385193 |
| 19 | 1.52137947083 | -0.00000140261 | 1.521377563 | 1.521381378 |
| 20 | 1.5213804245 | 0.00000426583 | 1.521379471 | 1.521381378 |
GATE tip
Bisection always converges if f is continuous and changes sign, but only linearly (one bit per step). GATE often asks for the root after 2 or 3 iterations; read those rows of the table.
More Numerical Methods Calculators
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Newton-Raphson Method Calculator
xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
quadratic convergence near a simple root
Example: f(x) = x³ − 2x − 5 starting at x₀ = 2: Root x ≈ 2.09455148154.
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Secant Method Calculator
xₙ₊₁ = xₙ − f(xₙ)(xₙ − xₙ₋₁) ÷ (f(xₙ) − f(xₙ₋₁))
order of convergence ≈ 1.618
Example: f(x) = x² − 612 starting from 10 and 30: Root x ≈ 24.7386337537.
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Lagrange Interpolation Calculator
P(x) = Σ yᵢ Π (x − xⱼ)/(xᵢ − xⱼ)
degree ≤ n − 1 through n points
Example: The points (1, 1), (2, 8), (3, 27), (4, 64) at x = 2.5: P(2.5) = 15.625.
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Newton Interpolation Calculator
P(x) = f[x₀] + f[x₀,x₁](x − x₀) + f[x₀,x₁,x₂](x − x₀)(x − x₁) + …
f[xᵢ,…,xᵢ₊ₖ] = (f[xᵢ₊₁,…] − f[xᵢ,…]) ÷ (xᵢ₊ₖ − xᵢ)
Example: The points (1, 1), (2, 8), (3, 27), (4, 64) at x = 2.5: P(2.5) = 15.625.
Frequently asked questions
How many bisection iterations are needed?
n ≥ log₂((b − a)/ε). Halving [1, 2] to an error below 0.001 needs 10 iterations.
Is regula falsi faster than bisection?
Usually, because it uses the function values, but one end can stay fixed for many steps, making it slow for strongly curved functions.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Bisection Method Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/bisection-method-calculator/
IEEE
Scientific Converters, "Bisection Method Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/bisection-method-calculator/