Enter a polynomial and an interval where it changes sign, and instantly see every bisection step narrow in on the root.
Bisection Method Calculator
How It’s Calculated
Bisection relies on the Intermediate Value Theorem: if f(a) and f(b) have opposite signs, a continuous function must cross zero somewhere between them. Each step halves the interval — checking the midpoint c, then keeping whichever half still brackets a sign change. It’s slower than Newton-Raphson but never fails to converge once a valid bracket is found, which is exactly why it’s often used to get a good starting guess for faster methods.
Worked Example
For f(x) = x³ − 4 on [1,2]: f(1) = −3, f(2) = 4, opposite signs. Midpoint c = 1.5, f(1.5) = −0.625, so the root is in [1.5, 2]. Next midpoint 1.75, f(1.75) = 1.359, so the root is in [1.5, 1.75]. This keeps narrowing toward ≈ 1.5874.
Frequently Asked Questions
Q: What if f(a) and f(b) have the same sign?
Bisection can’t guarantee a root exists in that interval — you need to choose a and b so the function genuinely changes sign between them first.
Q: Why is bisection considered more reliable than Newton-Raphson?
It only needs a sign change to guarantee convergence, with no risk of a zero derivative or a wildly bad initial guess sending it off course. The tradeoff is that it converges more slowly, halving the error each step rather than the much faster convergence Newton-Raphson typically achieves.
Bisection is a standard first numerical method across the VTU Engineering Mathematics syllabus. See how Daniel Classes teaches VTU Engineering Mathematics →