Newton's Method Calculator
Newton's method rapidly approximates a root of a function by repeatedly following the tangent line down to the x-axis.
Function f(x)
Converged in 5 iterations
How does this calculator work?
Newton's method finds a root of f(x) by iterating x_{n+1} = x_n − f(x_n)/f′(x_n) from an initial guess x₀. This tool uses a numerical derivative and stops when the step is below 1e-10 or after 50 iterations, reporting the approximate root and how many iterations it took.
Formula
How this is calculated
Pick a function f(x) and an initial guess x₀. The calculator iterates the Newton-Raphson update x_{n+1} = x_n − f(x_n)/f′(x_n), where the derivative f′(x_n) is estimated numerically with a central difference, f′(x) ≈ (f(x+h) − f(x−h)) / (2h) using h = 1e-6. This avoids needing a symbolic derivative while staying accurate.
Iteration continues until the step size |Δ| = |f(x_n)/f′(x_n)| falls below 1e-10 (convergence) or 50 steps have been taken. The final x is reported as the approximate root, along with the number of iterations and the residual f(root), which should be near zero for a good solution.
Newton's method converges very fast (quadratically) near a simple root, but it can fail or diverge if the initial guess is poor, if the derivative is zero at some iterate, or for functions with flat regions or multiple roots. Choosing x₀ close to the expected root and checking the residual guards against these edge cases.
Frequently asked questions
If the initial guess is far from a root, if f′(x) is zero or very small at an iterate, or if the function oscillates, the iteration can diverge or jump between values. Try a different starting point closer to where you expect the root.
It is approximated numerically with a central difference, f′(x) ≈ (f(x+h) − f(x−h)) / (2h) with h = 1e-6, so no hand-derived derivative is required.
The residual is f(root): the value of the function at the computed root. A residual very close to zero indicates the estimate is accurate.
Also known as
TG we-Calculate Editorial Team. (2026). Newton's Method Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/newtons-method-calculator
TG we-Calculate Editorial Team. "Newton's Method Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/newtons-method-calculator.
TG we-Calculate Editorial Team, "Newton's Method Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/newtons-method-calculator
@misc{wecalculate_newtons_method_calculator, title = {Newton's Method Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/newtons-method-calculator}}, year = {2026}, note = {TG we-Calculate} }
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