Newton–Raphson Iteration

Variables

• x_n: current estimate
• f: target function
• f′: derivative

How to use this formula

Iteratively approximates a root using the tangent line at the current estimate.

Important notes

• Convergence depends on the initial guess and a nonzero derivative.
• Safeguards are needed near flat derivatives or discontinuities.

Quick example

For f(x)=x²−2, the iteration converges to sqrt(2).

Applicability, worked calculation, and verification

Assumptions and domain checks

• Convergence depends on the initial guess and a nonzero derivative.
• For the Newton–Raphson Iteration, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
• The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region.

Worked example

For f(x)=x²−2, the iteration converges to sqrt(2).

Common mistakes

• When copying Newton–Raphson Iteration, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
• Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error.

Review and verification

Last reviewed: 2026-07-23

Automated quality check: Kept noindex until the missing evidence is supplied.

Formula references

• Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.

Frequently asked questions