Newton's Method Calculator
Find a root of any function by Newton-Raphson iteration. See every step in a table with xₙ, f(xₙ), f′(xₙ) and the shrinking step size.
Iteration table
| n | xₙ | f(xₙ) | f′(xₙ) | |xₙ₊₁ − xₙ| |
|---|---|---|---|---|
| 0 | 1 | -1 | 2 | 0.5 |
| 1 | 1.5 | 0.25 | 3 | 0.083333333333 |
| 2 | 1.416666666667 | 0.0069444444 | 2.8333333333 | 0.002450980392 |
| 3 | 1.414215686275 | 0.0000060073 | 2.8284313725 | 0.0000021239 |
| 4 | 1.414213562375 | 4.510614e-12 | 2.8284271247 | 1.594724e-12 |
Each row applies xn+1 = xn − f(xn) / f′(xn). Derivative used: 2*x (symbolic).
Newton's method finds one root near your starting guess, not all of them. A different x₀ can land on a different root, or diverge — that is a property of the method, not a bug.
What is the Newton's Method Calculator?
The ByteTools Newton's Method Calculator solves f(x) = 0 by iterating xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) from a starting guess.
- Symbolic derivative with a central-difference fallback
- Full iteration table with xₙ, f(xₙ), f′(xₙ) and step size
- Configurable tolerance and iteration limit
- Names the exact stopping reason — converged, diverged or zero derivative
- Guards the zero-derivative case instead of dividing by it
- Everything computed locally in your browser
How to use the Newton's Method Calculator
- 1
Enter the function whose root you want in the f(x) box, such as x^2 - 2 or cos(x) - x.
- 2
Type a starting guess x₀ reasonably close to the root you are after.
- 3
Set the tolerance exponent and the maximum number of iterations if you need different limits.
- 4
Read the root, f(root) and the iteration count, then follow the iteration table.
- 5
Click Copy result to save the root and the stopping reason.
About the Newton's Method Calculator
The ByteTools Newton's Method Calculator solves f(x) = 0 by iterating xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) from a starting guess. It differentiates your function symbolically for the derivative and falls back to a central difference where the symbolic form cannot be evaluated, so the iteration keeps going where it sensibly can.
Every iteration is listed with xₙ, f(xₙ), f′(xₙ) and the step size |xₙ₊₁ − xₙ|, so you can watch the quadratic convergence happen. The run stops on tolerance, on the iteration limit, on divergence, or on a zero derivative — and the page names which of those happened rather than just showing a number.
Newton's method finds one root near your guess, not all of them, and it can diverge; that is a property of the method and the tool is explicit about it. All computation happens in your browser and nothing is uploaded.
Frequently asked questions
How does Newton's method work?
It follows the tangent line at your current guess down to where it crosses the x-axis, and uses that crossing as the next guess. Repeating this converges very quickly — roughly doubling the number of correct digits each step — when you start near a root.
Why does Newton's method sometimes fail?
It fails when the derivative is zero at an iterate, because the tangent is horizontal and never meets the axis. It can also cycle or run away if the starting guess is poor or the function has an awkward shape. This calculator detects and names each of those cases.
How do I choose a good starting value?
Sketch or graph the function and pick a value near where it crosses the axis. The closer your guess, the faster it converges and the less likely it is to jump to a different root or diverge entirely.
Can Newton's method find all the roots of a function?
No. Each run converges to one root, chosen by where you start. To find several roots, run the calculator again from different starting guesses on either side of each crossing.
What is the difference between Newton's method and the bisection method?
Bisection repeatedly halves an interval that brackets a root — slow but guaranteed. Newton's method uses the derivative and converges far faster, but it offers no guarantee and can diverge from a bad starting point.
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