Newton-Raphson Root Calculator
The Newton-Raphson method finds a root of a function by following the tangent line: from a starting point it computes where the tangent crosses zero and uses that as the next estimate, repeating until the estimate settles. This calculator applies it to the cubic polynomial a x cubed plus b x squared plus c x plus d, whose exact derivative is known, so each step is precise. Enter the four coefficients and a starting guess to get the converged root, the function value there, and the number of iterations used.
Newton-Raphson formula
f(x) = a*x^3 + b*x^2 + c*x + d
f'(x) = 3a*x^2 + 2b*x + c
x_next = x - f(x) / f'(x)
Stop when |f(x)| is below the tolerance
The iteration starts at x0 and repeats until the function value is within a small tolerance or an iteration limit is reached. The exact derivative keeps each step accurate.
About the Newton-Raphson method
- It converges quadratically near a simple root, doubling correct digits each step.
- It needs the derivative, which is known exactly for a polynomial.
- A flat tangent (zero derivative) stalls the method and returns n/a.
- Different starting guesses can converge to different roots of the cubic.
- It is the workhorse behind many square-root and reciprocal hardware routines.
Newton-Raphson method: frequently asked questions
What is the Newton-Raphson method?
It is an iterative root-finding method. Starting from a guess x0, it repeatedly applies x_next = x - f(x) / f'(x), using the tangent line to step toward where the function crosses zero. It converges very quickly when the guess is near a simple root.
What function does this calculator solve?
It finds a root of the cubic polynomial f(x) = a x^3 + b x^2 + c x + d, with coefficients you enter. The derivative f'(x) = 3a x^2 + 2b x + c is computed exactly, so no numerical differentiation error enters.
Why does the method sometimes fail to converge?
If the derivative is zero at an iterate, the tangent is flat and the step is undefined. Poor starting guesses can also send iterates away from a root or cycle between values. The calculator reports n/a if it does not converge within the iteration limit.
How fast does Newton-Raphson converge?
Near a simple root it converges quadratically, meaning the number of correct digits roughly doubles each step. This is much faster than the bisection method, which only adds about one bit of accuracy per step.
How do I find a different root of the cubic?
Change the starting guess. A cubic can have up to three real roots, and Newton-Raphson converges to whichever root the chosen starting point leads to. Try guesses near different sign changes of the function.
Official sources
- NIST Digital Library of Mathematical Functions: Nonlinear Equations.
- National Institute of Standards and Technology DADS: Newton's method.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.