Newton-Raphson 1D
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The Newton-Raphson 1D calculator numerically solves equations of the form f(x)=0 using the iterative Newton-Raphson method. You provide the function f(x), its derivative f'(x), and an initial guess x₀. Each iteration computes x_{n+1} = x_n − f(x_n)/f'(x_n), approaching the root. The process stops when the difference between consecutive iterations is less than a defined tolerance (default 1e-7) or reaches the maximum number of iterations.
This method is widely used in engineering, physics, and finance to find roots of nonlinear equations. For example, to calculate the internal rate of return (IRR) of an investment, solve equations of state in thermodynamics, or determine equilibrium points in mathematical models. Convergence is fast (quadratic) if the initial guess is close to the root and the derivative is nonzero.
Important precautions: the method may fail if the derivative is zero or very small, or if the initial guess is far from the root (it may diverge or converge to another root). For functions with multiple roots, different guesses may lead to different results. It is recommended to test several guesses and verify that f(x) is close to zero. Additionally, the derivative must be provided analytically; for numerical derivatives, use other methods.
Frequently asked questions
What if the calculator does not converge?
Try a different initial guess closer to the expected root. Check if the derivative is zero in the region. Increase the maximum number of iterations or adjust the tolerance.
Can I use trigonometric or exponential functions?
Yes, as long as you provide the correct derivative. The calculator accepts functions like sin(x), cos(x), exp(x), log(x), etc.
Does the method work for multiple roots?
Yes, but convergence may be slower (linear). For multiple roots, the derivative is also zero at the root, which can cause issues. Modified methods are recommended in such cases.
What is the difference between Newton-Raphson and the bisection method?
Newton-Raphson converges faster (quadratic) but requires the derivative and a good initial guess. The bisection method is more robust but slower (linear) and does not need a derivative.
Do I have to type the derivative manually?
Yes, you must provide the derivative expression. If you cannot differentiate, use an online derivative calculator or numerical methods like the secant method.