Online Tool Store
Newton's Method Visualizer
Newton's Method Visualizer shows why the method is fast when it works; starting near a turning point shows just as clearly why it sometimes does not.
🔒 This tool runs entirely in your browser. Your files are never uploaded to a server.
Math & Science
Newton's Method Visualizer
Frontend preview — no upload or external service.
Iteration table
From x0 = 2: 2.1, 2.0946, 2.09455148, 2.09455148. Converged to eight decimals in four steps — roughly doubling the correct digits each iteration.
How the Newton's Method Visualizer works
- Enter the function and a starting point near the root you want.
- Step through and watch each tangent line hit the axis, which becomes the next estimate.
- Try a starting point near a turning point to see the method overshoot — that failure is as instructive as the convergence.
The method
Each step replaces the function with its tangent line at the current point and takes that line's root as the next estimate.
x_{n+1} = x_n - f(x_n) / f'(x_n)
Near a simple root convergence is quadratic, roughly doubling the number of correct digits per step. Near a point where the derivative is small, the tangent is nearly flat and the next estimate flies off.
FAQ
Why does Newton's method sometimes fail?
Because the tangent line can point anywhere. Start near a turning point and the near-zero derivative throws the next estimate far away. Some functions and starting points produce cycles that never converge.
How fast does it converge?
Quadratically near a simple root — the number of correct digits roughly doubles each step, which is why four iterations often reach machine precision. Near a repeated root it degrades to linear.
What makes a good starting point?
One close to the root and away from turning points. In practice a bracketing method like bisection is often used to get close, then Newton finishes quickly.
How we compare
| Feature | Online Tool Store | A graphing calculator | A stats package |
|---|---|---|---|
| Shows each tangent | ✓ | ✗ | ✗ |
| Demonstrates failure cases | ✓ | ✗ | ✗ |
| Iteration table | ✓ | ✓ | ✓ |
| No install | ✓ | ✗ | ✓ |
Newton's Method Visualizer shows why the method is fast when it works; starting near a turning point shows just as clearly why it sometimes does not.