Newton's method
Problem 3.507 · medium
Use Newton's method on \( \displaystyle f(x) = x^{3} - 3 x - 2 \) with \( \displaystyle x_0 = -2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
- \[ \frac{d}{d x} \left(x^{3} - 3 x - 2\right) = 3 x^{2} - 3 \]f′(x).✓ Proved
- \[ x - \frac{x^{3} - 3 x - 2}{3 x^{2} - 3} = \frac{2 x^{2} - 2 x + 2}{3 x - 3} \]Newton's formula x − f(x)/f′(x), simplified.✓ Proved
- \[ \left. \frac{2 x^{2} - 2 x + 2}{3 x - 3} \right|_{\substack{ x=-2 }} = - \frac{14}{9} \]x₁ = F(x₀).✓ Proved
- \[ \left. \frac{2 x^{2} - 2 x + 2}{3 x - 3} \right|_{\substack{ x=- \frac{14}{9} }} = - \frac{806}{621} \]x₂ = F(x₁).✓ Proved
Answer \( x_1 = - \frac{14}{9},\ x_2 = - \frac{806}{621} \approx -1.297907 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | two Newton steps in 30-digit floating point |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second Newton iterate is mis‑computed. The correct value is \(-65174/50229\), not \(-806/621\)."}qwen3.6:27b-mlx: pass — The solution correctly derives the Newton's method iteration function, handles the algebraic simplification accurately, and computes the exact values for x_1 and x_2 as requested.
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second Newton iterate is mis‑computed. The correct value is \(-65174/50229\), not \(-806/621\)."}qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly derives the Newton's method iteration function, handles the algebraic simplification accurately, and computes the exact values for x_1 and x_2 as requested.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly derives the Newton iteration function, handles the algebraic simplification accurately, and computes the exact values for x_1 and x_2 as requested.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/newtons_method, checked 2026-10-08 with SymPy 1.14.0.