Bisection Method Calculator
Find roots of continuous functions using the bisection (interval halving) method.
Function f(x) = 0
Bisection Algorithm
1. Check f(a) and f(b) have opposite signs
2. Compute midpoint c = (a + b) / 2
3. If f(c) ≈ 0 or interval small, done
4. If f(a)·f(c) < 0: b = c, else a = c
5. Repeat from step 2
Properties
Guaranteed convergence (if conditions met)
Linear convergence: error halves each step
After n iterations: error ≤ (b-a)/2^n
Root Found
x = 1.5214233398
f(x) = 2.5935e-4
Iterations
14
Final Error Bound
6.1035e-5
Iteration History
| n | a | b | mid | f(mid) |
|---|---|---|---|---|
| 0 | 1.000000 | 2.000000 | 1.500000 | -1.250e-1 |
| 1 | 1.500000 | 2.000000 | 1.750000 | 1.609e+0 |
| 2 | 1.500000 | 1.750000 | 1.625000 | 6.660e-1 |
| 3 | 1.500000 | 1.625000 | 1.562500 | 2.522e-1 |
| 4 | 1.500000 | 1.562500 | 1.531250 | 5.911e-2 |
| 5 | 1.500000 | 1.531250 | 1.515625 | -3.405e-2 |
| 6 | 1.515625 | 1.531250 | 1.523438 | 1.225e-2 |
| 7 | 1.515625 | 1.523438 | 1.519531 | -1.097e-2 |
| 8 | 1.519531 | 1.523438 | 1.521484 | 6.222e-4 |
| 9 | 1.519531 | 1.521484 | 1.520508 | -5.179e-3 |
| 10 | 1.520508 | 1.521484 | 1.520996 | -2.279e-3 |
| 11 | 1.520996 | 1.521484 | 1.521240 | -8.289e-4 |
| 12 | 1.521240 | 1.521484 | 1.521362 | -1.034e-4 |
| 13 | 1.521362 | 1.521484 | 1.521423 | 2.594e-4 |
Error Analysis
Iteration 0:error ≤ 5.0000e-1
Iteration 1:error ≤ 2.5000e-1
Iteration 2:error ≤ 1.2500e-1
Iteration 3:error ≤ 6.2500e-2
Iteration 4:error ≤ 3.1250e-2
Iteration 5:error ≤ 1.5625e-2
Iteration 6:error ≤ 7.8125e-3
Iteration 7:error ≤ 3.9063e-3
Iteration 8:error ≤ 1.9531e-3
Iteration 9:error ≤ 9.7656e-4
Error halves with each iteration (linear convergence)