Understand that knowing an equation does not mean you can solve it by hand — iterative methods converge toward a root, and the method you choose changes how many iterations that takes for a given tolerance.
Status
ConvergedIterations Used
15Estimated Root
1.521393Last iterations
| n | x | |f(x)| |
|---|---|---|
| 10 | 1.520508 | 5.179e-3 |
| 11 | 1.520996 | 2.279e-3 |
| 12 | 1.521240 | 8.289e-4 |
| 13 | 1.521362 | 1.034e-4 |
| 14 | 1.521423 | 2.594e-4 |
| 15 | 1.521393 | 7.796e-5 |
tolerance = 1e-4
f(x) = x³ − x − 2 (root near x ≈ 1.5214)Bisection: shrink [a,b] in half each step while f(a)·f(mid) changes signNewton-Raphson: x_{n+1} = x_n − f(x_n) / f'(x_n)Using Bisection with tolerance 10⁻⁴, confirm the method converges within 20 iterations.
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