Newton method for root finding

Mathematical statement

For a differentiable real function, the Newton update x_(k+1)=x_k−f(x_k)/f_prime(x_k) requires the current iterate in the domain and a nonzero derivative. The exact polynomial families use coefficients in low-to-high degree order and bounded rational iterates, recalculating f, its derivative and every update. If f(x_k)=0, stop as root_found before any division, even if the derivative also vanishes. A finite orbit, decreasing residual or one iterate near a root does not prove general or quadratic convergence, a global root basin or root uniqueness. Residual |f(x)| is not forward error to an unprovided true root.

Representative formula (exact TeX)

N(f,x)=y

Assumptions

• Coefficients are ordered from low to high degree and include a nonzero nonconstant term; iterates use bounded exact rationals.
• At every step test f(x)=0 before requiring a nonzero derivative; a step budget supplies a finite trace only.
• The square-root family requires a positive radicand and positive starting iterate, and still supplies only a finite exact orbit.

Symbol scope

• N — Newton-update operator
• f — differentiable root function
• x — current iterate
• y — next iterate or stopping status

Intuition, definition, and misconceptions

Domain, root-first stopping and the derivative condition precede division.

An exact finite orbit is an algorithm trace, not a convergence proof.

Keep residual, forward error, local rate and global basin separate.

A few rapidly decreasing residuals do not prove inevitable or quadratic convergence, and a point with f=f_prime=0 must not trigger0/0.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real derivatives at a point and finite difference quotients — Require the point derivative for an update.
• Numerical approximation and error — Separate residual from forward error.
• Algorithm correctness — Retain step preconditions and stopping statuses.
• Equation — Specify the target f(x)=0.
• Sequence — Represent an ordered finite prefix or infinite iterate sequence.
• Differentiating polynomial sums, products, and compositions — Recalculate the exact polynomial derivative.

Verification boundary

These families do not prove a general Newton convergence theorem, root uniqueness or floating-point stability.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add Kantorovich or simple-root local quadratic-convergence conditions and roundoff analysis.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.