Bisection Method Calculator – Bracketing Roots

Bisection exploits the intermediate value theorem: if a continuous function changes sign over [a, b], a root lies inside, and testing the midpoint tells which half keeps it. For f(x) = x² − 2 on [1, 2] (f(1) = −1, f(2) = 2): midpoint 1.5 gives f = 0.25 > 0, so the root sits in [1, 1.5]; next midpoint 1.25 has f = −0.4375, moving the bracket to [1.25, 1.5]; then 1.375, and so on toward √2 ≈ 1.41421. Each step halves the bracket — 10 iterations shrink uncertainty below 1/1024 ≈ 0.00098, and the required count is exactly ⌈log₂((b−a)/tolerance)⌉. The calculator tabulates every midpoint, its sign, and the surviving interval, and rejects brackets whose endpoints do not straddle zero (f(a)·f(b) > 0). Slow but unstoppable: unlike Newton, bisection cannot diverge once a valid bracket is given.

1. 1

   Enter f(x) and bracket [a, b]

   Endpoints must straddle the root: f(a)·f(b) < 0

2. 2

   Test the midpoint

   Sign of f((a+b)/2) picks the surviving half

3. 3

   Repeat

   Each iteration halves the interval width

4. 4

   Stop at tolerance

   Error ≤ (b−a)/2ⁿ⁺¹ after n steps

• 1

  One binary digit of accuracy per iteration — 10 steps ≈ 3 decimal digits

• 2

  Invalid bracket (no sign change) = no guarantee

• 3

  Slow but certain: convergence never fails on continuous f

• Passing a bracket without a sign change and expecting convergence

• Keeping the wrong half by comparing f(m) with 0 sloppily

• Forgetting the error bound (b−a)/2ⁿ⁺¹ when choosing iterations

• Mixing bisection guarantees with Newton’s speed claims