Newton-Raphson Calculator
Find a root of f(x)=0 using Newton-Raphson iteration with tolerance and iteration control.
Inputs
Function whose root you seek.
Root
2.094551481542
Iterations
5
|f(root)|
8.8818 × 10⁻¹⁶
Status
Converged
Step by step
Values used
f(x) = x^3 - 2*x - 5; Initial guess x₀ = 2; Tolerance = 0.0000; Max iterations = 100
Newton-Raphson iteration
x(n+1) = x(n) - f(x(n)) / f'(x(n))
Root
= 2.094551481542
Iterations
= 5
|f(root)|
= 0.0000
Status
= Converged
How it works
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
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.