S01 · implies
Before
f(x)=x^2-2,\quad f'(x)=2x
After
\operatorname{dom}f=\operatorname{dom}f'=\mathbb RConditions: none
Rule: polynomial · Uses: G1
Leaf basis: polynomial
The full polynomial supplies domain qualification for every rational iterate.
Qualified Newton updates and a finite trace
On this page
derivation-mko-newton-method-01 · ocme-new142-reader-adoption-20261021
Goal and scope
Take two exact Newton steps for x^2-2 from one, and audit root-first stopping and a cycle boundary.
A finite rational orbit only, without general convergence, quadratic rate, root uniqueness or floating-point stability.
Givens and symbols
algebrapolynomialnewton_update
G1— f(x)=x^2-2, f'(x)=2x, x_0=1, using exact rationals; the requested trace has two steps.
x_k— exact finite iterate|f(x_k)|— residual magnitude
Rules and sources
mko_definition· cited — 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. · mko-newton-methodalgebra· prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.polynomial· prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.newton_update· prerequisite — At each step first stop as root_found when f(x)=0; otherwise require the function/derivative domain and f'(x)≠0 before updating x-f(x)/f'(x).
Main derivation
S02 · implies
Before
x_0=1,\quad f(x_0)=-1\ne0,\quad f'(x_0)=2\ne0
After
x_1=1-(-1)/2=3/2
Conditions: none
Rule: newton_update · Uses: G1, S01
Leaf basis: newton_update
Perform the root test before the derivative test; divide only after both qualify.
S03 · implies
Before
f(3/2)=1/4\ne0,\quad f'(3/2)=3\ne0
After
x_2=3/2-(1/4)/3=17/12
Conditions: none
Rule: newton_update · Uses: S02
Leaf basis: newton_update
Recompute the new function and derivative values instead of reusing the previous step's data.
S04 · implies
Before
f(x_0)=-1,\quad f(x_1)=1/4
After
f(x_2)=289/144-2=1/144
Conditions: none
Rule: algebra · Uses: G1, S02, S03
Leaf basis: algebra
The residual magnitudes decrease in this finite trace, but no forward error to an unprovided root is asserted.
S05 · equivalent
Before
x>0
After
x-\frac{x^2-2}{2x}=\tfrac12(x+2/x)>0Conditions: x>0
Rule: algebra · Uses: G1
Leaf basis: algebra
The square-root special form preserves positive iterates and hence nonzero derivatives, without yet proving convergence.
S06 · implies
Before
g(x)=x^3,\quad x=0,\quad g(0)=g'(0)=0
After
\text{root_found; do not evaluate }0-0/0Conditions: none
Rule: newton_update · Uses: G1
Leaf basis: newton_update
Stopping at the root first avoids an undefined 0/0 update at a multiple root.
S07 · implies
Before
p(x)=x^3-2x+2,\quad p'(x)=3x^2-2,\quad z_0=0
After
z_1=0-2/(-2)=1,\quad z_2=1-1/1=0
Conditions: none
Rule: newton_update · Uses: G1
Leaf basis: newton_update
Both updates are legal but return to the start; deterministic repetition therefore cycles forever between zero and one.
S08 · implies
Before
\text{finite exact Newton traces and a nonconvergent cycle}After
\text{valid updates do not by themselves prove convergence}Conditions: none
Rule: algebra · Uses: S04, S06, S07
Leaf basis: algebra
Report the two exact iterates and residuals for the main example; the cycle refutes unconditional convergence while rates and stability remain unclaimed.
Substeps and lemmas
No substeps in this guide.
Reverse audits
S08· premise_audit · S04, S06, S07 — A successful trace is not promoted to a general theorem.S07· premise_audit · G1 — The two steps plus deterministic repetition prove the whole cycle, not a guess from a finite observation.S06· premise_audit · G1 — A zero derivative must not override an already verified root stop.S05· inverse_equivalence · G1 — Positivity qualifies updates but does not supply a local rate or global basin.S04· premise_audit · G1, S02, S03 — Residual is not the distance from x to a true root.S03· premise_audit · S02 — The denominator is exactly twelve, not a floating approximation.S02· premise_audit · G1, S01 — Division does not precede checking its denominator.S01· premise_audit · G1 — Domain membership does not ensure a nonzero derivative, which remains a per-step check.
Error controls and negative controls
Stopping order
After domain qualification test root first, then nonzero derivative, then divide.
Finite-trace boundary
An exact orbit is not a convergence or quadratic-rate proof.
- At f=f'=0 still update via 0/0. →
reject— Stop first at the verified root. - Residual is forward error. →
reject— No true root or residual-to-distance bound is provided. - Legal Newton updates must converge. →
reject— The 0-to-1 cycle at S07 does not converge.
Result and limitations
The two iterates are 3/2 and 17/12 with residual magnitudes 1/4 and 1/144; root-first stopping is qualified and a separate legal cycle exists.
Verification and source binding
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