Newton's Method Root Finder

xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ). Iteratively find roots of polynomial.

Result

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How to use this calculator

Enter polynomial coefficients + initial guess.

About this calculator

Newton's method for finding roots: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Quadratic convergence near a root (each iteration roughly doubles correct digits). Requires good initial guess + non-zero derivative. Can fail: jumping between roots, oscillating, dividing by zero (at critical points). Used everywhere in numerical analysis, optimization (find zero of gradient), implicit equation solving. Source: Wolfram MathWorld - Newton's Method.

Frequently asked

Convergence?+

Quadratic near a simple root: error squares each step. Slow / fails near multiple roots or far from any root.

Multiple roots?+

Try multiple initial guesses to find different roots. Newton finds the nearest root to x₀.

Failure modes?+

Initial guess with f'(x₀) = 0 → division by zero. Cycling between two values (rare). Diverging (poor guess).

When to prefer bisection?+

When you need guaranteed convergence (slower). Newton is faster when it works.

Higher dimensions?+

Generalized Newton uses Jacobian inverse. Used in optimization (gradient descent variants).

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