Newton-Raphson Calculator
Find a root of f(x)=0 using Newton-Raphson iteration with tolerance and iteration control.
About Newton-Raphson Calculator
Newton-Raphson finds a root of f(x) = 0 by iterating x_{n+1} = x_n - f(x_n)/f'(x_n). It converges quadratically near simple roots, meaning the number of correct digits roughly doubles each iteration. The derivative is computed numerically via central differences. The method may fail if the derivative is zero or the initial guess is too far from a root.
Formula
Newton-Raphson iteration
x(n+1) = x(n) - f(x(n)) / f'(x(n))
- x(n)
- Current approximation
- f(x)
- Function
- f'(x)
- Derivative of f
Worked example
With the default values below, Newton-Raphson Calculator returns Root of 2.094551481542. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x)
- x^3 - 2*x - 5
- Initial guess x₀
- 2
- Tolerance
- 0
- Max iterationsaccepts 1–1,000
- 100
Results
- Root
- 2.094551481542
- Iterations
- 5
- |f(root)|
- 8.8818 × 10⁻¹⁶
- Status
- Converged
Step by step
- 1Values used
- 2Newton-Raphson iteration
- 3Root= 2.094551481542
- 4Iterations= 5
- 5|f(root)|= 0.0000
- 6Status= Converged
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 12 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What if Newton's method doesn't converge?
Try a different initial guess closer to the root, or use bisection which always converges if you can bracket the root.
How fast does it converge?
Quadratically for simple roots: once close, digits of accuracy double per iteration. It may be slower or fail for multiple roots.