Newton's Method Root Calculator
Find a root of a cubic with Newton's method — fast quadratic convergence from a starting guess, with iteration count and residual.
How it works
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ) · repeated until |f(x)| or the step size falls below tolerance
Newton's method follows the tangent line at the current guess down to where it crosses zero, then repeats. When it works it converges quadratically — the number of correct digits roughly doubles each step. The catch is that it needs the derivative and a decent starting guess: near a flat spot (f′ ≈ 0) or far from any root it can stall or fly off, which this calculator detects and reports.
Worked example
To solve x² − 2 = 0 from x₀ = 1: step 1 gives 1.5, step 2 gives 1.41667, step 3 gives 1.414216 — already six correct digits, converging to √2 ≈ 1.41421356 in about five iterations.
Sources: Newton's method — Wikipedia
Frequently asked questions
What is quadratic convergence?
Once you are close to a simple root, each Newton step roughly squares the error — so the count of correct decimal places doubles per iteration. That is why it typically needs only a handful of steps to reach machine precision.
Why might Newton's method fail?
If the derivative is near zero at a guess, the tangent is almost flat and the next point jumps far away. A poor starting guess can also send it toward the wrong root or diverge. This tool flags a zero derivative and non-convergence instead of returning a bogus number.
How is this different from bisection?
Bisection is slower but bulletproof: give it a sign change and it always converges. Newton is much faster but can diverge. A common strategy is to bracket a root with bisection, then polish it with Newton.
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This calculator is for educational purposes. Double-check important results before acting on them.