Newton-Raphson Method Calculator – Roots Fast

Newton’s method slides down the tangent line: at each guess xₖ it follows the tangent’s slope f′(xₖ) to where that tangent hits zero, which lands closer to the actual root. Solving f(x) = x² − 2 from x₀ = 1: the tangent update gives x₁ = 1 − (−1)/2 = 1.5, then x₂ = 1.41667, then x₃ = 1.41422 — √2 to five digits in three steps. Convergence is quadratic near a simple root: correct digits double each iteration (error 0.086 → 0.0025 → 0.0000031 in the run above). The calculator prints f and f′ at every iterate, the tangent step, and the residual, and warns about the failure modes — a zero derivative mid-run, or oscillation when the starting point sits far from the root on a curvy function (x₀ = −1 on this parabola bounces between −1 and 1; choosing x₀ = 2 fixes it). Compare slower but safer methods: bisection halves the interval every step.

1. 1

   Enter f(x) and f′(x)

   The function and its derivative (or pick a preset)

2. 2

   Choose the starting guess

   Close guesses converge fastest

3. 3

   Iterate the tangent step

   Each xₖ, f(xₖ), f′(xₖ) and update is listed

4. 4

   Stop at tolerance

   Run ends when |f(xₖ)| < your threshold

• 1

  Quadratic convergence: digits double per step near a simple root

• 2

  Each step costs one function plus one derivative evaluation

• 3

  A derivative near zero stalls the method — pick another start

• Using f′(x₀) for every step instead of recomputing f′(xₖ)

• Starting far from the root on flat regions and diverging

• Forgetting multiplicity: double roots converge only linearly

• Skipping the residual check |f(x)| before accepting the answer