Bisection Method Root Calculator
The bisection method is the most dependable way to find a root of a continuous function. Given two points where the function has opposite signs, it repeatedly takes the midpoint, checks which half still contains the sign change, and discards the other half, shrinking the bracket until the root is pinned down. This calculator applies it to the cubic a x cubed plus b x squared plus c x plus d. Enter the coefficients and a bracketing interval to get the root, the function value, and the iteration count.
Bisection method steps
f(x) = a*x^3 + b*x^2 + c*x + d
Require f(lo) and f(hi) to have opposite signs
mid = (lo + hi) / 2
If f(lo) and f(mid) differ in sign, set hi = mid, else lo = mid
Repeat until the bracket is narrower than the tolerance
Each step halves the interval, so convergence is steady and guaranteed once a sign change is bracketed. The reported root is the final midpoint.
About the bisection method
- It relies on the intermediate value theorem for continuous functions.
- It always converges when the endpoints bracket a single sign change.
- The bracket width halves each iteration, about one binary digit per step.
- It needs no derivative, unlike Newton-Raphson.
- It is often used to safeguard faster methods that can otherwise diverge.
Bisection method: frequently asked questions
What is the bisection method?
The bisection method finds a root inside an interval where the function changes sign. It repeatedly halves the interval, keeping the half that still brackets the sign change, until the interval is small enough. It is slow but very reliable.
What function does this calculator solve?
It finds a root of the cubic f(x) = a x^3 + b x^2 + c x + d, with coefficients you enter. You also provide a left endpoint and a right endpoint that bracket a root.
Why must the endpoints bracket a sign change?
The method relies on the intermediate value theorem: if a continuous function has opposite signs at the two endpoints, it must cross zero between them. If f(left) and f(right) have the same sign, the calculator shows n/a.
How fast does bisection converge?
The interval width halves every step, so it gains about one binary digit of accuracy per iteration. This is slower than Newton-Raphson, but it always converges when a sign change is bracketed.
When should I prefer bisection over Newton-Raphson?
Use bisection when reliability matters more than speed, when you have a bracketing interval, or when the derivative is hard to obtain. It cannot diverge or stall the way Newton-Raphson can.
Official sources
- NIST Digital Library of Mathematical Functions: Nonlinear Equations.
- National Institute of Standards and Technology DADS: Bisection.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.