What is the Newton-Raphson method?

The Newton-Raphson method is an iterative root-finding algorithm. Starting from an initial guess x₀, it applies the update x_{n+1} = x_n - f(x_n) / f'(x_n) repeatedly until |f(x_n)| or |x_{n+1} - x_n| falls below a chosen tolerance. When it converges, it does so quadratically — each iteration roughly doubles the number of correct decimal digits.

What happens if I leave the derivative field blank?

The tool computes a numerical derivative using the symmetric finite-difference approximation: f'(x) ≈ (f(x + h) - f(x - h)) / (2h) with h = 1e-7. This is accurate enough for most smooth functions, but providing the exact analytic derivative improves both speed and numerical stability.

Why does iteration diverge or return NaN?

Divergence usually means f'(x) is near zero at some iterate (the update overshoots), the initial guess is far from a root, or the function has no real root in that region. Try a different initial guess or a smaller step. If f'(x) = 0 exactly, the method is undefined at that point.

What expression syntax is supported?

Standard JavaScript math: use x as the variable, * for multiplication, ** or Math.pow() for exponentiation, Math.sin(), Math.cos(), Math.tan(), Math.exp(), Math.log(), Math.sqrt(), Math.abs(), Math.PI, Math.E. Example: Math.exp(x) - 3*x or x**3 - 2*x - 5.

How many iterations will the tool run?

The solver stops when |f(x_n)| < tolerance AND |x_{n+1} - x_n| < tolerance, or when the maximum iteration count (default 50, max 200) is reached. It also stops if a NaN or Infinity is produced.

Can this find complex roots?

No. This tool works only over the real numbers. For complex roots, you need a solver that operates in the complex plane.

Newton Raphson Solver

Name: Newton Raphson Solver
Availability: InStock
Author: Nham Vu

Numerically solve f(x) = 0 using the Newton-Raphson method, with step-by-step iterations and a convergence chart.

Parameters

f(x)

Use x, Math.sin(), Math.exp(), ** for power.

f'(x) (optional — leave blank for numeric)

Initial guess x₀

Tolerance

Max iterations

Enter a function and click Solve to find the root.

Step-by-Step Iterations

n	x_n	f(x_n)	f'(x_n)	x_{n+1}	\|error\|

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Summary