# Newton method for root finding

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses.

mko-newton-method · algorithm · v0.10.0



## 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

[Qualified Newton updates and a finite trace](https://ocme.evemisslab.com/en/derivations/derivation-mko-newton-method-01/)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](https://ocme.evemisslab.com/en/mko/mko-derivative/) — Require the point derivative for an update.
- [Numerical approximation and error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/) — Separate residual from forward error.
- [Algorithm correctness](https://ocme.evemisslab.com/en/mko/mko-algorithm-correctness/) — Retain step preconditions and stopping statuses.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — Specify the target f(x)=0.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Represent an ordered finite prefix or infinite iterate sequence.
- [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/mko/mko-derivative-rules/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/newton-method.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-newton-method/index.md)