What is Newton's method?
An iterative root-finding algorithm using f and f′.
Newton's Method Calculator
Start from a guess x₀ and apply xₖ₊₁ = xₖ − f(xₖ)/f′(xₖ) for a fixed number of iterations.
Set up the iteration
Choose the equation, enter coefficients if needed, then x₀ and iteration count.
About this calculator
How to use Newton's Method Calculator, plus the formulas and logic behind the results.
Tool description
Newton's method uses the tangent line at the current guess to produce a better root estimate.
How to use
- Select x²−2 or a general quadratic.
- Enter x₀ and the number of iterations.
- Select Calculate to view successive approximations.
- Watch for division by zero if f′(xₖ) ≈ 0.
Formulas & reference
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)For x² − 2: xₙ₊₁ = (xₙ + 2/xₙ) / 2
Newton tips
- Classic demo: approximate √2 from x₀ = 1
- Stop if f′(x) = 0 at an iterate
- More iterations refine the estimate when converging
- Bad guesses can diverge or cycle
Calculation logic
Iterates the Newton update, recording each step until n iterations or a zero derivative stops the process.
Convergence depends on a good initial guess and a non-zero derivative at each iterate.
Newton's Method Calculator FAQ
Common questions about this calculator and how to use the results.
Why use x² − 2?
It famously approximates √2 and is a standard classroom example.
What if f′ is zero?
The update is undefined; the tool reports an error at that step.
How many iterations?
A small positive integer (e.g. 5–10) is usually enough for demos.
Does it always converge?
No—convergence is local and depends on the function and start point.
Can I solve cubics?
This educational tool supports quadratics and x²−2 only.